{"Polynomial.coeff":{"signature":"def Polynomial.coeff {R : Type u} [Semiring R] :Polynomial R \u2192 \u2115 \u2192 R","doc":"coeff p n (often denoted p.coeff n) is the coefficient of X^n in p.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L563-L565"},"Polynomial.support":{"signature":"def Polynomial.support {R : Type u} [Semiring R] :Polynomial R \u2192 Finset \u2115","doc":"The set of all n such that X^n has a non-zero coefficient.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L354-L356"},"Polynomial.degree":{"signature":"def Polynomial.degree {R : Type u} [Semiring R] (p : Polynomial R) :WithBot \u2115","doc":"degree p is the degree of the polynomial p, i.e. the largest X-exponent in p. degree p = some n when p \u2260 0 and n is the highest power of X that appears in p, otherwise degree 0 = \u22a5.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L49-L53"},"Polynomial.natDegree":{"signature":"def Polynomial.natDegree {R : Type u} [Semiring R] (p : Polynomial R) :\u2115","doc":"natDegree p forces degree p to \u2115, by defining natDegree 0 = 0.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L55-L57"},"Polynomial.leadingCoeff":{"signature":"def Polynomial.leadingCoeff {R : Type u} [Semiring R] (p : Polynomial R) :R","doc":"leadingCoeff p gives the coefficient of the highest power of X in p.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L59-L61"},"Polynomial.Monic":{"signature":"def Polynomial.Monic {R : Type u} [Semiring R] (p : Polynomial R) :Prop","doc":"a polynomial is Monic if its leading coefficient is 1","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L63-L65"},"Polynomial.sum":{"signature":"def Polynomial.sum {R : Type u} [Semiring R] {S : Type u_1} [AddCommMonoid S] (p : Polynomial R) (f : \u2115 \u2192 R \u2192 S) :S","doc":"Summing the values of a function applied to the coefficients of a polynomial","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L824-L826"},"Polynomial.eval\u2082":{"signature":"def Polynomial.eval\u2082 {R : Type u_1} {S : Type u_2} [Semiring R] [Semiring S] (f : R \u2192+* S) (x : S) (p : Polynomial R) :S","doc":"Evaluate a polynomial p given a ring hom f from the scalar ring to the target and a value x for the variable in the target","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L55-L58"},"Polynomial.instAdd":{"signature":"instance Polynomial.instAdd {R : Type u} [Semiring R] :Add (Polynomial R)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L117-L118"},"Polynomial.toFinsupp_injective":{"signature":"theorem Polynomial.toFinsupp_injective {R : Type u} [Semiring R] :Function.Injective toFinsupp","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L224-L225"},"Polynomial.toFinsupp_add":{"signature":"theorem Polynomial.toFinsupp_add {R : Type u} [Semiring R] (a b : Polynomial R) :(a + b).toFinsupp = a.toFinsupp + b.toFinsupp","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L189-L191"},"Polynomial.instZero":{"signature":"instance Polynomial.instZero {R : Type u} [Semiring R] :Zero (Polynomial R)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L111-L112"},"Polynomial.toFinsupp_zero":{"signature":"theorem Polynomial.toFinsupp_zero {R : Type u} [Semiring R] :toFinsupp 0 = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L181-L183"},"Polynomial.instNSMul":{"signature":"instance Polynomial.instNSMul {R : Type u} [Semiring R] :SMul \u2115 (Polynomial R)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L129-L130"},"Polynomial.toFinsupp_nsmul":{"signature":"theorem Polynomial.toFinsupp_nsmul {R : Type u} [Semiring R] (a : \u2115) (b : Polynomial R) :(a \u2022 b).toFinsupp = a \u2022 b.toFinsupp","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L207-L209"},"Polynomial.instMul":{"signature":"instance Polynomial.instMul {R : Type u} [Semiring R] :Mul (Polynomial R)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L126-L127"},"Polynomial.toFinsupp_mul":{"signature":"theorem Polynomial.toFinsupp_mul {R : Type u} [Semiring R] (a b : Polynomial R) :(a * b).toFinsupp = a.toFinsupp * b.toFinsupp","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L203-L205"},"Polynomial.instOne":{"signature":"instance Polynomial.instOne {R : Type u} [Semiring R] :One (Polynomial R)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L114-L115"},"Polynomial.instNatCast":{"signature":"instance Polynomial.instNatCast {R : Type u} [Semiring R] :NatCast (Polynomial R)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L253-L253"},"Polynomial.toFinsupp_one":{"signature":"theorem Polynomial.toFinsupp_one {R : Type u} [Semiring R] :toFinsupp 1 = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L185-L187"},"Polynomial.pow":{"signature":"instance Polynomial.pow {R : Type u} [Semiring R] :Pow (Polynomial R) \u2115","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L137-L137"},"Polynomial.ofFinsupp_mul":{"signature":"theorem Polynomial.ofFinsupp_mul {R : Type u} [Semiring R] (a b : AddMonoidAlgebra R \u2115) :{ toFinsupp := a * b } = { toFinsupp := a } * { toFinsupp := b }","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L160-L162"},"Polynomial.ofFinsupp_pow":{"signature":"theorem Polynomial.ofFinsupp_pow {R : Type u} [Semiring R] (a : AddMonoidAlgebra R \u2115) (n : \u2115) :{ toFinsupp := a ^ n } = { toFinsupp := a } ^ n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L174-L179"},"Polynomial.toFinsupp_pow":{"signature":"theorem Polynomial.toFinsupp_pow {R : Type u} [Semiring R] (a : Polynomial R) (n : \u2115) :(a ^ n).toFinsupp = a.toFinsupp ^ n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L216-L218"},"Polynomial.semiring":{"signature":"instance Polynomial.semiring {R : Type u} [Semiring R] :Semiring (Polynomial R)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L267-L270"},"Polynomial.smulZeroClass":{"signature":"instance Polynomial.smulZeroClass {R : Type u} [Semiring R] {S : Type u_1} [SMulZeroClass S R] :SMulZeroClass S (Polynomial R)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L132-L134"},"Polynomial.toFinsupp_smul":{"signature":"theorem Polynomial.toFinsupp_smul {R : Type u} [Semiring R] {S : Type u_1} [SMulZeroClass S R] (a : S) (b : Polynomial R) :(a \u2022 b).toFinsupp = a \u2022 b.toFinsupp","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L211-L214"},"Polynomial.distribMulAction":{"signature":"instance Polynomial.distribMulAction {R : Type u} [Semiring R] {S : Type u_1} [Monoid S] [DistribMulAction S R] :DistribMulAction S (Polynomial R)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L276-L278"},"Polynomial.module":{"signature":"instance Polynomial.module {R : Type u} [Semiring R] {S : Type u_1} [Semiring S] [Module S R] :Module S (Polynomial R)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L284-L286"},"Polynomial.ofFinsupp_add":{"signature":"theorem Polynomial.ofFinsupp_add {R : Type u} [Semiring R] {a b : AddMonoidAlgebra R \u2115} :{ toFinsupp := a + b } = { toFinsupp := a } + { toFinsupp := b }","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L147-L149"},"Polynomial.monomial":{"signature":"def Polynomial.monomial {R : Type u} [Semiring R] (n : \u2115) :R \u2192\u2097[R] Polynomial R","doc":"monomial s a is the monomial a * X^s","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L377-L381"},"Polynomial.toFinsupp_monomial":{"signature":"theorem Polynomial.toFinsupp_monomial {R : Type u} [Semiring R] (n : \u2115) (r : R) :((monomial n) r).toFinsupp = AddMonoidAlgebra.single n r","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L383-L385"},"Polynomial.monomial_mul_monomial":{"signature":"theorem Polynomial.monomial_mul_monomial {R : Type u} [Semiring R] (n m : \u2115) (r s : R) :(monomial n) r * (monomial m) s = (monomial (n + m)) (r * s)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L402-L405"},"Polynomial.monomial_zero_right":{"signature":"theorem Polynomial.monomial_zero_right {R : Type u} [Semiring R] (n : \u2115) :(monomial n) 0 = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L390-L392"},"Polynomial.C":{"signature":"def Polynomial.C {R : Type u} [Semiring R] :R \u2192+* Polynomial R","doc":"C a is the constant polynomial a. C is provided as a ring homomorphism.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L431-L438"},"Polynomial.X":{"signature":"def Polynomial.X {R : Type u} [Semiring R] :Polynomial R","doc":"X is the polynomial variable (aka indeterminate).","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L480-L482"},"Polynomial.map":{"signature":"def Polynomial.map {R : Type u} {S : Type v} [Semiring R] [Semiring S] (f : R \u2192+* S) :Polynomial R \u2192 Polynomial S","doc":"map f p maps a polynomial p across a ring hom f","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L481-L483"},"Polynomial.eval\u2082_def":{"signature":"theorem Polynomial.eval\u2082_def {R : Type u_1} {S : Type u_2} [Semiring R] [Semiring S] (f : R \u2192+* S) (x : S) (p : Polynomial R) :eval\u2082 f x p = p.sum fun (e : \u2115) (a : R) => f a * x ^ e","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L55-L58"},"Polynomial.coeff_add":{"signature":"theorem Polynomial.coeff_add {R : Type u} [Semiring R] (p q : Polynomial R) (n : \u2115) :(p + q).coeff n = p.coeff n + q.coeff n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L40-L45"},"Polynomial.coeff_smul":{"signature":"theorem Polynomial.coeff_smul {R : Type u} {S : Type v} [Semiring R] [SMulZeroClass S R] (r : S) (p : Polynomial R) (n : \u2115) :(r \u2022 p).coeff n = r \u2022 p.coeff n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L47-L50"},"Polynomial.lcoeff":{"signature":"def Polynomial.lcoeff (R : Type u) [Semiring R] (n : \u2115) :Polynomial R \u2192\u2097[R] R","doc":"The nth coefficient, as a linear map.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L78-L82"},"Polynomial.finset_sum_coeff":{"signature":"theorem Polynomial.finset_sum_coeff {R : Type u} [Semiring R] {\u03b9 : Type u_1} (s : Finset \u03b9) (f : \u03b9 \u2192 Polynomial R) (n : \u2115) :(\u2211 b \u2208 s, f b).coeff n = \u2211 b \u2208 s, (f b).coeff n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L88-L91"},"Polynomial.coeff_sum":{"signature":"theorem Polynomial.coeff_sum {R : Type u} {S : Type v} [Semiring R] {p : Polynomial R} [Semiring S] (n : \u2115) (f : \u2115 \u2192 R \u2192 Polynomial S) :(p.sum f).coeff n = p.sum fun (a : \u2115) (b : R) => (f a b).coeff n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L101-L104"},"Polynomial.ofFinsupp_single":{"signature":"theorem Polynomial.ofFinsupp_single {R : Type u} [Semiring R] (n : \u2115) (r : R) :{ toFinsupp := AddMonoidAlgebra.single n r } = (monomial n) r","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L387-L388"},"Polynomial.coeff_C_mul":{"signature":"theorem Polynomial.coeff_C_mul {R : Type u} {a : R} {n : \u2115} [Semiring R] (p : Polynomial R) :(C a * p).coeff n = a * p.coeff n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L151-L155"},"Polynomial.C_mul_X_pow_eq_monomial":{"signature":"theorem Polynomial.C_mul_X_pow_eq_monomial {R : Type u} {a : R} [Semiring R] {n : \u2115} :C a * X ^ n = (monomial n) a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L660-L663"},"Polynomial.coeff_monomial":{"signature":"theorem Polynomial.coeff_monomial {R : Type u} {a : R} {m n : \u2115} [Semiring R] :((monomial n) a).coeff m = if n = m then a else 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L583-L584"},"Polynomial.coeff_C_mul_X_pow":{"signature":"theorem Polynomial.coeff_C_mul_X_pow {R : Type u} [Semiring R] (x : R) (k n : \u2115) :(C x * X ^ k).coeff n = if n = k then x else 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L143-L146"},"Polynomial.coeff_X_pow":{"signature":"theorem Polynomial.coeff_X_pow {R : Type u} [Semiring R] (k n : \u2115) :(X ^ k).coeff n = if n = k then 1 else 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L185-L187"},"Polynomial.support_ofFinsupp":{"signature":"theorem Polynomial.support_ofFinsupp {R : Type u} [Semiring R] (p : AddMonoidAlgebra R \u2115) :{ toFinsupp := p }.support = p.support","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L358-L359"},"Polynomial.coeff_ofFinsupp":{"signature":"theorem Polynomial.coeff_ofFinsupp {R : Type u} [Semiring R] (p : AddMonoidAlgebra R \u2115) :{ toFinsupp := p }.coeff = \u21d1p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L567-L568"},"Polynomial.mem_support_iff":{"signature":"theorem Polynomial.mem_support_iff {R : Type u} {n : \u2115} [Semiring R] {p : Polynomial R} :n \u2208 p.support \u2194 p.coeff n \u2260 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L624-L627"},"Polynomial.coeff_map":{"signature":"theorem Polynomial.coeff_map {R : Type u} {S : Type v} [Semiring R] {p : Polynomial R} [Semiring S] (f : R \u2192+* S) (n : \u2115) :(map f p).coeff n = f (p.coeff n)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Coeff.lean#L78-L81"},"Polynomial.natDegree_eq_of_degree_eq":{"signature":"theorem Polynomial.natDegree_eq_of_degree_eq {R : Type u} {S : Type v} [Semiring R] {p : Polynomial R} [Semiring S] {q : Polynomial S} (h : p.degree = q.degree) :p.natDegree = q.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L121-L122"},"Polynomial.degree_le_iff_coeff_zero":{"signature":"theorem Polynomial.degree_le_iff_coeff_zero {R : Type u} [Semiring R] (f : Polynomial R) (n : WithBot \u2115) :f.degree \u2264 n \u2194 \u2200 (m : \u2115), n < \u2191m \u2192 f.coeff m = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L496-L499"},"Polynomial.degree_lt_iff_coeff_zero":{"signature":"theorem Polynomial.degree_lt_iff_coeff_zero {R : Type u} [Semiring R] (f : Polynomial R) (n : \u2115) :f.degree < \u2191n \u2194 \u2200 (m : \u2115), n \u2264 m \u2192 f.coeff m = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L501-L504"},"Polynomial.degree_map_le":{"signature":"theorem Polynomial.degree_map_le {R : Type u} {S : Type v} [Semiring R] [Semiring S] {f : R \u2192+* S} {p : Polynomial R} :(map f p).degree \u2264 p.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Degree.lean#L146-L149"},"Polynomial.ofFinsupp_zero":{"signature":"theorem Polynomial.ofFinsupp_zero {R : Type u} [Semiring R] :{ toFinsupp := 0 } = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L139-L141"},"Polynomial.ofFinsupp_inj":{"signature":"theorem Polynomial.ofFinsupp_inj {R : Type u} [Semiring R] {a b : AddMonoidAlgebra R \u2115} :{ toFinsupp := a } = { toFinsupp := b } \u2194 a = b","doc":"A more convenient spelling of Polynomial.ofFinsupp.injEq in terms of Iff.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L239-L241"},"Polynomial.ofFinsupp_eq_zero":{"signature":"theorem Polynomial.ofFinsupp_eq_zero {R : Type u} [Semiring R] {a : AddMonoidAlgebra R \u2115} :{ toFinsupp := a } = 0 \u2194 a = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L243-L245"},"Polynomial.support_eq_empty":{"signature":"theorem Polynomial.support_eq_empty {R : Type u} [Semiring R] {p : Polynomial R} :p.support = \u2205 \u2194 p = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L367-L370"},"Polynomial.degree_eq_bot":{"signature":"theorem Polynomial.degree_eq_bot {R : Type u} [Semiring R] {p : Polynomial R} :p.degree = \u22a5 \u2194 p = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L91-L93"},"Polynomial.degree_eq_natDegree":{"signature":"theorem Polynomial.degree_eq_natDegree {R : Type u} [Semiring R] {p : Polynomial R} (hp : p \u2260 0) :p.degree = \u2191p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L97-L100"},"Polynomial.leadingCoeff_zero":{"signature":"theorem Polynomial.leadingCoeff_zero {R : Type u} [Semiring R] :leadingCoeff 0 = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L352-L354"},"Polynomial.leadingCoeff_eq_zero":{"signature":"theorem Polynomial.leadingCoeff_eq_zero {R : Type u} [Semiring R] {p : Polynomial R} :p.leadingCoeff = 0 \u2194 p = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L356-L361"},"Polynomial.leadingCoeff_ne_zero":{"signature":"theorem Polynomial.leadingCoeff_ne_zero {R : Type u} [Semiring R] {p : Polynomial R} :p.leadingCoeff \u2260 0 \u2194 p \u2260 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L363-L363"},"Polynomial.le_degree_of_ne_zero":{"signature":"theorem Polynomial.le_degree_of_ne_zero {R : Type u} {n : \u2115} [Semiring R] {p : Polynomial R} (h : p.coeff n \u2260 0) :\u2191n \u2264 p.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L124-L126"},"Polynomial.degree_map_eq_of_leadingCoeff_ne_zero":{"signature":"theorem Polynomial.degree_map_eq_of_leadingCoeff_ne_zero {R : Type u} {S : Type v} [Semiring R] [Semiring S] {p : Polynomial R} (f : R \u2192+* S) (hf : f p.leadingCoeff \u2260 0) :(map f p).degree = p.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Degree.lean#L170-L178"},"Polynomial.natDegree_map_of_leadingCoeff_ne_zero":{"signature":"theorem Polynomial.natDegree_map_of_leadingCoeff_ne_zero {R : Type u} {S : Type v} [Semiring R] [Semiring S] {p : Polynomial R} (f : R \u2192+* S) (hf : f p.leadingCoeff \u2260 0) :(map f p).natDegree = p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Degree.lean#L180-L182"},"Polynomial.leadingCoeff_map_of_leadingCoeff_ne_zero":{"signature":"theorem Polynomial.leadingCoeff_map_of_leadingCoeff_ne_zero {R : Type u} {S : Type v} [Semiring R] [Semiring S] {p : Polynomial R} (f : R \u2192+* S) (hf : f p.leadingCoeff \u2260 0) :(map f p).leadingCoeff = f p.leadingCoeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Degree.lean#L184-L187"},"Polynomial.leadingCoeff_map_eq_of_isUnit_leadingCoeff":{"signature":"theorem Polynomial.leadingCoeff_map_eq_of_isUnit_leadingCoeff {R : Type u} {S : Type v} [Semiring R] {p : Polynomial R} [Semiring S] [Nontrivial S] (f : R \u2192+* S) (hp : IsUnit p.leadingCoeff) :(map f p).leadingCoeff = f p.leadingCoeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L285-L287"},"Polynomial.eval":{"signature":"def Polynomial.eval {R : Type u} [Semiring R] (x : R) (p : Polynomial R) :R","doc":"eval x p is the evaluation of the polynomial p at x","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L242-L244"},"Polynomial.C_0":{"signature":"theorem Polynomial.C_0 {R : Type u} [Semiring R] :C 0 = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L448-L448"},"Polynomial.toFinsuppIso":{"signature":"def Polynomial.toFinsuppIso (R : Type u) [Semiring R] :Polynomial R \u2243+* AddMonoidAlgebra R \u2115","doc":"Ring isomorphism between R[X] and R[\u2115]. This is just an implementation detail, but it can be useful to transfer results from Finsupp to polynomials.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L325-L332"},"Polynomial.ofFinsupp_sum":{"signature":"theorem Polynomial.ofFinsupp_sum {R : Type u} [Semiring R] {\u03b9 : Type u_1} (s : Finset \u03b9) (f : \u03b9 \u2192 AddMonoidAlgebra R \u2115) :{ toFinsupp := \u2211 i \u2208 s, f i } = \u2211 i \u2208 s, { toFinsupp := f i }","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L346-L348"},"Polynomial.sum_monomial_eq":{"signature":"theorem Polynomial.sum_monomial_eq {R : Type u} [Semiring R] (p : Polynomial R) :(p.sum fun (n : \u2115) (a : R) => (monomial n) a) = p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L889-L891"},"Polynomial.sum_C_mul_X_pow_eq":{"signature":"theorem Polynomial.sum_C_mul_X_pow_eq {R : Type u} [Semiring R] (p : Polynomial R) :(p.sum fun (n : \u2115) (a : R) => C a * X ^ n) = p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L893-L894"},"Polynomial.induction_on":{"signature":"theorem Polynomial.induction_on {R : Type u} [Semiring R] {motive : Polynomial R \u2192 Prop} (p : Polynomial R) (C : \u2200 (a : R), motive (C a)) (add : \u2200 (p q : Polynomial R), motive p \u2192 motive q \u2192 motive (p + q)) (monomial : \u2200 (n : \u2115) (a : R), motive (Polynomial.C a * X ^ n) \u2192 motive (Polynomial.C a * X ^ (n + 1))) :motive p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L896-L914"},"Polynomial.induction_on'":{"signature":"theorem Polynomial.induction_on' {R : Type u} [Semiring R] {motive : Polynomial R \u2192 Prop} (p : Polynomial R) (add : \u2200 (p q : Polynomial R), motive p \u2192 motive q \u2192 motive (p + q)) (monomial : \u2200 (n : \u2115) (a : R), motive ((monomial n) a)) :motive p","doc":"To prove something about polynomials, it suffices to show the condition is closed under taking sums, and it holds for monomials.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L916-L925"},"Polynomial.eval\u2082_eq_sum":{"signature":"theorem Polynomial.eval\u2082_eq_sum {R : Type u} {S : Type v} [Semiring R] {p : Polynomial R} [Semiring S] {f : R \u2192+* S} {x : S} :eval\u2082 f x p = p.sum fun (e : \u2115) (a : R) => f a * x ^ e","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L60-L61"},"Polynomial.sum_add_index":{"signature":"theorem Polynomial.sum_add_index {R : Type u} [Semiring R] {S : Type u_1} [AddCommMonoid S] (p q : Polynomial R) (f : \u2115 \u2192 R \u2192 S) (hf : \u2200 (i : \u2115), f i 0 = 0) (h_add : \u2200 (a : \u2115) (b\u2081 b\u2082 : R), f a (b\u2081 + b\u2082) = f a b\u2081 + f a b\u2082) :(p + q).sum f = p.sum f + q.sum f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L864-L868"},"Polynomial.eval\u2082_add":{"signature":"theorem Polynomial.eval\u2082_add {R : Type u} {S : Type v} [Semiring R] {p q : Polynomial R} [Semiring S] (f : R \u2192+* S) (x : S) :eval\u2082 f x (p + q) = eval\u2082 f x p + eval\u2082 f x q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L86-L89"},"Polynomial.map_add":{"signature":"theorem Polynomial.map_add {R : Type u} {S : Type v} [Semiring R] {p q : Polynomial R} [Semiring S] (f : R \u2192+* S) :map f (p + q) = map f p + map f q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L502-L504"},"Polynomial.eval_add":{"signature":"theorem Polynomial.eval_add {R : Type u} [Semiring R] {p q : Polynomial R} {x : R} :eval x (p + q) = eval x p + eval x q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L299-L301"},"Polynomial.sum_monomial_index":{"signature":"theorem Polynomial.sum_monomial_index {R : Type u} [Semiring R] {S : Type u_1} [AddCommMonoid S] {n : \u2115} (a : R) (f : \u2115 \u2192 R \u2192 S) (hf : f n 0 = 0) :((monomial n) a).sum f = f n a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L848-L851"},"Polynomial.eval\u2082_monomial":{"signature":"theorem Polynomial.eval\u2082_monomial {R : Type u} {S : Type v} [Semiring R] [Semiring S] (f : R \u2192+* S) (x : S) {n : \u2115} {r : R} :eval\u2082 f x ((monomial n) r) = f r * x ^ n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L76-L78"},"Polynomial.map_monomial":{"signature":"theorem Polynomial.map_monomial {R : Type u} {S : Type v} [Semiring R] [Semiring S] (f : R \u2192+* S) {n : \u2115} {a : R} :map f ((monomial n) a) = (monomial n) (f a)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L493-L496"},"Polynomial.eval_monomial":{"signature":"theorem Polynomial.eval_monomial {R : Type u} [Semiring R] {x : R} {n : \u2115} {a : R} :eval x ((monomial n) a) = a * x ^ n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L291-L293"},"Polynomial.eval\u2082_eq_eval_map":{"signature":"theorem Polynomial.eval\u2082_eq_eval_map {R : Type u} {S : Type v} [Semiring R] {p : Polynomial R} [Semiring S] (f : R \u2192+* S) {x : S} :eval\u2082 f x p = eval x (map f p)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L553-L556"},"Polynomial.ext_iff":{"signature":"theorem Polynomial.ext_iff {R : Type u} [Semiring R] {p q : Polynomial R} :p = q \u2194 \u2200 (n : \u2115), p.coeff n = q.coeff n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L700-L703"},"Polynomial.ext":{"signature":"theorem Polynomial.ext {R : Type u} [Semiring R] {p q : Polynomial R} :(\u2200 (n : \u2115), p.coeff n = q.coeff n) \u2192 p = q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L705-L707"},"Polynomial.map_map":{"signature":"theorem Polynomial.map_map {R : Type u} {S : Type v} {T : Type w} [Semiring R] [Semiring S] (f : R \u2192+* S) [Semiring T] (g : S \u2192+* T) (p : Polynomial R) :map g (map f p) = map (g.comp f) p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Coeff.lean#L86-L87"},"Polynomial.eval\u2082_map":{"signature":"theorem Polynomial.eval\u2082_map {R : Type u} {S : Type v} {T : Type w} [Semiring R] {p : Polynomial R} [Semiring S] (f : R \u2192+* S) [Semiring T] (g : S \u2192+* T) (x : T) :eval\u2082 g x (map f p) = eval\u2082 (g.comp f) x p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Coeff.lean#L138-L140"},"Polynomial.eval_eq_sum":{"signature":"theorem Polynomial.eval_eq_sum {R : Type u} [Semiring R] {p : Polynomial R} {x : R} :eval x p = p.sum fun (e : \u2115) (a : R) => a * x ^ e","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L249-L251"},"Polynomial.eval\u2082_at_apply":{"signature":"theorem Polynomial.eval\u2082_at_apply {R : Type u} [Semiring R] {p : Polynomial R} {S : Type u_1} [Semiring S] (f : R \u2192+* S) (r : R) :eval\u2082 f (f r) p = f (eval r p)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L253-L257"},"Polynomial.eval_map":{"signature":"theorem Polynomial.eval_map {R : Type u} {S : Type v} [Semiring R] {p : Polynomial R} [Semiring S] (f : R \u2192+* S) (x : S) :eval x (map f p) = eval\u2082 f x p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L565-L566"},"Polynomial.hom_eval\u2082":{"signature":"theorem Polynomial.hom_eval\u2082 {R : Type u} {S : Type v} {T : Type w} [Semiring R] (p : Polynomial R) [Semiring S] [Semiring T] (f : R \u2192+* S) (g : S \u2192+* T) (x : S) :g (eval\u2082 f x p) = eval\u2082 (g.comp f) (g x) p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Coeff.lean#L172-L173"},"Polynomial.toFinsupp_C":{"signature":"theorem Polynomial.toFinsupp_C {R : Type u} [Semiring R] (a : R) :(C a).toFinsupp = AddMonoidAlgebra.single 0 a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L444-L446"},"Polynomial.algebraOfAlgebra":{"signature":"instance Polynomial.algebraOfAlgebra {R : Type u} {A : Type z} [CommSemiring R] [Semiring A] [Algebra R A] :Algebra R (Polynomial A)","doc":"Note that this instance also provides Algebra R R[X].","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L54-L66"},"Polynomial.commSemiring":{"signature":"instance Polynomial.commSemiring {R : Type u} [CommSemiring R] :CommSemiring (Polynomial R)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L1059-L1061"},"Polynomial.C_1":{"signature":"theorem Polynomial.C_1 {R : Type u} [Semiring R] :C 1 = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L450-L451"},"Polynomial.sum_C_index":{"signature":"theorem Polynomial.sum_C_index {R : Type u} [Semiring R] {a : R} {\u03b2 : Type u_1} [AddCommMonoid \u03b2] {f : \u2115 \u2192 R \u2192 \u03b2} (h : f 0 0 = 0) :(C a).sum f = f 0 a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L853-L856"},"Polynomial.eval\u2082_C":{"signature":"theorem Polynomial.eval\u2082_C {R : Type u} {S : Type v} {a : R} [Semiring R] [Semiring S] (f : R \u2192+* S) (x : S) :eval\u2082 f x (C a) = f a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L70-L71"},"Polynomial.eval\u2082_one":{"signature":"theorem Polynomial.eval\u2082_one {R : Type u} {S : Type v} [Semiring R] [Semiring S] (f : R \u2192+* S) (x : S) :eval\u2082 f x 1 = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L91-L92"},"Polynomial.eval\u2082_ofFinsupp":{"signature":"theorem Polynomial.eval\u2082_ofFinsupp {R : Type u} {S : Type v} [Semiring R] [Semiring S] {f : R \u2192+* S} {x : S} {p : AddMonoidAlgebra R \u2115} :eval\u2082 f x { toFinsupp := p } = (AddMonoidAlgebra.liftNC \u2191f \u21d1((powersHom S) x)) p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L136-L139"},"Polynomial.eval\u2082_mul_noncomm":{"signature":"theorem Polynomial.eval\u2082_mul_noncomm {R : Type u} {S : Type v} [Semiring R] {p q : Polynomial R} [Semiring S] (f : R \u2192+* S) (x : S) (hf : \u2200 (k : \u2115), Commute (f (q.coeff k)) x) :eval\u2082 f x (p * q) = eval\u2082 f x p * eval\u2082 f x q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L141-L146"},"Polynomial.sum_zero_index":{"signature":"theorem Polynomial.sum_zero_index {R : Type u} [Semiring R] {S : Type u_1} [AddCommMonoid S] (f : \u2115 \u2192 R \u2192 S) :sum 0 f = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L844-L846"},"Polynomial.eval\u2082_zero":{"signature":"theorem Polynomial.eval\u2082_zero {R : Type u} {S : Type v} [Semiring R] [Semiring S] (f : R \u2192+* S) (x : S) :eval\u2082 f x 0 = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L67-L68"},"Polynomial.eval\u2082RingHom'":{"signature":"def Polynomial.eval\u2082RingHom' {R : Type u} {S : Type v} [Semiring R] [Semiring S] (f : R \u2192+* S) (x : S) (hf : \u2200 (a : R), Commute (f a) x) :Polynomial R \u2192+* S","doc":"eval\u2082 as a RingHom for noncommutative rings","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L174-L181"},"Polynomial.eval\u2082AlgHom'":{"signature":"def Polynomial.eval\u2082AlgHom' {R : Type u} {A : Type z} {B : Type u_2} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A \u2192\u2090[R] B) (b : B) (hf : \u2200 (a : A), Commute (f a) b) :Polynomial A \u2192\u2090[R] B","doc":"Polynomial.eval\u2082 as an AlgHom for noncommutative algebras. This is Polynomial.eval\u2082RingHom' for AlgHoms.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L154-L160"},"Polynomial.aevalTower":{"signature":"def Polynomial.aevalTower {R : Type u} {S : Type v} {A' : Type u_1} [CommSemiring R] [CommSemiring A'] [CommSemiring S] [Algebra S R] [Algebra S A'] (f : R \u2192\u2090[S] A') (x : A') :Polynomial R \u2192\u2090[S] A'","doc":"Version of aeval for defining algebra homs out of R[X] over a smaller base ring than R.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L520-L523"},"Polynomial.CAlgHom":{"signature":"def Polynomial.CAlgHom {R : Type u} {A : Type z} [CommSemiring R] [Semiring A] [Algebra R A] :A \u2192\u2090[R] Polynomial A","doc":"Polynomial.C as an AlgHom.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L93-L97"},"Polynomial.ringHom_ext":{"signature":"theorem Polynomial.ringHom_ext {R : Type u} [Semiring R] {S : Type u_1} [Semiring S] {f g : Polynomial R \u2192+* S} (h\u2081 : \u2200 (a : R), f (C a) = g (C a)) (h\u2082 : f X = g X) :f = g","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Monomial.lean#L54-L72"},"Polynomial.ringHom_ext'":{"signature":"theorem Polynomial.ringHom_ext' {R : Type u} [Semiring R] {S : Type u_1} [Semiring S] {f g : Polynomial R \u2192+* S} (h\u2081 : f.comp C = g.comp C) (h\u2082 : f X = g X) :f = g","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Monomial.lean#L74-L77"},"Polynomial.algHom_ext'":{"signature":"theorem Polynomial.algHom_ext' {R : Type u} {A : Type z} {B : Type u_2} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] {f g : Polynomial A \u2192\u2090[R] B} (hC : f.comp CAlgHom = g.comp CAlgHom) (hX : f X = g X) :f = g","doc":"Extensionality lemma for algebra maps out of A'[X] over a smaller base ring than A'","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L99-L105"},"Polynomial.aevalTower_C":{"signature":"theorem Polynomial.aevalTower_C {R : Type u} {S : Type v} {A' : Type u_1} [CommSemiring R] [CommSemiring A'] [CommSemiring S] [Algebra S R] [Algebra S A'] (g : R \u2192\u2090[S] A') (y : A') (x : R) :(aevalTower g y) (C x) = g x","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L531-L533"},"Polynomial.sum_X_index":{"signature":"theorem Polynomial.sum_X_index {R : Type u} [Semiring R] {S : Type u_1} [AddCommMonoid S] {f : \u2115 \u2192 R \u2192 S} (hf : f 1 0 = 0) :X.sum f = f 1 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L859-L862"},"Polynomial.eval\u2082_X":{"signature":"theorem Polynomial.eval\u2082_X {R : Type u} {S : Type v} [Semiring R] [Semiring S] (f : R \u2192+* S) (x : S) :eval\u2082 f x X = x","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L73-L74"},"Polynomial.aevalTower_X":{"signature":"theorem Polynomial.aevalTower_X {R : Type u} {S : Type v} {A' : Type u_1} [CommSemiring R] [CommSemiring A'] [CommSemiring S] [Algebra S R] [Algebra S A'] (g : R \u2192\u2090[S] A') (y : A') :(aevalTower g y) X = y","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L527-L529"},"Polynomial.degreeLE":{"signature":"def Polynomial.degreeLE (R : Type u) [Semiring R] (n : WithBot \u2115) :Submodule R (Polynomial R)","doc":"The R-submodule of R[X] consisting of polynomials of degree \u2264 n.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Basic.lean#L52-L54"},"Polynomial.C_mul'":{"signature":"theorem Polynomial.C_mul' {R : Type u} [Semiring R] (a : R) (f : Polynomial R) :C a * f = a \u2022 f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L157-L159"},"Polynomial.instNeg":{"signature":"instance Polynomial.instNeg {R : Type u} [Ring R] :Neg (Polynomial R)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L120-L121"},"Polynomial.instSub":{"signature":"instance Polynomial.instSub {R : Type u} [Ring R] :Sub (Polynomial R)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L123-L124"},"Polynomial.instZSMul":{"signature":"instance Polynomial.instZSMul {R : Type u} [Ring R] :SMul \u2124 (Polynomial R)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L1069-L1070"},"Polynomial.instIntCast":{"signature":"instance Polynomial.instIntCast {R : Type u} [Ring R] :IntCast (Polynomial R)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L1082-L1082"},"Polynomial.toFinsupp_neg":{"signature":"theorem Polynomial.toFinsupp_neg {R : Type u} [Ring R] (a : Polynomial R) :(-a).toFinsupp = -a.toFinsupp","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L193-L195"},"Polynomial.toFinsupp_sub":{"signature":"theorem Polynomial.toFinsupp_sub {R : Type u} [Ring R] (a b : Polynomial R) :(a - b).toFinsupp = a.toFinsupp - b.toFinsupp","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L197-L201"},"Polynomial.toFinsupp_zsmul":{"signature":"theorem Polynomial.toFinsupp_zsmul {R : Type u} [Ring R] (a : \u2124) (b : Polynomial R) :(a \u2022 b).toFinsupp = a \u2022 b.toFinsupp","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L1077-L1080"},"Polynomial.ring":{"signature":"instance Polynomial.ring {R : Type u} [Ring R] :Ring (Polynomial R)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L1090-L1094"},"Polynomial.instDecidableEq":{"signature":"instance Polynomial.instDecidableEq (R : Type u) [Semiring R] [DecidableEq R] :DecidableEq (Polynomial R)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L334-L335"},"Polynomial.mem_degreeLE":{"signature":"theorem Polynomial.mem_degreeLE {R : Type u} [Semiring R] {n : WithBot \u2115} {f : Polynomial R} :f \u2208 degreeLE R n \u2194 f.degree \u2264 n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Basic.lean#L62-L63"},"Polynomial.degree_zero":{"signature":"theorem Polynomial.degree_zero {R : Type u} [Semiring R] :degree 0 = \u22a5","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L79-L81"},"Polynomial.support_monomial":{"signature":"theorem Polynomial.support_monomial {R : Type u} [Semiring R] (n : \u2115) {a : R} (H : a \u2260 0) :((monomial n) a).support = {n}","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L741-L742"},"Polynomial.degree_monomial":{"signature":"theorem Polynomial.degree_monomial {R : Type u} {a : R} [Semiring R] (n : \u2115) (ha : a \u2260 0) :((monomial n) a).degree = \u2191n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L189-L191"},"Polynomial.degree_monomial_le":{"signature":"theorem Polynomial.degree_monomial_le {R : Type u} [Semiring R] (n : \u2115) (a : R) :((monomial n) a).degree \u2264 \u2191n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L200-L203"},"Polynomial.degree_C_mul_X_pow_le":{"signature":"theorem Polynomial.degree_C_mul_X_pow_le {R : Type u} [Semiring R] (n : \u2115) (a : R) :(C a * X ^ n).degree \u2264 \u2191n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L205-L207"},"Polynomial.degree_X_pow_le":{"signature":"theorem Polynomial.degree_X_pow_le {R : Type u} [Semiring R] (n : \u2115) :(X ^ n).degree \u2264 \u2191n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L240-L241"},"Polynomial.degreeLE_eq_span_X_pow":{"signature":"theorem Polynomial.degreeLE_eq_span_X_pow {R : Type u} [Semiring R] [DecidableEq R] {n : \u2115} :degreeLE R \u2191n = Submodule.span R \u2191(Finset.image (fun (n : \u2115) => X ^ n) (Finset.range (n + 1)))","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Basic.lean#L69-L87"},"Polynomial.coeff_eq_zero_of_degree_lt":{"signature":"theorem Polynomial.coeff_eq_zero_of_degree_lt {R : Type u} {n : \u2115} [Semiring R] {p : Polynomial R} (h : p.degree < \u2191n) :p.coeff n = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L82-L83"},"Polynomial.natDegree_le_iff_degree_le":{"signature":"theorem Polynomial.natDegree_le_iff_degree_le {R : Type u} [Semiring R] {p : Polynomial R} {n : \u2115} :p.natDegree \u2264 n \u2194 p.degree \u2264 \u2191n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L139-L140"},"Polynomial.natDegree_le_of_degree_le":{"signature":"theorem Polynomial.natDegree_le_of_degree_le {R : Type u} [Semiring R] {p : Polynomial R} {n : \u2115} :p.degree \u2264 \u2191n \u2192 p.natDegree \u2264 n","doc":"Alias of the reverse direction of Polynomial.natDegree_le_iff_degree_le.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L145-L145"},"Polynomial.support_toFinsupp":{"signature":"theorem Polynomial.support_toFinsupp {R : Type u} [Semiring R] (p : Polynomial R) :p.toFinsupp.support = p.support","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L361-L361"},"Polynomial.degree_mul_le":{"signature":"theorem Polynomial.degree_mul_le {R : Type u} [Semiring R] (p q : Polynomial R) :(p * q).degree \u2264 p.degree + q.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L398-L399"},"Polynomial.degree_le_natDegree":{"signature":"theorem Polynomial.degree_le_natDegree {R : Type u} [Semiring R] {p : Polynomial R} :p.degree \u2264 \u2191p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L117-L119"},"Polynomial.coeff_mul":{"signature":"theorem Polynomial.coeff_mul {R : Type u} [Semiring R] (p q : Polynomial R) (n : \u2115) :(p * q).coeff n = \u2211 x \u2208 Finset.antidiagonal n, p.coeff x.1 * q.coeff x.2","doc":"Decomposes the coefficient of the product p * q as a sum over antidiagonal. A version which sums over range (n + 1) can be obtained by using Finset.Nat.sum_antidiagonal_eq_sum_range_succ.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L106-L113"},"Polynomial.coeff_mul_X_pow":{"signature":"theorem Polynomial.coeff_mul_X_pow {R : Type u} [Semiring R] (p : Polynomial R) (n d : \u2115) :(p * X ^ n).coeff (d + n) = p.coeff d","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L224-L231"},"Polynomial.natDegree_eq_of_degree_eq_some":{"signature":"theorem Polynomial.natDegree_eq_of_degree_eq_some {R : Type u} [Semiring R] {p : Polynomial R} {n : \u2115} (h : p.degree = \u2191n) :p.natDegree = n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L111-L112"},"Polynomial.coeff_mul_degree_add_degree":{"signature":"theorem Polynomial.coeff_mul_degree_add_degree {R : Type u} [Semiring R] (p q : Polynomial R) :(p * q).coeff (p.natDegree + q.natDegree) = p.leadingCoeff * q.leadingCoeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L282-L311"},"Polynomial.degree_mul'":{"signature":"theorem Polynomial.degree_mul' {R : Type u} [Semiring R] {p q : Polynomial R} (h : p.leadingCoeff * q.leadingCoeff \u2260 0) :(p * q).degree = p.degree + q.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L313-L321"},"Polynomial.natDegree_mul'":{"signature":"theorem Polynomial.natDegree_mul' {R : Type u} [Semiring R] {p q : Polynomial R} (h : p.leadingCoeff * q.leadingCoeff \u2260 0) :(p * q).natDegree = p.natDegree + q.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L328-L333"},"Polynomial.leadingCoeff_mul'":{"signature":"theorem Polynomial.leadingCoeff_mul' {R : Type u} [Semiring R] {p q : Polynomial R} (h : p.leadingCoeff * q.leadingCoeff \u2260 0) :(p * q).leadingCoeff = p.leadingCoeff * q.leadingCoeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L335-L337"},"Polynomial.leadingCoeff_mul_monic":{"signature":"theorem Polynomial.leadingCoeff_mul_monic {R : Type u} [Semiring R] {p q : Polynomial R} (hq : q.Monic) :(p * q).leadingCoeff = p.leadingCoeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L372-L380"},"Polynomial.degree_C_mul_X_pow":{"signature":"theorem Polynomial.degree_C_mul_X_pow {R : Type u} {a : R} [Semiring R] (n : \u2115) (ha : a \u2260 0) :(C a * X ^ n).degree = \u2191n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L193-L195"},"Polynomial.natDegree_C_mul_X_pow":{"signature":"theorem Polynomial.natDegree_C_mul_X_pow {R : Type u} [Semiring R] (n : \u2115) (a : R) (ha : a \u2260 0) :(C a * X ^ n).natDegree = n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L212-L214"},"Polynomial.natDegree_monomial":{"signature":"theorem Polynomial.natDegree_monomial {R : Type u} [Semiring R] [DecidableEq R] (i : \u2115) (r : R) :((monomial i) r).natDegree = if r = 0 then 0 else i","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L220-L225"},"Polynomial.leadingCoeff_monomial":{"signature":"theorem Polynomial.leadingCoeff_monomial {R : Type u} [Semiring R] (a : R) (n : \u2115) :((monomial n) a).leadingCoeff = a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L416-L422"},"Polynomial.leadingCoeff_C_mul_X_pow":{"signature":"theorem Polynomial.leadingCoeff_C_mul_X_pow {R : Type u} [Semiring R] (a : R) (n : \u2115) :(C a * X ^ n).leadingCoeff = a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L424-L425"},"Polynomial.leadingCoeff_X_pow":{"signature":"theorem Polynomial.leadingCoeff_X_pow {R : Type u} [Semiring R] (n : \u2115) :(X ^ n).leadingCoeff = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L434-L435"},"Polynomial.monic_X_pow":{"signature":"theorem Polynomial.monic_X_pow {R : Type u} [Semiring R] (n : \u2115) :(X ^ n).Monic","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L440-L442"},"Polynomial.leadingCoeff_mul_X_pow":{"signature":"theorem Polynomial.leadingCoeff_mul_X_pow {R : Type u} [Semiring R] {p : Polynomial R} {n : \u2115} :(p * X ^ n).leadingCoeff = p.leadingCoeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L412-L414"},"Polynomial.monomial_one_one_eq_X":{"signature":"theorem Polynomial.monomial_one_one_eq_X {R : Type u} [Semiring R] :(monomial 1) 1 = X","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L484-L485"},"Polynomial.monomial_one_right_eq_X_pow":{"signature":"theorem Polynomial.monomial_one_right_eq_X_pow {R : Type u} [Semiring R] (n : \u2115) :(monomial n) 1 = X ^ n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L487-L490"},"Polynomial.X_pow_eq_monomial":{"signature":"theorem Polynomial.X_pow_eq_monomial {R : Type u} [Semiring R] (n : \u2115) :X ^ n = (monomial n) 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L791-L792"},"Polynomial.degree_X_pow":{"signature":"theorem Polynomial.degree_X_pow {R : Type u} [Semiring R] [Nontrivial R] (n : \u2115) :(X ^ n).degree = \u2191n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L515-L517"},"Polynomial.erase":{"signature":"def Polynomial.erase {R : Type u_1} [Semiring R] (n : \u2115) :Polynomial R \u2192 Polynomial R","doc":"erase p n is the polynomial p in which the X^n term has been erased.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L927-L929"},"Polynomial.erase_def":{"signature":"theorem Polynomial.erase_def {R : Type u_1} [Semiring R] (n : \u2115) (x\u271d : Polynomial R) :erase n x\u271d = match x\u271d with | { toFinsupp := p } => { toFinsupp := Finsupp.erase n p }","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L927-L929"},"Polynomial.toFinsupp_erase":{"signature":"theorem Polynomial.toFinsupp_erase {R : Type u} [Semiring R] (p : Polynomial R) (n : \u2115) :(erase n p).toFinsupp = Finsupp.erase n p.toFinsupp","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L931-L933"},"Polynomial.monomial_add_erase":{"signature":"theorem Polynomial.monomial_add_erase {R : Type u} [Semiring R] (p : Polynomial R) (n : \u2115) :(monomial n) (p.coeff n) + erase n p = p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L944-L947"},"Polynomial.degree_add_le":{"signature":"theorem Polynomial.degree_add_le {R : Type u} [Semiring R] (p q : Polynomial R) :(p + q).degree \u2264 max p.degree q.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L329-L331"},"Polynomial.ofFinsupp_neg":{"signature":"theorem Polynomial.ofFinsupp_neg {R : Type u} [Ring R] {a : AddMonoidAlgebra R \u2115} :{ toFinsupp := -a } = -{ toFinsupp := a }","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L151-L153"},"Polynomial.support_neg":{"signature":"theorem Polynomial.support_neg {R : Type u} [Ring R] {p : Polynomial R} :(-p).support = p.support","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L1114-L1117"},"Polynomial.degree_neg":{"signature":"theorem Polynomial.degree_neg {R : Type u} [Ring R] (p : Polynomial R) :(-p).degree = p.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L277-L278"},"Polynomial.degree_erase_le":{"signature":"theorem Polynomial.degree_erase_le {R : Type u} [Semiring R] (p : Polynomial R) (n : \u2115) :(erase n p).degree \u2264 p.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L371-L374"},"Polynomial.support_erase":{"signature":"theorem Polynomial.support_erase {R : Type u} [Semiring R] (p : Polynomial R) (n : \u2115) :(erase n p).support = p.support.erase n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L940-L942"},"Polynomial.degree_erase_lt":{"signature":"theorem Polynomial.degree_erase_lt {R : Type u} [Semiring R] {p : Polynomial R} (hp : p \u2260 0) :(erase p.natDegree p).degree < p.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L376-L379"},"Polynomial.degree_sub_lt":{"signature":"theorem Polynomial.degree_sub_lt {R : Type u} [Ring R] {p q : Polynomial R} (hd : p.degree = q.degree) (hp0 : p \u2260 0) (hlc : p.leadingCoeff = q.leadingCoeff) :(p - q).degree < p.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L543-L558"},"Polynomial.commRing":{"signature":"instance Polynomial.commRing {R : Type u} [CommRing R] :CommRing (Polynomial R)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L1129-L1132"},"Polynomial.isNoetherianRing":{"signature":"instance Polynomial.isNoetherianRing {R : Type u} [CommRing R] [inst : IsNoetherianRing R] :IsNoetherianRing (Polynomial R)","doc":"Hilbert basis theorem: a polynomial ring over a Noetherian ring is a Noetherian ring.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Basic.lean#L822-L894"},"Polynomial.degree_lt_wf":{"signature":"theorem Polynomial.degree_lt_wf {R : Type u} [Semiring R] :WellFounded fun (p q : Polynomial R) => p.degree < q.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L42-L43"},"Polynomial.instWellFoundedRelation":{"signature":"instance Polynomial.instWellFoundedRelation {R : Type u} [Semiring R] :WellFoundedRelation (Polynomial R)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L45-L46"},"Polynomial.inhabited":{"signature":"instance Polynomial.inhabited {R : Type u} [Semiring R] :Inhabited (Polynomial R)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L250-L251"},"Polynomial.unique":{"signature":"instance Polynomial.unique {R : Type u} [Semiring R] [Subsingleton R] :Unique (Polynomial R)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L316-L321"},"Polynomial.leadingCoeff_monic_mul":{"signature":"theorem Polynomial.leadingCoeff_monic_mul {R : Type u} [Semiring R] {p q : Polynomial R} (hp : p.Monic) :(p * q).leadingCoeff = q.leadingCoeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L365-L370"},"Polynomial.leadingCoeff_C":{"signature":"theorem Polynomial.leadingCoeff_C {R : Type u} [Semiring R] (a : R) :(C a).leadingCoeff = a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L430-L432"},"Polynomial.div_wf_lemma":{"signature":"theorem Polynomial.div_wf_lemma {R : Type u} [Ring R] {p q : Polynomial R} (h : q.degree \u2264 p.degree \u2227 p \u2260 0) (hq : q.Monic) :(p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))).degree < p.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L106-L117"},"Polynomial.divModByMonicAux":{"signature":"noncomputable def Polynomial.divModByMonicAux {R : Type u} [Ring R] (_p : Polynomial R) {q : Polynomial R} :q.Monic \u2192 Polynomial R \u00d7 Polynomial R","doc":"See divByMonic.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L119-L129"},"Polynomial.modByMonic":{"signature":"def Polynomial.modByMonic {R : Type u} [Ring R] (p q : Polynomial R) :Polynomial R","doc":"modByMonic, denoted as p %\u2098 q, gives the remainder of p by a monic polynomial q.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L136-L139"},"Polynomial.divByMonic":{"signature":"def Polynomial.divByMonic {R : Type u} [Ring R] (p q : Polynomial R) :Polynomial R","doc":"divByMonic, denoted as p /\u2098 q, gives the quotient of p by a monic polynomial q.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L131-L134"},"Polynomial.modByMonic_eq_sub_mul_div":{"signature":"theorem Polynomial.modByMonic_eq_sub_mul_div {R : Type u} [Ring R] (p : Polynomial R) {q : Polynomial R} (_hq : q.Monic) :p %\u2098 q = p - q * (p /\u2098 q)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L238-L257"},"Polynomial.modByMonic_add_div":{"signature":"theorem Polynomial.modByMonic_add_div {R : Type u} [Ring R] (p : Polynomial R) {q : Polynomial R} (hq : q.Monic) :p %\u2098 q + q * (p /\u2098 q) = p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L259-L260"},"Polynomial.degree_modByMonic_lt":{"signature":"theorem Polynomial.degree_modByMonic_lt {R : Type u} [Ring R] [Nontrivial R] (p : Polynomial R) {q : Polynomial R} (_hq : q.Monic) :(p %\u2098 q).degree < q.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L147-L169"},"Polynomial.modByMonic_eq_zero_iff_dvd":{"signature":"theorem Polynomial.modByMonic_eq_zero_iff_dvd {R : Type u} [Ring R] {p q : Polynomial R} (hq : q.Monic) :p %\u2098 q = 0 \u2194 q \u2223 p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L386-L400"},"Polynomial.decidableDvdMonic":{"signature":"def Polynomial.decidableDvdMonic {R : Type u} [Ring R] {q : Polynomial R} [DecidableEq R] (p : Polynomial R) (hq : q.Monic) :Decidable (q \u2223 p)","doc":"An algorithm for deciding polynomial divisibility. The algorithm is \"compute p %\u2098 q and compare to 0\". See Polynomial.modByMonic for the algorithm that computes %\u2098.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L480-L485"},"Polynomial.monic_one":{"signature":"theorem Polynomial.monic_one {R : Type u} [Semiring R] :Monic 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L451-L453"},"Polynomial.C_neg":{"signature":"theorem Polynomial.C_neg {R : Type u} {a : R} [Ring R] :C (-a) = -C a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L1121-L1122"},"Polynomial.monic_of_subsingleton":{"signature":"theorem Polynomial.monic_of_subsingleton {R : Type u} [Semiring R] [Subsingleton R] (p : Polynomial R) :p.Monic","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L48-L50"},"Polynomial.le_natDegree_of_ne_zero":{"signature":"theorem Polynomial.le_natDegree_of_ne_zero {R : Type u} {n : \u2115} [Semiring R] {p : Polynomial R} (h : p.coeff n \u2260 0) :n \u2264 p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L60-L64"},"Polynomial.natDegree_eq_of_le_of_coeff_ne_zero":{"signature":"theorem Polynomial.natDegree_eq_of_le_of_coeff_ne_zero {R : Type u} {n : \u2115} [Semiring R] {p : Polynomial R} (pn : p.natDegree \u2264 n) (p1 : p.coeff n \u2260 0) :p.natDegree = n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L69-L71"},"Polynomial.monic_of_natDegree_le_of_coeff_eq_one":{"signature":"theorem Polynomial.monic_of_natDegree_le_of_coeff_eq_one {R : Type u} [Semiring R] {p : Polynomial R} (n : \u2115) (pn : p.natDegree \u2264 n) (p1 : p.coeff n = 1) :p.Monic","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L250-L255"},"Polynomial.monic_of_degree_le":{"signature":"theorem Polynomial.monic_of_degree_le {R : Type u} [Semiring R] {p : Polynomial R} (n : \u2115) (pn : p.degree \u2264 \u2191n) (p1 : p.coeff n = 1) :p.Monic","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L257-L258"},"Polynomial.monic_X_pow_add":{"signature":"theorem Polynomial.monic_X_pow_add {R : Type u} [Semiring R] {p : Polynomial R} {n : \u2115} (H : p.degree < \u2191n) :(X ^ n + p).Monic","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Monic.lean#L75-L78"},"Polynomial.monomial_zero_left":{"signature":"theorem Polynomial.monomial_zero_left {R : Type u} [Semiring R] (a : R) :(monomial 0) a = C a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L440-L442"},"Polynomial.degree_C":{"signature":"theorem Polynomial.degree_C {R : Type u} {a : R} [Semiring R] (ha : a \u2260 0) :(C a).degree = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L151-L154"},"Polynomial.degree_C_le":{"signature":"theorem Polynomial.degree_C_le {R : Type u} {a : R} [Semiring R] :(C a).degree \u2264 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L156-L160"},"Polynomial.monic_X_pow_add_C":{"signature":"theorem Polynomial.monic_X_pow_add_C {R : Type u} (a : R) [Semiring R] {n : \u2115} (h : n \u2260 0) :(X ^ n + C a).Monic","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Monic.lean#L81-L83"},"Polynomial.monic_X_add_C":{"signature":"theorem Polynomial.monic_X_add_C {R : Type u} [Semiring R] (x : R) :(X + C x).Monic","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Monic.lean#L85-L86"},"Polynomial.monic_X_sub_C":{"signature":"theorem Polynomial.monic_X_sub_C {R : Type u} [Ring R] (x : R) :(X - C x).Monic","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Monic.lean#L434-L435"},"Polynomial.leadingCoeff_pow'":{"signature":"theorem Polynomial.leadingCoeff_pow' {R : Type u} {n : \u2115} [Semiring R] {p : Polynomial R} :p.leadingCoeff ^ n \u2260 0 \u2192 (p ^ n).leadingCoeff = p.leadingCoeff ^ n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L339-L343"},"Polynomial.natDegree_C":{"signature":"theorem Polynomial.natDegree_C {R : Type u} [Semiring R] (a : R) :(C a).natDegree = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L167-L172"},"Polynomial.natDegree_one":{"signature":"theorem Polynomial.natDegree_one {R : Type u} [Semiring R] :natDegree 1 = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L174-L176"},"Polynomial.degree_pow'":{"signature":"theorem Polynomial.degree_pow' {R : Type u} [Semiring R] {p : Polynomial R} {n : \u2115} :p.leadingCoeff ^ n \u2260 0 \u2192 (p ^ n).degree = n \u2022 p.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L345-L351"},"Polynomial.natDegree_pow'":{"signature":"theorem Polynomial.natDegree_pow' {R : Type u} [Semiring R] {p : Polynomial R} {n : \u2115} (h : p.leadingCoeff ^ n \u2260 0) :(p ^ n).natDegree = n * p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L353-L363"},"Polynomial.finiteMultiplicity_of_degree_pos_of_monic":{"signature":"theorem Polynomial.finiteMultiplicity_of_degree_pos_of_monic {R : Type u} [Semiring R] {p q : Polynomial R} (hp : 0 < p.degree) (hmp : p.Monic) (hq : q \u2260 0) :FiniteMultiplicity p q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L60-L81"},"Polynomial.degree_X":{"signature":"theorem Polynomial.degree_X {R : Type u} [Semiring R] [Nontrivial R] :X.degree = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L263-L265"},"Polynomial.degree_le_degree":{"signature":"theorem Polynomial.degree_le_degree {R : Type u} [Semiring R] {p q : Polynomial R} (h : q.coeff p.natDegree \u2260 0) :p.degree \u2264 q.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L132-L137"},"Polynomial.coeff_natDegree_eq_zero_of_degree_lt":{"signature":"theorem Polynomial.coeff_natDegree_eq_zero_of_degree_lt {R : Type u} [Semiring R] {p q : Polynomial R} (h : p.degree < q.degree) :p.coeff q.natDegree = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L135-L137"},"Polynomial.ne_zero_of_degree_gt":{"signature":"theorem Polynomial.ne_zero_of_degree_gt {R : Type u} [Semiring R] {p : Polynomial R} {n : WithBot \u2115} (h : n < p.degree) :p \u2260 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L139-L140"},"Polynomial.degree_add_eq_left_of_degree_lt":{"signature":"theorem Polynomial.degree_add_eq_left_of_degree_lt {R : Type u} [Semiring R] {p q : Polynomial R} (h : q.degree < p.degree) :(p + q).degree = p.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L175-L179"},"Polynomial.degree_X_add_C":{"signature":"theorem Polynomial.degree_X_add_C {R : Type u} [Nontrivial R] [Semiring R] (a : R) :(X + C a).degree = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L599-L606"},"Polynomial.degree_X_sub_C":{"signature":"theorem Polynomial.degree_X_sub_C {R : Type u} [Ring R] [Nontrivial R] (a : R) :(X - C a).degree = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L756-L758"},"Polynomial.finiteMultiplicity_X_sub_C":{"signature":"theorem Polynomial.finiteMultiplicity_X_sub_C {R : Type u} [Ring R] {p : Polynomial R} (a : R) (h0 : p \u2260 0) :FiniteMultiplicity (X - C a) p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L487-L491"},"Polynomial.rootMultiplicity":{"signature":"def Polynomial.rootMultiplicity {R : Type u} [Ring R] (a : R) (p : Polynomial R) :\u2115","doc":"The largest power of X - C a which divides p. This could be computable via the divisibility algorithm Polynomial.decidableDvdMonic, as shown by Polynomial.rootMultiplicity_eq_nat_find_of_nonzero which has a computable RHS.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L495-L505"},"Polynomial.IsRoot":{"signature":"def Polynomial.IsRoot {R : Type u} [Semiring R] (p : Polynomial R) (a : R) :Prop","doc":"IsRoot p x implies x is a root of p. The evaluation of p at x is zero","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L342-L344"},"Polynomial.coeff_C_zero":{"signature":"theorem Polynomial.coeff_C_zero {R : Type u} {a : R} [Semiring R] :(C a).coeff 0 = a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L636-L638"},"Polynomial.coeff_C":{"signature":"theorem Polynomial.coeff_C {R : Type u} {a : R} {n : \u2115} [Semiring R] :(C a).coeff n = if n = 0 then a else 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L631-L634"},"Polynomial.eq_C_of_degree_le_zero":{"signature":"theorem Polynomial.eq_C_of_degree_le_zero {R : Type u} [Semiring R] {p : Polynomial R} (h : p.degree \u2264 0) :p = C (p.coeff 0)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L163-L167"},"Polynomial.eval_C":{"signature":"theorem Polynomial.eval_C {R : Type u} {a : R} [Semiring R] {x : R} :eval x (C a) = a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L275-L277"},"Polynomial.degree_pos_of_root":{"signature":"theorem Polynomial.degree_pos_of_root {R : Type u} {a : R} [Semiring R] {p : Polynomial R} (hp : p \u2260 0) (h : p.IsRoot a) :0 < p.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L72-L77"},"Polynomial.eval\u2082_mul":{"signature":"theorem Polynomial.eval\u2082_mul {R : Type u} {S : Type v} [Semiring R] {p q : Polynomial R} [CommSemiring S] (f : R \u2192+* S) (x : S) :eval\u2082 f x (p * q) = eval\u2082 f x p * eval\u2082 f x q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L198-L200"},"Polynomial.eval_mul":{"signature":"theorem Polynomial.eval_mul {R : Type u} [CommSemiring R] {p q : Polynomial R} {x : R} :eval x (p * q) = eval x p * eval x q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L599-L601"},"Polynomial.eval\u2082_neg":{"signature":"theorem Polynomial.eval\u2082_neg {R : Type u} [Ring R] {p : Polynomial R} {S : Type u_1} [Ring S] (f : R \u2192+* S) {x : S} :eval\u2082 f x (-p) = -eval\u2082 f x p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L738-L740"},"Polynomial.eval\u2082_sub":{"signature":"theorem Polynomial.eval\u2082_sub {R : Type u} [Ring R] {p q : Polynomial R} {S : Type u_1} [Ring S] (f : R \u2192+* S) {x : S} :eval\u2082 f x (p - q) = eval\u2082 f x p - eval\u2082 f x q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L742-L745"},"Polynomial.eval_sub":{"signature":"theorem Polynomial.eval_sub {R : Type u} [Ring R] (p q : Polynomial R) (x : R) :eval x (p - q) = eval x p - eval x q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L751-L753"},"Polynomial.eval_X":{"signature":"theorem Polynomial.eval_X {R : Type u} [Semiring R] {x : R} :eval x X = x","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L287-L289"},"Polynomial.modByMonic_X_sub_C_eq_C_eval":{"signature":"theorem Polynomial.modByMonic_X_sub_C_eq_C_eval {R : Type u} [CommRing R] (p : Polynomial R) (a : R) :p %\u2098 (X - C a) = C (eval a p)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L573-L587"},"Polynomial.mul_divByMonic_eq_iff_isRoot":{"signature":"theorem Polynomial.mul_divByMonic_eq_iff_isRoot {R : Type u} {a : R} [CommRing R] {p : Polynomial R} :(X - C a) * (p /\u2098 (X - C a)) = p \u2194 p.IsRoot a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L589-L596"},"Polynomial.modByMonic_eq_self_iff":{"signature":"theorem Polynomial.modByMonic_eq_self_iff {R : Type u} [Ring R] {p q : Polynomial R} [Nontrivial R] (hq : q.Monic) :p %\u2098 q = p \u2194 p.degree < q.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L210-L214"},"Polynomial.divByMonic_eq_zero_iff":{"signature":"theorem Polynomial.divByMonic_eq_zero_iff {R : Type u} [Ring R] {p q : Polynomial R} [Nontrivial R] (hq : q.Monic) :p /\u2098 q = 0 \u2194 p.degree < q.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L262-L269"},"Polynomial.degree_of_subsingleton":{"signature":"theorem Polynomial.degree_of_subsingleton {R : Type u} [Semiring R] {p : Polynomial R} [Subsingleton R] :p.degree = \u22a5","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L52-L54"},"Polynomial.degree_add_eq_right_of_degree_lt":{"signature":"theorem Polynomial.degree_add_eq_right_of_degree_lt {R : Type u} [Semiring R] {p q : Polynomial R} (h : p.degree < q.degree) :(p + q).degree = q.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L181-L182"},"Polynomial.degree_add_divByMonic":{"signature":"theorem Polynomial.degree_add_divByMonic {R : Type u} [Ring R] {p q : Polynomial R} (hq : q.Monic) (h : q.degree \u2264 p.degree) :q.degree + (p /\u2098 q).degree = p.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L271-L287"},"Polynomial.degree_divByMonic_lt":{"signature":"theorem Polynomial.degree_divByMonic_lt {R : Type u} [Ring R] (p : Polynomial R) {q : Polynomial R} (hq : q.Monic) (hp0 : p \u2260 0) (h0q : 0 < q.degree) :(p /\u2098 q).degree < p.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L304-L317"},"Polynomial.rootMultiplicity_eq_multiplicity":{"signature":"theorem Polynomial.rootMultiplicity_eq_multiplicity {R : Type u} [Ring R] [DecidableEq R] (p : Polynomial R) (a : R) :rootMultiplicity a p = if p = 0 then 0 else multiplicity (X - C a) p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L517-L527"},"Polynomial.instIsLeftCancelMulZeroOfIsCancelAdd":{"signature":"instance Polynomial.instIsLeftCancelMulZeroOfIsCancelAdd {R : Type u} [Semiring R] [IsCancelAdd R] [IsLeftCancelMulZero R] :IsLeftCancelMulZero (Polynomial R)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L1151-L1152"},"Polynomial.instIsRightCancelMulZeroOfIsCancelAdd":{"signature":"instance Polynomial.instIsRightCancelMulZeroOfIsCancelAdd {R : Type u} [Semiring R] [IsCancelAdd R] [IsRightCancelMulZero R] :IsRightCancelMulZero (Polynomial R)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L1154-L1155"},"Polynomial.instIsCancelMulZeroOfIsCancelAdd":{"signature":"instance Polynomial.instIsCancelMulZeroOfIsCancelAdd {R : Type u} [Semiring R] [IsCancelAdd R] [IsCancelMulZero R] :IsCancelMulZero (Polynomial R)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L1157-L1157"},"Polynomial.degree_X_pow_add_C":{"signature":"theorem Polynomial.degree_X_pow_add_C {R : Type u} [Nontrivial R] [Semiring R] {n : \u2115} (hn : 0 < n) (a : R) :(X ^ n + C a).degree = \u2191n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L616-L619"},"Polynomial.X_pow_add_C_ne_zero":{"signature":"theorem Polynomial.X_pow_add_C_ne_zero {R : Type u} [Nontrivial R] [Semiring R] {n : \u2115} (hn : 0 < n) (a : R) :X ^ n + C a \u2260 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L621-L624"},"Polynomial.X_pow_sub_C_ne_zero":{"signature":"theorem Polynomial.X_pow_sub_C_ne_zero {R : Type u} [Ring R] [Nontrivial R] {n : \u2115} (hn : 0 < n) (a : R) :X ^ n - C a \u2260 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L770-L772"},"Polynomial.X_sub_C_ne_zero":{"signature":"theorem Polynomial.X_sub_C_ne_zero {R : Type u} [Ring R] [Nontrivial R] (r : R) :X - C r \u2260 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L774-L775"},"Polynomial.natDegree_mul_le":{"signature":"theorem Polynomial.natDegree_mul_le {R : Type u} [Semiring R] {p q : Polynomial R} :(p * q).natDegree \u2264 p.natDegree + q.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L472-L476"},"Polynomial.degree_sub_le":{"signature":"theorem Polynomial.degree_sub_le {R : Type u} [Ring R] (p q : Polynomial R) :(p - q).degree \u2264 max p.degree q.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L529-L530"},"Polynomial.degree_X_le":{"signature":"theorem Polynomial.degree_X_le {R : Type u} [Semiring R] :X.degree \u2264 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L243-L244"},"Polynomial.degree_X_sub_C_le":{"signature":"theorem Polynomial.degree_X_sub_C_le {R : Type u} [Ring R] (r : R) :(X - C r).degree \u2264 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L560-L561"},"Polynomial.natDegree_X_sub_C_le":{"signature":"theorem Polynomial.natDegree_X_sub_C_le {R : Type u} [Ring R] (r : R) :(X - C r).natDegree \u2264 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L563-L564"},"Polynomial.le_natDegree_of_mem_supp":{"signature":"theorem Polynomial.le_natDegree_of_mem_supp {R : Type u} [Semiring R] {p : Polynomial R} (a : \u2115) :a \u2208 p.support \u2192 a \u2264 p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Support.lean#L46-L47"},"Polynomial.supp_subset_range":{"signature":"theorem Polynomial.supp_subset_range {R : Type u} {m : \u2115} [Semiring R] {p : Polynomial R} (h : p.natDegree < m) :p.support \u2286 Finset.range m","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Support.lean#L49-L50"},"Polynomial.sum_over_range'":{"signature":"theorem Polynomial.sum_over_range' {R : Type u} {S : Type v} [Semiring R] [AddCommMonoid S] (p : Polynomial R) {f : \u2115 \u2192 R \u2192 S} (h : \u2200 (n : \u2115), f n 0 = 0) (n : \u2115) (hn : p.natDegree < n) :p.sum f = \u2211 a \u2208 Finset.range n, f a (p.coeff a)","doc":"We can reexpress a sum over p.support as a sum over range n, for any n satisfying p.natDegree < n.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Support.lean#L61-L68"},"Polynomial.ofFinsupp_sub":{"signature":"theorem Polynomial.ofFinsupp_sub {R : Type u} [Ring R] {a b : AddMonoidAlgebra R \u2115} :{ toFinsupp := a - b } = { toFinsupp := a } - { toFinsupp := b }","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L155-L158"},"Polynomial.coeff_sub":{"signature":"theorem Polynomial.coeff_sub {R : Type u} [Ring R] (p q : Polynomial R) (n : \u2115) :(p - q).coeff n = p.coeff n - q.coeff n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L1101-L1105"},"Polynomial.coeff_mul_X":{"signature":"theorem Polynomial.coeff_mul_X {R : Type u} [Semiring R] (p : Polynomial R) (n : \u2115) :(p * X).coeff (n + 1) = p.coeff n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L250-L252"},"Polynomial.coeff_mul_C":{"signature":"theorem Polynomial.coeff_mul_C {R : Type u} [Semiring R] (p : Polynomial R) (n : \u2115) (a : R) :(p * C a).coeff n = p.coeff n * a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L161-L165"},"Polynomial.coeff_mul_X_sub_C":{"signature":"theorem Polynomial.coeff_mul_X_sub_C {R : Type u} [Ring R] {p : Polynomial R} {r : R} {a : \u2115} :(p * (X - C r)).coeff (a + 1) = p.coeff a - p.coeff (a + 1) * r","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L123-L124"},"Polynomial.coeff_eq_zero_of_natDegree_lt":{"signature":"theorem Polynomial.coeff_eq_zero_of_natDegree_lt {R : Type u} [Semiring R] {p : Polynomial R} {n : \u2115} (h : p.natDegree < n) :p.coeff n = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L85-L91"},"Polynomial.coeff_natDegree_succ_eq_zero":{"signature":"theorem Polynomial.coeff_natDegree_succ_eq_zero {R : Type u} [Semiring R] {p : Polynomial R} :p.coeff (p.natDegree + 1) = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L106-L108"},"Polynomial.mul_coeff_zero":{"signature":"theorem Polynomial.mul_coeff_zero {R : Type u} [Semiring R] (p q : Polynomial R) :(p * q).coeff 0 = p.coeff 0 * q.coeff 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L115-L116"},"Polynomial.coeff_X_zero":{"signature":"theorem Polynomial.coeff_X_zero {R : Type u} [Semiring R] :X.coeff 0 = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L610-L612"},"Polynomial.eval_mul_X_sub_C":{"signature":"theorem Polynomial.eval_mul_X_sub_C {R : Type u} [Ring R] {p : Polynomial R} (r : R) :eval r (p * (X - C r)) = 0","doc":"The evaluation map is not generally multiplicative when the coefficient ring is noncommutative, but nevertheless any polynomial of the form p * (X - monomial 0 r) is sent to zero when evaluated at r. This is the key step in our proof of the Cayley-Hamilton theorem.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L608-L630"},"Polynomial.eval_one":{"signature":"theorem Polynomial.eval_one {R : Type u} [Semiring R] {x : R} :eval x 1 = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L303-L305"},"Polynomial.not_isUnit_X_sub_C":{"signature":"theorem Polynomial.not_isUnit_X_sub_C {R : Type u} [Ring R] [Nontrivial R] (r : R) :\u00acIsUnit (X - C r)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L632-L633"},"Polynomial.monomial_injective":{"signature":"theorem Polynomial.monomial_injective {R : Type u} [Semiring R] (n : \u2115) :Function.Injective \u21d1(monomial n)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L417-L418"},"Polynomial.C_injective":{"signature":"theorem Polynomial.C_injective {R : Type u} [Semiring R] :Function.Injective \u21d1C","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L675-L676"},"Polynomial.C_inj":{"signature":"theorem Polynomial.C_inj {R : Type u} {a b : R} [Semiring R] :C a = C b \u2194 a = b","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L678-L680"},"Polynomial.dvd_iff_isRoot":{"signature":"theorem Polynomial.dvd_iff_isRoot {R : Type u} {a : R} [CommRing R] {p : Polynomial R} :X - C a \u2223 p \u2194 p.IsRoot a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L598-L602"},"Polynomial.prime_X_sub_C":{"signature":"theorem Polynomial.prime_X_sub_C {R : Type u} [CommRing R] [IsDomain R] (r : R) :Prime (X - C r)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/RingDivision.lean#L203-L206"},"Polynomial.rootMultiplicity_mul":{"signature":"theorem Polynomial.rootMultiplicity_mul {R : Type u} [CommRing R] [IsDomain R] {p q : Polynomial R} {x : R} (hpq : p * q \u2260 0) :rootMultiplicity x (p * q) = rootMultiplicity x p + rootMultiplicity x q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/RingDivision.lean#L275-L282"},"Polynomial.instNoZeroDivisors":{"signature":"instance Polynomial.instNoZeroDivisors {R : Type u} [Semiring R] [NoZeroDivisors R] :NoZeroDivisors (Polynomial R)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L1148-L1149"},"Polynomial.degree_le_of_natDegree_le":{"signature":"theorem Polynomial.degree_le_of_natDegree_le {R : Type u} [Semiring R] {p : Polynomial R} {n : \u2115} :p.natDegree \u2264 n \u2192 p.degree \u2264 \u2191n","doc":"Alias of the forward direction of Polynomial.natDegree_le_iff_degree_le.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L145-L145"},"Polynomial.eq_C_of_natDegree_le_zero":{"signature":"theorem Polynomial.eq_C_of_natDegree_le_zero {R : Type u} [Semiring R] {p : Polynomial R} (h : p.natDegree \u2264 0) :p = C (p.coeff 0)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L479-L480"},"Polynomial.eq_C_of_natDegree_eq_zero":{"signature":"theorem Polynomial.eq_C_of_natDegree_eq_zero {R : Type u} [Semiring R] {p : Polynomial R} (h : p.natDegree = 0) :p = C (p.coeff 0)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L482-L483"},"Polynomial.eq_one_of_monic_natDegree_zero":{"signature":"theorem Polynomial.eq_one_of_monic_natDegree_zero {R : Type u} [Semiring R] {p : Polynomial R} (hf : p.Monic) (hfd : p.natDegree = 0) :p = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L491-L493"},"Polynomial.natDegree_eq_zero_iff_degree_le_zero":{"signature":"theorem Polynomial.natDegree_eq_zero_iff_degree_le_zero {R : Type u} [Semiring R] {p : Polynomial R} :p.natDegree = 0 \u2194 p.degree \u2264 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L491-L492"},"Polynomial.rootMultiplicity_eq_nat_find_of_nonzero":{"signature":"theorem Polynomial.rootMultiplicity_eq_nat_find_of_nonzero {R : Type u} [Ring R] [DecidableEq R] {p : Polynomial R} (p0 : p \u2260 0) {a : R} :rootMultiplicity a p = Nat.find \u22ef","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L507-L515"},"Polynomial.le_rootMultiplicity_iff":{"signature":"theorem Polynomial.le_rootMultiplicity_iff {R : Type u} [CommRing R] {p : Polynomial R} (p0 : p \u2260 0) {a : R} {n : \u2115} :n \u2264 rootMultiplicity a p \u2194 (X - C a) ^ n \u2223 p","doc":"The multiplicity of a as root of a nonzero polynomial p is at least n iff (X - a) ^ n divides p.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L714-L724"},"Polynomial.rootMultiplicity_le_iff":{"signature":"theorem Polynomial.rootMultiplicity_le_iff {R : Type u} [CommRing R] {p : Polynomial R} (p0 : p \u2260 0) (a : R) (n : \u2115) :rootMultiplicity a p \u2264 n \u2194 \u00ac(X - C a) ^ (n + 1) \u2223 p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L726-L728"},"Polynomial.natDegree_le_natDegree":{"signature":"theorem Polynomial.natDegree_le_natDegree {R : Type u} {S : Type v} [Semiring R] {p : Polynomial R} [Semiring S] {q : Polynomial S} (hpq : p.degree \u2264 q.degree) :p.natDegree \u2264 q.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L147-L149"},"Polynomial.degree_smul_le":{"signature":"theorem Polynomial.degree_smul_le {R : Type u} [Semiring R] {S : Type u_2} [SMulZeroClass S R] (a : S) (p : Polynomial R) :(a \u2022 p).degree \u2264 p.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L456-L460"},"Polynomial.natDegree_smul_le":{"signature":"theorem Polynomial.natDegree_smul_le {R : Type u} [Semiring R] {S : Type u_2} [SMulZeroClass S R] (a : S) (p : Polynomial R) :(a \u2022 p).natDegree \u2264 p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L462-L464"},"Polynomial.smul_eq_C_mul":{"signature":"theorem Polynomial.smul_eq_C_mul {R : Type u} [Semiring R] {p : Polynomial R} (a : R) :a \u2022 p = C a * p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L330-L330"},"Polynomial.coeff_zero":{"signature":"theorem Polynomial.coeff_zero {R : Type u} [Semiring R] (n : \u2115) :coeff 0 n = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L593-L595"},"Polynomial.coeff_natDegree":{"signature":"theorem Polynomial.coeff_natDegree {R : Type u} [Semiring R] {p : Polynomial R} :p.coeff p.natDegree = p.leadingCoeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L87-L89"},"Polynomial.eq_zero_of_mul_eq_zero_of_smul":{"signature":"theorem Polynomial.eq_zero_of_mul_eq_zero_of_smul {R : Type u} [CommSemiring R] (P : Polynomial R) (h : \u2200 (r : R), r \u2022 P = 0 \u2192 r = 0) (Q : Polynomial R) (hQ : P * Q = 0) :Q = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L718-L745"},"Polynomial.C_eq_zero":{"signature":"theorem Polynomial.C_eq_zero {R : Type u} {a : R} [Semiring R] :C a = 0 \u2194 a = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L682-L684"},"Polynomial.notMem_nonZeroDivisors_iff":{"signature":"theorem Polynomial.notMem_nonZeroDivisors_iff {R : Type u} [CommSemiring R] {P : Polynomial R} :P \u2209 nonZeroDivisors (Polynomial R) \u2194 \u2203 (a : R), a \u2260 0 \u2227 a \u2022 P = 0","doc":"McCoy theorem: a polynomial P : R[X] is a zerodivisor if and only if there is a : R such that a \u2260 0 and a \u2022 P = 0.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L749-L757"},"Polynomial.mem_nonZeroDivisors_iff":{"signature":"theorem Polynomial.mem_nonZeroDivisors_iff {R : Type u} [CommSemiring R] {P : Polynomial R} :P \u2208 nonZeroDivisors (Polynomial R) \u2194 \u2200 (a : R), a \u2022 P = 0 \u2192 a = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L759-L760"},"Polynomial.mem_nonzeroDivisors_of_coeff_mem":{"signature":"theorem Polynomial.mem_nonzeroDivisors_of_coeff_mem {R : Type u} [CommSemiring R] {p : Polynomial R} (n : \u2115) (hp : p.coeff n \u2208 nonZeroDivisors R) :p \u2208 nonZeroDivisors (Polynomial R)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L762-L764"},"Polynomial.pow_rootMultiplicity_not_dvd":{"signature":"theorem Polynomial.pow_rootMultiplicity_not_dvd {R : Type u} [CommRing R] {p : Polynomial R} (p0 : p \u2260 0) (a : R) :\u00ac(X - C a) ^ (rootMultiplicity a p + 1) \u2223 p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L746-L747"},"Polynomial.rootMultiplicity_zero":{"signature":"theorem Polynomial.rootMultiplicity_zero {R : Type u} [Ring R] {x : R} :rootMultiplicity x 0 = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L529-L531"},"Polynomial.pow_rootMultiplicity_dvd":{"signature":"theorem Polynomial.pow_rootMultiplicity_dvd {R : Type u} [Ring R] (p : Polynomial R) (a : R) :(X - C a) ^ rootMultiplicity a p \u2223 p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L544-L549"},"Polynomial.rootMultiplicity_mul_X_sub_C_pow":{"signature":"theorem Polynomial.rootMultiplicity_mul_X_sub_C_pow {R : Type u} [CommRing R] {p : Polynomial R} {a : R} {n : \u2115} (h : p \u2260 0) :rootMultiplicity a (p * (X - C a) ^ n) = rootMultiplicity a p + n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/RingDivision.lean#L152-L160"},"Polynomial.nontrivial":{"signature":"instance Polynomial.nontrivial {R : Type u} [Semiring R] [Nontrivial R] :Nontrivial (Polynomial R)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L1138-L1142"},"Polynomial.C_add":{"signature":"theorem Polynomial.C_add {R : Type u} {a b : R} [Semiring R] :C (a + b) = C a + C b","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L459-L460"},"Polynomial.natDegree_add_eq_left_of_degree_lt":{"signature":"theorem Polynomial.natDegree_add_eq_left_of_degree_lt {R : Type u} [Semiring R] {p q : Polynomial R} (h : q.degree < p.degree) :(p + q).natDegree = p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L184-L186"},"Polynomial.ne_zero_of_natDegree_gt":{"signature":"theorem Polynomial.ne_zero_of_natDegree_gt {R : Type u} [Semiring R] {p : Polynomial R} {n : \u2115} (h : n < p.natDegree) :p \u2260 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L147-L148"},"Polynomial.degree_lt_degree":{"signature":"theorem Polynomial.degree_lt_degree {R : Type u} [Semiring R] {p q : Polynomial R} (h : p.natDegree < q.natDegree) :p.degree < q.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L150-L156"},"Polynomial.natDegree_add_eq_left_of_natDegree_lt":{"signature":"theorem Polynomial.natDegree_add_eq_left_of_natDegree_lt {R : Type u} [Semiring R] {p q : Polynomial R} (h : q.natDegree < p.natDegree) :(p + q).natDegree = p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L188-L190"},"Polynomial.natDegree_add_C":{"signature":"theorem Polynomial.natDegree_add_C {R : Type u} [Semiring R] {p : Polynomial R} {a : R} :(p + C a).natDegree = p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L203-L209"},"Polynomial.natDegree_sub_C":{"signature":"theorem Polynomial.natDegree_sub_C {R : Type u} [Ring R] {p : Polynomial R} {a : R} :(p - C a).natDegree = p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L551-L553"},"Polynomial.natDegree_X":{"signature":"theorem Polynomial.natDegree_X {R : Type u} [Semiring R] [Nontrivial R] :X.natDegree = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L267-L269"},"Polynomial.rootMultiplicity_C":{"signature":"theorem Polynomial.rootMultiplicity_C {R : Type u} [Ring R] (r a : R) :rootMultiplicity a (C r) = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L533-L542"},"Polynomial.rootMultiplicity_X_sub_C_pow":{"signature":"theorem Polynomial.rootMultiplicity_X_sub_C_pow {R : Type u} [CommRing R] [Nontrivial R] (a : R) (n : \u2115) :rootMultiplicity a ((X - C a) ^ n) = n","doc":"The multiplicity of a as root of (X - a) ^ n is n.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/RingDivision.lean#L162-L166"},"Polynomial.rootMultiplicity_X_sub_C_self":{"signature":"theorem Polynomial.rootMultiplicity_X_sub_C_self {R : Type u} [CommRing R] [Nontrivial R] {x : R} :rootMultiplicity x (X - C x) = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/RingDivision.lean#L168-L170"},"Polynomial.rootMultiplicity_eq_zero_iff":{"signature":"theorem Polynomial.rootMultiplicity_eq_zero_iff {R : Type u} [CommRing R] {p : Polynomial R} {x : R} :rootMultiplicity x p = 0 \u2194 p.IsRoot x \u2192 p = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L624-L629"},"Polynomial.rootMultiplicity_eq_zero":{"signature":"theorem Polynomial.rootMultiplicity_eq_zero {R : Type u} [CommRing R] {p : Polynomial R} {x : R} (h : \u00acp.IsRoot x) :rootMultiplicity x p = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L631-L632"},"Polynomial.root_X_sub_C":{"signature":"theorem Polynomial.root_X_sub_C {R : Type u} {a b : R} [Ring R] :(X - C a).IsRoot b \u2194 a = b","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L755-L756"},"Polynomial.rootMultiplicity_X_sub_C":{"signature":"theorem Polynomial.rootMultiplicity_X_sub_C {R : Type u} [CommRing R] [Nontrivial R] [DecidableEq R] {x y : R} :rootMultiplicity x (X - C y) = if x = y then 1 else 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/RingDivision.lean#L172-L177"},"Polynomial.exists_multiset_roots":{"signature":"theorem Polynomial.exists_multiset_roots {R : Type u} [CommRing R] [IsDomain R] [DecidableEq R] {p : Polynomial R} :p \u2260 0 \u2192 \u2203 (s : Multiset R), \u2191s.card \u2264 p.degree \u2227 \u2200 (a : R), Multiset.count a s = rootMultiplicity a p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/RingDivision.lean#L285-L326"},"Polynomial.roots":{"signature":"noncomputable def Polynomial.roots {R : Type u} [CommRing R] [IsDomain R] (p : Polynomial R) :Multiset R","doc":"roots p noncomputably gives a multiset containing all the roots of p, including their multiplicities.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L55-L60"},"Polynomial.nthRoots":{"signature":"def Polynomial.nthRoots {R : Type u} [CommRing R] [IsDomain R] (n : \u2115) (a : R) :Multiset R","doc":"nthRoots n a noncomputably returns the solutions to x ^ n = a.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L319-L321"},"Polynomial.roots_zero":{"signature":"theorem Polynomial.roots_zero {R : Type u} [CommRing R] [IsDomain R] :roots 0 = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L69-L71"},"Polynomial.roots_def":{"signature":"theorem Polynomial.roots_def {R : Type u} [CommRing R] [IsDomain R] [DecidableEq R] (p : Polynomial R) [Decidable (p = 0)] :p.roots = if h : p = 0 then \u2205 else Classical.choose \u22ef","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L62-L67"},"Polynomial.count_roots":{"signature":"theorem Polynomial.count_roots {R : Type u} {a : R} [CommRing R] [IsDomain R] [DecidableEq R] (p : Polynomial R) :Multiset.count a p.roots = rootMultiplicity a p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L97-L103"},"Polynomial.rootMultiplicity_pos'":{"signature":"theorem Polynomial.rootMultiplicity_pos' {R : Type u} [CommRing R] {p : Polynomial R} {x : R} :0 < rootMultiplicity x p \u2194 p \u2260 0 \u2227 p.IsRoot x","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L634-L637"},"Polynomial.mem_roots'":{"signature":"theorem Polynomial.mem_roots' {R : Type u} {a : R} [CommRing R] [IsDomain R] {p : Polynomial R} :a \u2208 p.roots \u2194 p \u2260 0 \u2227 p.IsRoot a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L105-L108"},"Polynomial.mem_roots":{"signature":"theorem Polynomial.mem_roots {R : Type u} {a : R} [CommRing R] [IsDomain R] {p : Polynomial R} (hp : p \u2260 0) :a \u2208 p.roots \u2194 p.IsRoot a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L110-L111"},"Polynomial.eval\u2082AddMonoidHom":{"signature":"def Polynomial.eval\u2082AddMonoidHom {R : Type u} {S : Type v} [Semiring R] [Semiring S] (f : R \u2192+* S) (x : S) :Polynomial R \u2192+ S","doc":"eval\u2082AddMonoidHom (f : R \u2192+* S) (x : S) is the AddMonoidHom from R[X] to S obtained by evaluating the pushforward of p along f at x.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L94-L100"},"Polynomial.eval\u2082RingHom":{"signature":"def Polynomial.eval\u2082RingHom {R : Type u} {S : Type v} [Semiring R] [CommSemiring S] (f : R \u2192+* S) (x : S) :Polynomial R \u2192+* S","doc":"eval\u2082 as a RingHom","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L210-L214"},"Polynomial.eval\u2082_pow":{"signature":"theorem Polynomial.eval\u2082_pow {R : Type u} {S : Type v} [Semiring R] {p : Polynomial R} [CommSemiring S] (f : R \u2192+* S) (x : S) (n : \u2115) :eval\u2082 f x (p ^ n) = eval\u2082 f x p ^ n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L220-L221"},"Polynomial.eval_pow":{"signature":"theorem Polynomial.eval_pow {R : Type u} [CommSemiring R] {p : Polynomial R} {x : R} (n : \u2115) :eval x (p ^ n) = eval x p ^ n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L611-L613"},"Polynomial.mem_nthRoots":{"signature":"theorem Polynomial.mem_nthRoots {R : Type u} [CommRing R] [IsDomain R] {n : \u2115} (hn : 0 < n) {a x : R} :x \u2208 nthRoots n a \u2194 x ^ n = a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L323-L326"},"Polynomial.card_roots":{"signature":"theorem Polynomial.card_roots {R : Type u} [CommRing R] [IsDomain R] {p : Polynomial R} (hp0 : p \u2260 0) :\u2191p.roots.card \u2264 p.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L73-L77"},"Polynomial.degree_X_pow_sub_C":{"signature":"theorem Polynomial.degree_X_pow_sub_C {R : Type u} [Ring R] [Nontrivial R] {n : \u2115} (hn : 0 < n) (a : R) :(X ^ n - C a).degree = \u2191n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L767-L768"},"Polynomial.card_nthRoots":{"signature":"theorem Polynomial.card_nthRoots {R : Type u} [CommRing R] [IsDomain R] (n : \u2115) (a : R) :(nthRoots n a).card \u2264 n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L337-L351"},"Polynomial.aeval":{"signature":"def Polynomial.aeval {R : Type u} {A : Type z} [CommSemiring R] [Semiring A] [Algebra R A] (x : A) :Polynomial R \u2192\u2090[R] A","doc":"Given a valuation x of the variable in an R-algebra A, aeval R A x is the unique R-algebra homomorphism from R[X] to A sending X to x. This is a stronger variant of the linear map Polynomial.leval.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L242-L247"},"Polynomial.smul_monomial":{"signature":"theorem Polynomial.smul_monomial {R : Type u} [Semiring R] {S : Type u_1} [SMulZeroClass S R] (a : S) (n : \u2115) (b : R) :a \u2022 (monomial n) b = (monomial n) (a \u2022 b)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L413-L415"},"Polynomial.smul_X_eq_monomial":{"signature":"theorem Polynomial.smul_X_eq_monomial {R : Type u} {a : R} [Semiring R] {n : \u2115} :a \u2022 X ^ n = (monomial n) a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L798-L799"},"Polynomial.adjoin_X":{"signature":"theorem Polynomial.adjoin_X {R : Type u} [CommSemiring R] :Algebra.adjoin R {X} = \u22a4","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Adjoin/Polynomial.lean#L41-L46"},"Polynomial.aeval_X":{"signature":"theorem Polynomial.aeval_X {R : Type u} {A : Type z} [CommSemiring R] [Semiring A] [Algebra R A] (x : A) :(aeval x) X = x","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L275-L277"},"Polynomial.aeval_def":{"signature":"theorem Polynomial.aeval_def {R : Type u} {A : Type z} [CommSemiring R] [Semiring A] [Algebra R A] (x : A) (p : Polynomial R) :(aeval x) p = eval\u2082 (algebraMap R A) x p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L259-L260"},"Polynomial.eval\u2082_sum":{"signature":"theorem Polynomial.eval\u2082_sum {R : Type u} {S : Type v} {T : Type w} [Semiring R] [Semiring S] (f : R \u2192+* S) [Semiring T] (p : Polynomial T) (g : \u2115 \u2192 T \u2192 Polynomial R) (x : S) :eval\u2082 f x (p.sum g) = p.sum fun (n : \u2115) (a : T) => eval\u2082 f x (g n a)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L115-L123"},"Polynomial.sum_def":{"signature":"theorem Polynomial.sum_def {R : Type u} [Semiring R] {S : Type u_1} [AddCommMonoid S] (p : Polynomial R) (f : \u2115 \u2192 R \u2192 S) :p.sum f = \u2211 n \u2208 p.support, f n (p.coeff n)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L828-L830"},"Polynomial.natDegree_zero":{"signature":"theorem Polynomial.natDegree_zero {R : Type u} [Semiring R] :natDegree 0 = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L83-L85"},"Polynomial.natDegree_lt_natDegree":{"signature":"theorem Polynomial.natDegree_lt_natDegree {R : Type u} {S : Type v} [Semiring R] [Semiring S] {p : Polynomial R} {q : Polynomial S} (hp : p \u2260 0) (hpq : p.degree < q.degree) :p.natDegree < q.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L73-L77"},"Polynomial.natDegree_modByMonic_lt":{"signature":"theorem Polynomial.natDegree_modByMonic_lt {R : Type u} [Ring R] (p : Polynomial R) {q : Polynomial R} (hmq : q.Monic) (hq : q \u2260 1) :(p %\u2098 q).natDegree < q.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L171-L178"},"Polynomial.monic_multiset_prod_of_monic":{"signature":"theorem Polynomial.monic_multiset_prod_of_monic {R : Type u} {\u03b9 : Type y} [CommSemiring R] (t : Multiset \u03b9) (f : \u03b9 \u2192 Polynomial R) (ht : \u2200 i \u2208 t, (f i).Monic) :(Multiset.map f t).prod.Monic","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Monic.lean#L280-L286"},"Polynomial.monic_prod_of_monic":{"signature":"theorem Polynomial.monic_prod_of_monic {R : Type u} {\u03b9 : Type y} [CommSemiring R] (s : Finset \u03b9) (f : \u03b9 \u2192 Polynomial R) (hs : \u2200 i \u2208 s, (f i).Monic) :(\u220f i \u2208 s, f i).Monic","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Monic.lean#L288-L290"},"Polynomial.leadingCoeff_add_of_degree_lt":{"signature":"theorem Polynomial.leadingCoeff_add_of_degree_lt {R : Type u} [Semiring R] {p q : Polynomial R} (h : p.degree < q.degree) :(p + q).leadingCoeff = q.leadingCoeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L263-L267"},"Polynomial.degree_one":{"signature":"theorem Polynomial.degree_one {R : Type u} [Semiring R] [Nontrivial R] :degree 1 = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L259-L261"},"Polynomial.degree_eq_iff_natDegree_eq":{"signature":"theorem Polynomial.degree_eq_iff_natDegree_eq {R : Type u} [Semiring R] {p : Polynomial R} {n : \u2115} (hp : p \u2260 0) :p.degree = \u2191n \u2194 p.natDegree = n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L102-L103"},"Polynomial.degree_eq_iff_natDegree_eq_of_pos":{"signature":"theorem Polynomial.degree_eq_iff_natDegree_eq_of_pos {R : Type u} [Semiring R] {p : Polynomial R} {n : \u2115} (hn : 0 < n) :p.degree = \u2191n \u2194 p.natDegree = n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L105-L109"},"Polynomial.leadingCoeff_multiset_prod'":{"signature":"theorem Polynomial.leadingCoeff_multiset_prod' {R : Type u} [CommSemiring R] (t : Multiset (Polynomial R)) (h : (Multiset.map leadingCoeff t).prod \u2260 0) :t.prod.leadingCoeff = (Multiset.map leadingCoeff t).prod","doc":"The leading coefficient of a product of polynomials is equal to the product of the leading coefficients, provided that this product is nonzero. See Polynomial.leadingCoeff_multiset_prod (without the ') for a version for integral domains, where this condition is automatically satisfied.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/BigOperators.lean#L146-L163"},"Polynomial.natDegree_multiset_prod'":{"signature":"theorem Polynomial.natDegree_multiset_prod' {R : Type u} [CommSemiring R] (t : Multiset (Polynomial R)) (h : (Multiset.map (fun (f : Polynomial R) => f.leadingCoeff) t).prod \u2260 0) :t.prod.natDegree = (Multiset.map (fun (f : Polynomial R) => f.natDegree) t).sum","doc":"The degree of a product of polynomials is equal to the sum of the degrees, provided that the product of leading coefficients is nonzero. See Polynomial.natDegree_multiset_prod (without the ') for a version for integral domains, where this condition is automatically satisfied.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/BigOperators.lean#L175-L189"},"Polynomial.natDegree_prod'":{"signature":"theorem Polynomial.natDegree_prod' {R : Type u} {\u03b9 : Type w} (s : Finset \u03b9) [CommSemiring R] (f : \u03b9 \u2192 Polynomial R) (h : \u220f i \u2208 s, (f i).leadingCoeff \u2260 0) :(\u220f i \u2208 s, f i).natDegree = \u2211 i \u2208 s, (f i).natDegree","doc":"The degree of a product of polynomials is equal to the sum of the degrees, provided that the product of leading coefficients is nonzero. See Polynomial.natDegree_prod (without the ') for a version for integral domains, where this condition is automatically satisfied.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/BigOperators.lean#L191-L199"},"Polynomial.natDegree_X_sub_C":{"signature":"theorem Polynomial.natDegree_X_sub_C {R : Type u} [Ring R] [Nontrivial R] (x : R) :(X - C x).natDegree = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L760-L761"},"Polynomial.degreeLT":{"signature":"def Polynomial.degreeLT (R : Type u) [Semiring R] (n : \u2115) :Submodule R (Polynomial R)","doc":"The R-submodule of R[X] consisting of polynomials of degree < n.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Basic.lean#L56-L58"},"Polynomial.mem_degreeLT":{"signature":"theorem Polynomial.mem_degreeLT {R : Type u} [Semiring R] {n : \u2115} {f : Polynomial R} :f \u2208 degreeLT R n \u2194 f.degree < \u2191n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Basic.lean#L89-L90"},"Polynomial.C_eq_intCast":{"signature":"theorem Polynomial.C_eq_intCast {R : Type u} [Ring R] (n : \u2124) :C \u2191n = \u2191n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L1119-L1119"},"Polynomial.natDegree_list_prod_le":{"signature":"theorem Polynomial.natDegree_list_prod_le {S : Type u_1} [Semiring S] (l : List (Polynomial S)) :l.prod.natDegree \u2264 (List.map natDegree l).sum","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/BigOperators.lean#L104-L105"},"Polynomial.natDegree_multiset_prod_le":{"signature":"theorem Polynomial.natDegree_multiset_prod_le {R : Type u} [CommSemiring R] (t : Multiset (Polynomial R)) :t.prod.natDegree \u2264 (Multiset.map natDegree t).sum","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/BigOperators.lean#L131-L132"},"Polynomial.natDegree_prod_le":{"signature":"theorem Polynomial.natDegree_prod_le {R : Type u} {\u03b9 : Type w} (s : Finset \u03b9) [CommSemiring R] (f : \u03b9 \u2192 Polynomial R) :(\u220f i \u2208 s, f i).natDegree \u2264 \u2211 i \u2208 s, (f i).natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/BigOperators.lean#L134-L135"},"Polynomial.natDegree_X_le":{"signature":"theorem Polynomial.natDegree_X_le {R : Type u} [Semiring R] :X.natDegree \u2264 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L246-L247"},"Polynomial.natDegree_neg":{"signature":"theorem Polynomial.natDegree_neg {R : Type u} [Ring R] (p : Polynomial R) :(-p).natDegree = p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L283-L284"},"Polynomial.coeff_one":{"signature":"theorem Polynomial.coeff_one {R : Type u} [Semiring R] {n : \u2115} :coeff 1 n = if n = 0 then 1 else 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L597-L600"},"Polynomial.coeff_prod_mem_ideal_pow_tsub":{"signature":"theorem Polynomial.coeff_prod_mem_ideal_pow_tsub {R : Type u} [CommSemiring R] {\u03b9 : Type u_2} (s : Finset \u03b9) (f : \u03b9 \u2192 Polynomial R) (I : Ideal R) (n : \u03b9 \u2192 \u2115) (h : \u2200 i \u2208 s, \u2200 (k : \u2115), (f i).coeff k \u2208 I ^ (n i - k)) (k : \u2115) :(s.prod f).coeff k \u2208 I ^ (s.sum n - k)","doc":"If I is an ideal, and p\u1d62 is a finite family of polynomials each satisfying \u2200 k, (p\u1d62)\u2096 \u2208 I\u207f\u2071\u207b\u1d4f for some n\u1d62, then p = \u220f p\u1d62 also satisfies \u2200 k, p\u2096 \u2208 I\u207f\u207b\u1d4f with n = \u2211 n\u1d62.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Basic.lean#L586-L604"},"Polynomial.coeff_X_mul_zero":{"signature":"theorem Polynomial.coeff_X_mul_zero {R : Type u} [Semiring R] (p : Polynomial R) :(X * p).coeff 0 = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L141-L141"},"Polynomial.aeval_C":{"signature":"theorem Polynomial.aeval_C {R : Type u} {A : Type z} [CommSemiring R] [Semiring A] [Algebra R A] (x : A) (r : R) :(aeval x) (C r) = (algebraMap R A) r","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L279-L281"},"Polynomial.aeval_algHom_apply":{"signature":"theorem Polynomial.aeval_algHom_apply {R : Type u} {A : Type z} {B : Type u_2} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] {F : Type u_3} [FunLike F A B] [AlgHomClass F R A B] (f : F) (x : A) (p : Polynomial R) :(aeval (f x)) p = f ((aeval x) p)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L396-L401"},"Polynomial.aeval_subalgebra_coe":{"signature":"theorem Polynomial.aeval_subalgebra_coe {R : Type u} [CommSemiring R] (g : Polynomial R) {A : Type u_3} [Semiring A] [Algebra R A] (s : Subalgebra R A) (f : \u21a5s) :\u2191((aeval f) g) = (aeval \u2191f) g","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L465-L468"},"Polynomial.smulCommClass":{"signature":"instance Polynomial.smulCommClass {R : Type u} [Semiring R] {S\u2081 : Type u_1} {S\u2082 : Type u_2} [SMulZeroClass S\u2081 R] [SMulZeroClass S\u2082 R] [SMulCommClass S\u2081 S\u2082 R] :SMulCommClass S\u2081 S\u2082 (Polynomial R)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L288-L292"},"Polynomial.isScalarTower":{"signature":"instance Polynomial.isScalarTower {R : Type u} [Semiring R] {S\u2081 : Type u_1} {S\u2082 : Type u_2} [SMul S\u2081 S\u2082] [SMulZeroClass S\u2081 R] [SMulZeroClass S\u2082 R] [IsScalarTower S\u2081 S\u2082 R] :IsScalarTower S\u2081 S\u2082 (Polynomial R)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L294-L298"},"Polynomial.map_one":{"signature":"theorem Polynomial.map_one {R : Type u} {S : Type v} [Semiring R] [Semiring S] (f : R \u2192+* S) :map f 1 = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L506-L508"},"Polynomial.X_mul":{"signature":"theorem Polynomial.X_mul {R : Type u} [Semiring R] {p : Polynomial R} :X * p = p * X","doc":"X commutes with everything, even when the coefficients are noncommutative.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L500-L505"},"Polynomial.commute_X":{"signature":"theorem Polynomial.commute_X {R : Type u} [Semiring R] (p : Polynomial R) :Commute X p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L538-L539"},"Polynomial.map_mul":{"signature":"theorem Polynomial.map_mul {R : Type u} {S : Type v} [Semiring R] {p q : Polynomial R} [Semiring S] (f : R \u2192+* S) :map f (p * q) = map f p * map f q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L510-L513"},"Polynomial.map_zero":{"signature":"theorem Polynomial.map_zero {R : Type u} {S : Type v} [Semiring R] [Semiring S] (f : R \u2192+* S) :map f 0 = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L498-L500"},"Polynomial.mapRingHom":{"signature":"def Polynomial.mapRingHom {R : Type u} {S : Type v} [Semiring R] [Semiring S] (f : R \u2192+* S) :Polynomial R \u2192+* Polynomial S","doc":"Polynomial.map as a RingHom.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L521-L527"},"Polynomial.map_sum":{"signature":"theorem Polynomial.map_sum {R : Type u} {S : Type v} [Semiring R] [Semiring S] (f : R \u2192+* S) {\u03b9 : Type u_1} (g : \u03b9 \u2192 Polynomial R) (s : Finset \u03b9) :map f (\u2211 i \u2208 s, g i) = \u2211 i \u2208 s, map f (g i)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L571-L573"},"Polynomial.distribSMul":{"signature":"instance Polynomial.distribSMul {R : Type u} [Semiring R] {S : Type u_1} [DistribSMul S R] :DistribSMul S (Polynomial R)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L272-L274"},"Polynomial.aeval_one":{"signature":"theorem Polynomial.aeval_one {R : Type u} {A : Type z} [CommSemiring R] [Semiring A] [Algebra R A] (x : A) :(aeval x) 1 = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L293-L294"},"Polynomial.coeff_X":{"signature":"theorem Polynomial.coeff_X {R : Type u} {n : \u2115} [Semiring R] :X.coeff n = if 1 = n then 1 else 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L617-L619"},"Polynomial.coeff_neg":{"signature":"theorem Polynomial.coeff_neg {R : Type u} [Ring R] (p : Polynomial R) (n : \u2115) :(-p).coeff n = -p.coeff n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L1096-L1099"},"Polynomial.sum_smul_index":{"signature":"theorem Polynomial.sum_smul_index {R : Type u} [Semiring R] {S : Type u_1} [AddCommMonoid S] (p : Polynomial R) (b : R) (f : \u2115 \u2192 R \u2192 S) (hf : \u2200 (i : \u2115), f i 0 = 0) :(b \u2022 p).sum f = p.sum fun (n : \u2115) (a : R) => f n (b * a)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L877-L879"},"Polynomial.derivative":{"signature":"def Polynomial.derivative {R : Type u} [Semiring R] :Polynomial R \u2192\u2097[R] Polynomial R","doc":"derivative p is the formal derivative of the polynomial p","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L45-L53"},"Polynomial.Separable":{"signature":"def Polynomial.Separable {R : Type u} [CommSemiring R] (f : Polynomial R) :Prop","doc":"A polynomial is separable iff it is coprime with its derivative.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Separable.lean#L47-L50"},"Polynomial.Splits":{"signature":"def Polynomial.Splits {R : Type u_1} [Semiring R] (f : Polynomial R) :Prop","doc":"A polynomial Splits if it is a product of constant and monic linear polynomials.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L35-L36"},"Polynomial.monic_X_pow_sub_C":{"signature":"theorem Polynomial.monic_X_pow_sub_C {R : Type u} [Ring R] (a : R) {n : \u2115} (h : n \u2260 0) :(X ^ n - C a).Monic","doc":"X ^ n - a is monic.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Monic.lean#L440-L443"},"Polynomial.eval\u2082_X_pow":{"signature":"theorem Polynomial.eval\u2082_X_pow {R : Type u} {S : Type v} [Semiring R] [Semiring S] (f : R \u2192+* S) (x : S) {n : \u2115} :eval\u2082 f x (X ^ n) = x ^ n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L80-L84"},"Polynomial.algHom_ext":{"signature":"theorem Polynomial.algHom_ext {R : Type u} {B : Type u_2} [CommSemiring R] [Semiring B] [Algebra R B] {f g : Polynomial R \u2192\u2090[R] B} (hX : f X = g X) :f = g","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L254-L257"},"Polynomial.aeval_algHom":{"signature":"theorem Polynomial.aeval_algHom {R : Type u} {A : Type z} {B : Type u_2} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A \u2192\u2090[R] B) (x : A) :aeval (f x) = f.comp (aeval x)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L379-L380"},"Polynomial.leadingCoeff_mul":{"signature":"theorem Polynomial.leadingCoeff_mul {R : Type u} [Semiring R] [NoZeroDivisors R] (p q : Polynomial R) :(p * q).leadingCoeff = p.leadingCoeff * q.leadingCoeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L692-L699"},"Polynomial.monic_mul_leadingCoeff_inv":{"signature":"theorem Polynomial.monic_mul_leadingCoeff_inv {K : Type u_1} [DivisionRing K] {p : Polynomial K} (h : p \u2260 0) :(p * C p.leadingCoeff\u207b\u00b9).Monic","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L449-L451"},"Polynomial.derivative_apply":{"signature":"theorem Polynomial.derivative_apply {R : Type u} [Semiring R] (p : Polynomial R) :derivative p = p.sum fun (n : \u2115) (a : R) => C (a * \u2191n) * X ^ (n - 1)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L55-L56"},"Polynomial.derivative_monomial":{"signature":"theorem Polynomial.derivative_monomial {R : Type u} [Semiring R] (a : R) (n : \u2115) :derivative ((monomial n) a) = (monomial (n - 1)) (a * \u2191n)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L79-L82"},"Polynomial.derivative_mul":{"signature":"theorem Polynomial.derivative_mul {R : Type u} [Semiring R] {f g : Polynomial R} :derivative (f * g) = derivative f * g + f * derivative g","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L246-L261"},"Polynomial.map_C":{"signature":"theorem Polynomial.map_C {R : Type u} {S : Type v} {a : R} [Semiring R] [Semiring S] (f : R \u2192+* S) :map f (C a) = C (f a)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L485-L487"},"Polynomial.map_natCast":{"signature":"theorem Polynomial.map_natCast {R : Type u} {S : Type v} [Semiring R] [Semiring S] (f : R \u2192+* S) (n : \u2115) :map f \u2191n = \u2191n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L534-L536"},"Polynomial.map_pow":{"signature":"theorem Polynomial.map_pow {R : Type u} {S : Type v} [Semiring R] {p : Polynomial R} [Semiring S] (f : R \u2192+* S) (n : \u2115) :map f (p ^ n) = map f p ^ n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L561-L563"},"Polynomial.map_X":{"signature":"theorem Polynomial.map_X {R : Type u} {S : Type v} [Semiring R] [Semiring S] (f : R \u2192+* S) :map f X = X","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L489-L491"},"Polynomial.derivative_map":{"signature":"theorem Polynomial.derivative_map {R : Type u} {S : Type v} [Semiring R] [Semiring S] (p : Polynomial R) (f : R \u2192+* S) :derivative (map f p) = map f (derivative p)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L267-L276"},"Polynomial.degree_mul":{"signature":"theorem Polynomial.degree_mul {R : Type u} [Semiring R] [NoZeroDivisors R] {p q : Polynomial R} :(p * q).degree = p.degree + q.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L673-L679"},"Polynomial.degree_mul_C":{"signature":"theorem Polynomial.degree_mul_C {R : Type u} [Semiring R] {p : Polynomial R} {a : R} [NoZeroDivisors R] (a0 : a \u2260 0) :(p * C a).degree = p.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L371-L372"},"Polynomial.degree_mul_leadingCoeff_inv":{"signature":"theorem Polynomial.degree_mul_leadingCoeff_inv {K : Type u_1} [DivisionRing K] (p : Polynomial K) {q : Polynomial K} (h : q \u2260 0) :(p * C q.leadingCoeff\u207b\u00b9).degree = p.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L458-L461"},"Polynomial.aeval_modByMonic_eq_self_of_root":{"signature":"theorem Polynomial.aeval_modByMonic_eq_self_of_root {R : Type u} {S : Type v} [CommRing R] [Ring S] [Algebra R S] {p q : Polynomial R} (hq : q.Monic) {x : S} (hx : (aeval x) q = 0) :(aeval x) (p %\u2098 q) = (aeval x) p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/RingDivision.lean#L79-L83"},"Polynomial.aeval_zero":{"signature":"theorem Polynomial.aeval_zero {R : Type u} {A : Type z} [CommSemiring R] [Semiring A] [Algebra R A] (x : A) :(aeval x) 0 = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L272-L273"},"Polynomial.aeval_map_algebraMap":{"signature":"theorem Polynomial.aeval_map_algebraMap {R : Type u_1} (A : Type u_2) {B : Type u_3} [CommSemiring R] [CommSemiring A] [Semiring B] [Algebra R A] [Algebra A B] [Algebra R B] [IsScalarTower R A B] (x : B) (p : Polynomial R) :(aeval x) (map (algebraMap R A) p) = (aeval x) p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Tower.lean#L38-L40"},"Polynomial.coeffs":{"signature":"def Polynomial.coeffs {R : Type u} [Semiring R] (p : Polynomial R) :Finset R","doc":"The finset of nonzero coefficients of a polynomial.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L1013-L1016"},"Polynomial.coeff_mem_coeffs":{"signature":"theorem Polynomial.coeff_mem_coeffs {R : Type u} [Semiring R] {p : Polynomial R} {n : \u2115} (h : p.coeff n \u2260 0) :p.coeff n \u2208 p.coeffs","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L1030-L1033"},"Polynomial.toSubring":{"signature":"def Polynomial.toSubring {R : Type u} [Ring R] (p : Polynomial R) (T : Subring R) (hp : \u2191p.coeffs \u2286 \u2191T) :Polynomial \u21a5T","doc":"Given a polynomial p and a subring T that contains the coefficients of p, return the corresponding polynomial whose coefficients are in T.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Basic.lean#L362-L369"},"Polynomial.coeff_toSubring":{"signature":"theorem Polynomial.coeff_toSubring {R : Type u} [Ring R] (p : Polynomial R) (T : Subring R) (hp : \u2191p.coeffs \u2286 \u2191T) {n : \u2115} :\u2191((p.toSubring T hp).coeff n) = p.coeff n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Basic.lean#L373-L381"},"Polynomial.map_toSubring":{"signature":"theorem Polynomial.map_toSubring {R : Type u} [Ring R] (p : Polynomial R) (T : Subring R) (hp : \u2191p.coeffs \u2286 \u2191T) :map T.subtype (p.toSubring T hp) = p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Basic.lean#L418-L421"},"Polynomial.support_toSubring":{"signature":"theorem Polynomial.support_toSubring {R : Type u} [Ring R] (p : Polynomial R) (T : Subring R) (hp : \u2191p.coeffs \u2286 \u2191T) :(p.toSubring T hp).support = p.support","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Basic.lean#L386-L391"},"Polynomial.degree_toSubring":{"signature":"theorem Polynomial.degree_toSubring {R : Type u} [Ring R] (p : Polynomial R) (T : Subring R) (hp : \u2191p.coeffs \u2286 \u2191T) :(p.toSubring T hp).degree = p.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Basic.lean#L393-L394"},"Polynomial.natDegree_toSubring":{"signature":"theorem Polynomial.natDegree_toSubring {R : Type u} [Ring R] (p : Polynomial R) (T : Subring R) (hp : \u2191p.coeffs \u2286 \u2191T) :(p.toSubring T hp).natDegree = p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Basic.lean#L396-L397"},"Polynomial.monic_toSubring":{"signature":"theorem Polynomial.monic_toSubring {R : Type u} [Ring R] (p : Polynomial R) (T : Subring R) (hp : \u2191p.coeffs \u2286 \u2191T) :(p.toSubring T hp).Monic \u2194 p.Monic","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Basic.lean#L399-L402"},"Polynomial.derivative_zero":{"signature":"theorem Polynomial.derivative_zero {R : Type u} [Semiring R] :derivative 0 = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L72-L74"},"Polynomial.not_separable_zero":{"signature":"theorem Polynomial.not_separable_zero {R : Type u} [CommSemiring R] [Nontrivial R] :\u00acSeparable 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Separable.lean#L58-L60"},"Polynomial.div":{"signature":"def Polynomial.div {R : Type u} [Field R] (p q : Polynomial R) :Polynomial R","doc":"Division of polynomials. See Polynomial.divByMonic for more details.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L308-L310"},"Polynomial.instDiv":{"signature":"instance Polynomial.instDiv {R : Type u} [Field R] :Div (Polynomial R)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L328-L329"},"Polynomial.monic_zero_iff_subsingleton":{"signature":"theorem Polynomial.monic_zero_iff_subsingleton {R : Type u} [Semiring R] :Monic 0 \u2194 Subsingleton R","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Monic.lean#L37-L38"},"Polynomial.divByMonic_zero":{"signature":"theorem Polynomial.divByMonic_zero {R : Type u} [Ring R] (p : Polynomial R) :p /\u2098 0 = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L196-L202"},"Polynomial.mod":{"signature":"def Polynomial.mod {R : Type u} [Field R] (p q : Polynomial R) :Polynomial R","doc":"Remainder of polynomial division. See Polynomial.modByMonic for more details.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L312-L314"},"Polynomial.instMod":{"signature":"instance Polynomial.instMod {R : Type u} [Field R] :Mod (Polynomial R)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L331-L332"},"Polynomial.modByMonic_zero":{"signature":"theorem Polynomial.modByMonic_zero {R : Type u} [Ring R] (p : Polynomial R) :p %\u2098 0 = p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L188-L194"},"Polynomial.degree_le_mul_left":{"signature":"theorem Polynomial.degree_le_mul_left {R : Type u} [Semiring R] [NoZeroDivisors R] {q : Polynomial R} (p : Polynomial R) (hq : q \u2260 0) :p.degree \u2264 (p * q).degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L720-L725"},"Polynomial.instEuclideanDomain":{"signature":"instance Polynomial.instEuclideanDomain {R : Type u} [Field R] :EuclideanDomain (Polynomial R)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L355-L365"},"Polynomial.mod_def":{"signature":"theorem Polynomial.mod_def {R : Type u} [Field R] {p q : Polynomial R} :p % q = p %\u2098 (q * C q.leadingCoeff\u207b\u00b9)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L337-L337"},"Polynomial.degree_map_eq_iff":{"signature":"theorem Polynomial.degree_map_eq_iff {R : Type u} {S : Type v} [Semiring R] [Semiring S] {f : R \u2192+* S} {p : Polynomial R} :(map f p).degree = p.degree \u2194 f p.leadingCoeff \u2260 0 \u2228 p = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L258-L268"},"Polynomial.degree_map_eq_of_injective":{"signature":"theorem Polynomial.degree_map_eq_of_injective {R : Type u} {S : Type v} [Semiring R] [Semiring S] {f : R \u2192+* S} (hf : Function.Injective \u21d1f) (p : Polynomial R) :(map f p).degree = p.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L293-L295"},"Polynomial.natDegree_map_eq_of_injective":{"signature":"theorem Polynomial.natDegree_map_eq_of_injective {R : Type u} {S : Type v} [Semiring R] [Semiring S] {f : R \u2192+* S} (hf : Function.Injective \u21d1f) (p : Polynomial R) :(map f p).natDegree = p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L297-L299"},"Polynomial.leadingCoeff_map_of_injective":{"signature":"theorem Polynomial.leadingCoeff_map_of_injective {R : Type u} {S : Type v} [Semiring R] [Semiring S] {f : R \u2192+* S} (hf : Function.Injective \u21d1f) (p : Polynomial R) :(map f p).leadingCoeff = f p.leadingCoeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L301-L303"},"Polynomial.leadingCoeff_map":{"signature":"theorem Polynomial.leadingCoeff_map {R : Type u} {S : Type v} [Ring R] [IsSimpleRing R] [Semiring S] [Nontrivial S] {p : Polynomial R} (f : R \u2192+* S) :(map f p).leadingCoeff = f p.leadingCoeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L282-L284"},"Polynomial.div_modByMonic_unique":{"signature":"theorem Polynomial.div_modByMonic_unique {R : Type u} [Ring R] {f g : Polynomial R} (q r : Polynomial R) (hg : g.Monic) (h : r + g * q = f \u2227 r.degree < g.degree) :f /\u2098 g = q \u2227 f %\u2098 g = r","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L338-L360"},"Polynomial.map_mod_divByMonic":{"signature":"theorem Polynomial.map_mod_divByMonic {R : Type u} {S : Type v} [Ring R] {p q : Polynomial R} [Ring S] (f : R \u2192+* S) (hq : q.Monic) :map f (p /\u2098 q) = map f p /\u2098 map f q \u2227 map f (p %\u2098 q) = map f p %\u2098 map f q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L362-L376"},"Polynomial.map_modByMonic":{"signature":"theorem Polynomial.map_modByMonic {R : Type u} {S : Type v} [Ring R] {p q : Polynomial R} [Ring S] (f : R \u2192+* S) (hq : q.Monic) :map f (p %\u2098 q) = map f p %\u2098 map f q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L382-L384"},"Polynomial.map_mod":{"signature":"theorem Polynomial.map_mod {R : Type u} {k : Type y} [Field R] {p q : Polynomial R} [Field k] (f : R \u2192+* k) :map f (p % q) = map f p % map f q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L416-L420"},"Polynomial.gcd_map":{"signature":"theorem Polynomial.gcd_map {R : Type u} {k : Type y} [Field R] {p q : Polynomial R} [Field k] [DecidableEq R] [DecidableEq k] (f : R \u2192+* k) :EuclideanDomain.gcd (map f p) (map f q) = map f (EuclideanDomain.gcd p q)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L456-L459"},"Polynomial.natDegree_of_subsingleton":{"signature":"theorem Polynomial.natDegree_of_subsingleton {R : Type u} [Semiring R] {p : Polynomial R} [Subsingleton R] :p.natDegree = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L56-L58"},"Polynomial.natDegree_mul":{"signature":"theorem Polynomial.natDegree_mul {R : Type u} [Semiring R] [NoZeroDivisors R] {p q : Polynomial R} (hp : p \u2260 0) (hq : q \u2260 0) :(p * q).natDegree = p.natDegree + q.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Domain.lean#L37-L39"},"Polynomial.natDegree_eq_zero_of_isUnit":{"signature":"theorem Polynomial.natDegree_eq_zero_of_isUnit {R : Type u} [Semiring R] [NoZeroDivisors R] {p : Polynomial R} (h : IsUnit p) :p.natDegree = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Units.lean#L31-L36"},"Polynomial.zero_le_degree_iff":{"signature":"theorem Polynomial.zero_le_degree_iff {R : Type u} [Semiring R] {p : Polynomial R} :0 \u2264 p.degree \u2194 p \u2260 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/SmallDegree.lean#L59-L60"},"Polynomial.degree_eq_zero_of_isUnit":{"signature":"theorem Polynomial.degree_eq_zero_of_isUnit {R : Type u} [Semiring R] [NoZeroDivisors R] {p : Polynomial R} [Nontrivial R] (h : IsUnit p) :p.degree = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Units.lean#L38-L40"},"Polynomial.C_mul":{"signature":"theorem Polynomial.C_mul {R : Type u} {a b : R} [Semiring R] :C (a * b) = C a * C b","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L456-L457"},"Polynomial.isUnit_iff_degree_eq_zero":{"signature":"theorem Polynomial.isUnit_iff_degree_eq_zero {R : Type u} [Field R] {p : Polynomial R} :IsUnit p \u2194 p.degree = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L298-L306"},"Polynomial.degree_map":{"signature":"theorem Polynomial.degree_map {R : Type u} {S : Type v} [Ring R] [IsSimpleRing R] [Semiring S] [Nontrivial S] (p : Polynomial R) (f : R \u2192+* S) :(map f p).degree = p.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L274-L276"},"Polynomial.isUnit_map":{"signature":"theorem Polynomial.isUnit_map {R : Type u} {k : Type y} [Field R] {p : Polynomial R} [Field k] (f : R \u2192+* k) :IsUnit (map f p) \u2194 IsUnit p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L407-L408"},"Polynomial.isCoprime_map":{"signature":"theorem Polynomial.isCoprime_map {R : Type u} {k : Type y} [Field R] {p q : Polynomial R} [Field k] (f : R \u2192+* k) :IsCoprime (map f p) (map f q) \u2194 IsCoprime p q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L492-L494"},"Polynomial.separable_def":{"signature":"theorem Polynomial.separable_def {R : Type u} [CommSemiring R] (f : Polynomial R) :f.Separable \u2194 IsCoprime f (derivative f)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Separable.lean#L52-L53"},"Polynomial.separable_map":{"signature":"theorem Polynomial.separable_map {F : Type u} [Field F] {S : Type u_1} [CommRing S] [Nontrivial S] (f : F \u2192+* S) {p : Polynomial F} :(map f p).Separable \u2194 p.Separable","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Separable.lean#L326-L331"},"Polynomial.factor":{"signature":"def Polynomial.factor {K : Type v} [Field K] (f : Polynomial K) :Polynomial K","doc":"Non-computably choose an irreducible factor from a polynomial.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/SplittingField/Construction.lean#L51-L53"},"Polynomial.isScalarTower_right":{"signature":"instance Polynomial.isScalarTower_right {\u03b1 : Type u_1} {K : Type u_2} [Semiring K] [DistribSMul \u03b1 K] [IsScalarTower \u03b1 K K] :IsScalarTower \u03b1 (Polynomial K) (Polynomial K)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L300-L304"},"Polynomial.smul_C":{"signature":"theorem Polynomial.smul_C {R : Type u} [Semiring R] {S : Type u_1} [SMulZeroClass S R] (s : S) (r : R) :s \u2022 C r = C (s \u2022 r)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L462-L464"},"Polynomial.nnqsmul_eq_C_mul":{"signature":"theorem Polynomial.nnqsmul_eq_C_mul {R : Type u} [DivisionSemiring R] (q : \u211a\u22650) (f : Polynomial R) :q \u2022 f = C \u2191q * f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L1173-L1174"},"Polynomial.qsmul_eq_C_mul":{"signature":"theorem Polynomial.qsmul_eq_C_mul {R : Type u} [DivisionRing R] (a : \u211a) (f : Polynomial R) :a \u2022 f = C \u2191a * f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L1182-L1183"},"Polynomial.prime_X":{"signature":"theorem Polynomial.prime_X {R : Type u} [CommRing R] [IsDomain R] :Prime X","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/RingDivision.lean#L208-L210"},"Polynomial.irreducible_X":{"signature":"theorem Polynomial.irreducible_X {R : Type u} [CommRing R] [IsDomain R] :Irreducible X","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/RingDivision.lean#L219-L220"},"Polynomial.irreducible_factor":{"signature":"theorem Polynomial.irreducible_factor {K : Type v} [Field K] (f : Polynomial K) :Irreducible f.factor","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/SplittingField/Construction.lean#L55-L59"},"Polynomial.fact_irreducible_factor":{"signature":"theorem Polynomial.fact_irreducible_factor {K : Type v} [Field K] (f : Polynomial K) :Fact (Irreducible f.factor)","doc":"See note [fact non-instances].","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/SplittingField/Construction.lean#L61-L63"},"Polynomial.removeFactor":{"signature":"def Polynomial.removeFactor {K : Type v} [Field K] (f : Polynomial K) :Polynomial (AdjoinRoot f.factor)","doc":"Divide a polynomial f by X - C r where r is a root of f in a bigger field extension.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/SplittingField/Construction.lean#L89-L91"},"Polynomial.SplittingFieldAuxAux":{"signature":"def Polynomial.SplittingFieldAuxAux (n : \u2115) {K : Type u} [Field K] :Polynomial K \u2192 (L : Type u) \u00d7 (x : Field L) \u00d7 Algebra K L","doc":"Auxiliary construction to a splitting field of a polynomial, which removes n (arbitrarily-chosen) factors. It constructs the type, proves that is a field and algebra over the base field. Uses recursion on the degree.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/SplittingField/Construction.lean#L106-L121"},"Polynomial.SplittingFieldAux":{"signature":"def Polynomial.SplittingFieldAux (n : \u2115) {K : Type u} [Field K] (f : Polynomial K) :Type u","doc":"Auxiliary construction to a splitting field of a polynomial, which removes n (arbitrarily-chosen) factors. It is the type constructed in SplittingFieldAuxAux.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/SplittingField/Construction.lean#L123-L127"},"Polynomial.SplittingField":{"signature":"def Polynomial.SplittingField {K : Type v} [Field K] (f : Polynomial K) :Type v","doc":"A splitting field of a polynomial.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/SplittingField/Construction.lean#L213-L218"},"Polynomial.instCommRingSplittingField":{"signature":"instance Polynomial.instCommRingSplittingField {K : Type u_1} [Field K] (f : Polynomial K) :CommRing f.SplittingField","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/SplittingField/Construction.lean#L218-L218"},"Polynomial.natDegree_map_le":{"signature":"theorem Polynomial.natDegree_map_le {R : Type u} {S : Type v} [Semiring R] [Semiring S] {f : R \u2192+* S} {p : Polynomial R} :(map f p).natDegree \u2264 p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Degree.lean#L151-L151"},"Polynomial.splits_iff_exists_multiset'":{"signature":"theorem Polynomial.splits_iff_exists_multiset' {R : Type u_1} [CommSemiring R] {f : Polynomial R} :f.Splits \u2194 \u2203 (m : Multiset R), f = C f.leadingCoeff * (Multiset.map (fun (x : R) => X + C x) m).prod","doc":"See splits_iff_exists_multiset for the version with subtraction.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L142-L159"},"Polynomial.splits_iff_exists_multiset":{"signature":"theorem Polynomial.splits_iff_exists_multiset {R : Type u_1} [CommRing R] {f : Polynomial R} :f.Splits \u2194 \u2203 (m : Multiset R), f = C f.leadingCoeff * (Multiset.map (fun (x : R) => X - C x) m).prod","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L255-L258"},"Polynomial.roots_mul":{"signature":"theorem Polynomial.roots_mul {R : Type u} [CommRing R] [IsDomain R] {p q : Polynomial R} (hpq : p * q \u2260 0) :(p * q).roots = p.roots + q.roots","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L160-L163"},"Polynomial.not_isRoot_C":{"signature":"theorem Polynomial.not_isRoot_C {R : Type u} [Semiring R] (r a : R) (hr : r \u2260 0) :\u00ac(C r).IsRoot a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L360-L360"},"Polynomial.roots_C":{"signature":"theorem Polynomial.roots_C {R : Type u} [CommRing R] [IsDomain R] (x : R) :(C x).roots = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L188-L194"},"Polynomial.roots_C_mul":{"signature":"theorem Polynomial.roots_C_mul {R : Type u} {a : R} [CommRing R] [IsDomain R] (p : Polynomial R) (ha : a \u2260 0) :(C a * p).roots = p.roots","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L200-L204"},"Polynomial.roots_one":{"signature":"theorem Polynomial.roots_one {R : Type u} [CommRing R] [IsDomain R] :roots 1 = \u2205","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L196-L198"},"Polynomial.roots_list_prod":{"signature":"theorem Polynomial.roots_list_prod {R : Type u} [CommRing R] [IsDomain R] (L : List (Polynomial R)) :0 \u2209 L \u2192 L.prod.roots = (\u2191L).bind roots","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L231-L236"},"Polynomial.roots_multiset_prod":{"signature":"theorem Polynomial.roots_multiset_prod {R : Type u} [CommRing R] [IsDomain R] (m : Multiset (Polynomial R)) :0 \u2209 m \u2192 m.prod.roots = m.bind roots","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L238-L240"},"Polynomial.roots_X_sub_C":{"signature":"theorem Polynomial.roots_X_sub_C {R : Type u} [CommRing R] [IsDomain R] (r : R) :(X - C r).roots = {r}","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L176-L180"},"Polynomial.roots_multiset_prod_X_sub_C":{"signature":"theorem Polynomial.roots_multiset_prod_X_sub_C {R : Type u} [CommRing R] [IsDomain R] (s : Multiset R) :(Multiset.map (fun (a : R) => X - C a) s).prod.roots = s","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L273-L280"},"Polynomial.degree_C_mul":{"signature":"theorem Polynomial.degree_C_mul {R : Type u} [Semiring R] {p : Polynomial R} {a : R} [NoZeroDivisors R] (a0 : a \u2260 0) :(C a * p).degree = p.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L374-L375"},"Polynomial.natDegree_C_mul":{"signature":"theorem Polynomial.natDegree_C_mul {R : Type u} [Semiring R] {p : Polynomial R} {a : R} [NoZeroDivisors R] (a0 : a \u2260 0) :(C a * p).natDegree = p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L380-L381"},"Polynomial.natDegree_multiset_prod_of_monic":{"signature":"theorem Polynomial.natDegree_multiset_prod_of_monic {R : Type u} [CommSemiring R] (t : Multiset (Polynomial R)) (h : \u2200 f \u2208 t, f.Monic) :t.prod.natDegree = (Multiset.map natDegree t).sum","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/BigOperators.lean#L201-L205"},"Polynomial.natDegree_multiset_prod_X_sub_C_eq_card":{"signature":"theorem Polynomial.natDegree_multiset_prod_X_sub_C_eq_card {R : Type u} [CommRing R] [Nontrivial R] (s : Multiset R) :(Multiset.map (fun (x : R) => X - C x) s).prod.natDegree = s.card","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/BigOperators.lean#L286-L292"},"Polynomial.eq_mul_leadingCoeff_of_monic_of_dvd_of_natDegree_le":{"signature":"theorem Polynomial.eq_mul_leadingCoeff_of_monic_of_dvd_of_natDegree_le {R : Type u} [Semiring R] {p q : Polynomial R} (hp : p.Monic) (hdvd : p \u2223 q) (hdeg : q.natDegree \u2264 p.natDegree) :q = p * C q.leadingCoeff","doc":"See Polynomial.eq_leadingCoeff_mul_of_monic_of_dvd_of_natDegree_le for the other multiplication order. That version, unlike this one, requires commutativity.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L83-L93"},"Polynomial.eq_leadingCoeff_mul_of_monic_of_dvd_of_natDegree_le":{"signature":"theorem Polynomial.eq_leadingCoeff_mul_of_monic_of_dvd_of_natDegree_le {R : Type u_1} [CommSemiring R] {p q : Polynomial R} (hp : p.Monic) (hdvd : p \u2223 q) (hdeg : q.natDegree \u2264 p.natDegree) :q = C q.leadingCoeff * p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L837-L840"},"Polynomial.monic_multisetProd_X_sub_C":{"signature":"theorem Polynomial.monic_multisetProd_X_sub_C {R : Type u} [CommRing R] (s : Multiset R) :(Multiset.map (fun (a : R) => X - C a) s).prod.Monic","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L749-L750"},"Polynomial.map_injective":{"signature":"theorem Polynomial.map_injective {R : Type u} {S : Type v} [Semiring R] [Semiring S] (f : R \u2192+* S) (hf : Function.Injective \u21d1f) :Function.Injective (map f)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Coeff.lean#L103-L104"},"Polynomial.map_dvd_map":{"signature":"theorem Polynomial.map_dvd_map {R : Type u} {S : Type v} [Ring R] [Ring S] (f : R \u2192+* S) (hf : Function.Injective \u21d1f) {x y : Polynomial R} (hx : x.Monic) :map f x \u2223 map f y \u2194 x \u2223 y","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L410-L416"},"Polynomial.prod_multiset_root_eq_finset_root":{"signature":"theorem Polynomial.prod_multiset_root_eq_finset_root {R : Type u} [CommRing R] [IsDomain R] {p : Polynomial R} [DecidableEq R] :(Multiset.map (fun (a : R) => X - C a) p.roots).prod = \u220f a \u2208 p.roots.toFinset, (X - C a) ^ rootMultiplicity a p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L761-L764"},"Polynomial.map_prod":{"signature":"theorem Polynomial.map_prod {R : Type u} {S : Type v} [CommSemiring R] [CommSemiring S] (f : R \u2192+* S) {\u03b9 : Type u_1} (g : \u03b9 \u2192 Polynomial R) (s : Finset \u03b9) :map f (\u220f i \u2208 s, g i) = \u220f i \u2208 s, map f (g i)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L711-L713"},"Polynomial.map_sub":{"signature":"theorem Polynomial.map_sub {R : Type u} [Ring R] {p q : Polynomial R} {S : Type u_1} [Ring S] (f : R \u2192+* S) :map f (p - q) = map f p - map f q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L723-L725"},"Polynomial.C_sub":{"signature":"theorem Polynomial.C_sub {R : Type u} {a b : R} [Ring R] :C (a - b) = C a - C b","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L1124-L1125"},"Polynomial.isCoprime_X_sub_C_of_isUnit_sub":{"signature":"theorem Polynomial.isCoprime_X_sub_C_of_isUnit_sub {R : Type u_1} [CommRing R] {a b : R} (h : IsUnit (a - b)) :IsCoprime (X - C a) (X - C b)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/RingDivision.lean#L262-L268"},"Polynomial.pairwise_coprime_X_sub_C":{"signature":"theorem Polynomial.pairwise_coprime_X_sub_C {K : Type u_1} [Field K] {I : Type v} {s : I \u2192 K} (H : Function.Injective s) :Pairwise (Function.onFun IsCoprime fun (i : I) => X - C (s i))","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/RingDivision.lean#L271-L273"},"Polynomial.map_dvd":{"signature":"theorem Polynomial.map_dvd {R : Type u} {S : Type v} [Semiring R] [Semiring S] (f : R \u2192+* S) {x y : Polynomial R} :x \u2223 y \u2192 map f x \u2223 map f y","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L544-L545"},"Polynomial.prod_multiset_X_sub_C_dvd":{"signature":"theorem Polynomial.prod_multiset_X_sub_C_dvd {R : Type u} [CommRing R] [IsDomain R] (p : Polynomial R) :(Multiset.map (fun (a : R) => X - C a) p.roots).prod \u2223 p","doc":"The product \u220f (X - a) for a inside the multiset p.roots divides p.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L766-L775"},"Polynomial.C_leadingCoeff_mul_prod_multiset_X_sub_C":{"signature":"theorem Polynomial.C_leadingCoeff_mul_prod_multiset_X_sub_C {R : Type u} [CommRing R] [IsDomain R] {p : Polynomial R} (hroots : p.roots.card = p.natDegree) :C p.leadingCoeff * (Multiset.map (fun (a : R) => X - C a) p.roots).prod = p","doc":"A polynomial p that has as many roots as its degree can be written p = p.leadingCoeff * \u220f(X - a), for a in p.roots.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L807-L813"},"Polynomial.splits_iff_card_roots":{"signature":"theorem Polynomial.splits_iff_card_roots {R : Type u_1} [CommRing R] {f : Polynomial R} [IsDomain R] :f.Splits \u2194 f.roots.card = f.natDegree","doc":"A polynomial splits if and only if it has as many roots as its degree.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L369-L372"},"Polynomial.natDegree_eq_zero":{"signature":"theorem Polynomial.natDegree_eq_zero {R : Type u} [Semiring R] {p : Polynomial R} :p.natDegree = 0 \u2194 \u2203 (x : R), C x = p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L485-L486"},"Polynomial.splits_of_natDegree_eq_zero":{"signature":"theorem Polynomial.splits_of_natDegree_eq_zero {R : Type u_1} [Semiring R] {f : Polynomial R} (hf : f.natDegree = 0) :f.Splits","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L88-L90"},"Polynomial.eval_zero":{"signature":"theorem Polynomial.eval_zero {R : Type u} [Semiring R] {x : R} :eval x 0 = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L295-L297"},"Polynomial.degree_ne_of_natDegree_ne":{"signature":"theorem Polynomial.degree_ne_of_natDegree_ne {R : Type u} [Semiring R] {p : Polynomial R} {n : \u2115} :p.natDegree \u2260 n \u2192 p.degree \u2260 \u2191n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L114-L115"},"Polynomial.natDegree_X_pow":{"signature":"theorem Polynomial.natDegree_X_pow {R : Type u} [Semiring R] [Nontrivial R] (n : \u2115) :(X ^ n).natDegree = n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L519-L521"},"Polynomial.natDegree_X_pow_add_C":{"signature":"theorem Polynomial.natDegree_X_pow_add_C {R : Type u} [Nontrivial R] [Semiring R] {n : \u2115} {r : R} :(X ^ n + C r).natDegree = n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L634-L635"},"Polynomial.coeff_X_pow_self":{"signature":"theorem Polynomial.coeff_X_pow_self {R : Type u} [Semiring R] (n : \u2115) :(X ^ n).coeff n = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L189-L189"},"Polynomial.leadingCoeff_X_pow_add_C":{"signature":"theorem Polynomial.leadingCoeff_X_pow_add_C {R : Type u} [Semiring R] {n : \u2115} (hn : 0 < n) {r : R} :(X ^ n + C r).leadingCoeff = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L651-L656"},"Polynomial.leadingCoeff_X_add_C":{"signature":"theorem Polynomial.leadingCoeff_X_add_C {S : Type v} [Semiring S] (r : S) :(X + C r).leadingCoeff = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L658-L660"},"Polynomial.leadingCoeff_X_sub_C":{"signature":"theorem Polynomial.leadingCoeff_X_sub_C {S : Type v} [Ring S] (r : S) :(X - C r).leadingCoeff = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L785-L787"},"Polynomial.splits_X_sub_C_mul_iff":{"signature":"theorem Polynomial.splits_X_sub_C_mul_iff {R : Type u_1} [CommRing R] {f : Polynomial R} [IsDomain R] {a : R} :((X - C a) * f).Splits \u2194 f.Splits","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L450-L459"},"Polynomial.splits_mul_iff":{"signature":"theorem Polynomial.splits_mul_iff {R : Type u_1} [CommRing R] {f g : Polynomial R} [IsDomain R] (hf\u2080 : f \u2260 0) (hg\u2080 : g \u2260 0) :(f * g).Splits \u2194 f.Splits \u2227 g.Splits","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L461-L481"},"Polynomial.splits_prod_iff":{"signature":"theorem Polynomial.splits_prod_iff {R : Type u_1} [CommRing R] [IsDomain R] {\u03b9 : Type u_4} {f : \u03b9 \u2192 Polynomial R} {s : Finset \u03b9} (hf : \u2200 i \u2208 s, f i \u2260 0) :(\u220f x \u2208 s, f x).Splits \u2194 \u2200 x \u2208 s, (f x).Splits","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L490-L493"},"Polynomial.map_eq_zero_iff":{"signature":"theorem Polynomial.map_eq_zero_iff {R : Type u} {S : Type v} [Semiring R] {p : Polynomial R} [Semiring S] {f : R \u2192+* S} (hf : Function.Injective \u21d1f) :map f p = 0 \u2194 p = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Coeff.lean#L121-L122"},"Polynomial.map_eq_zero":{"signature":"theorem Polynomial.map_eq_zero {R : Type u} {S : Type v} [Ring R] [IsSimpleRing R] [Semiring S] [Nontrivial S] {p : Polynomial R} (f : R \u2192+* S) :map f p = 0 \u2194 p = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L267-L269"},"Polynomial.map_ne_zero":{"signature":"theorem Polynomial.map_ne_zero {R : Type u} {S : Type v} [Ring R] [IsSimpleRing R] [Semiring S] [Nontrivial S] {p : Polynomial R} {f : R \u2192+* S} (hp : p \u2260 0) :map f p \u2260 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L271-L272"},"Polynomial.aroots":{"signature":"noncomputable abbrev Polynomial.aroots {T : Type w} [CommRing T] (p : Polynomial T) (S : Type u_1) [CommRing S] [IsDomain S] [Algebra T S] :Multiset S","doc":"Given a polynomial p with coefficients in a ring T and a T-algebra S, aroots p S is the multiset of roots of p regarded as a polynomial over S.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L437-L440"},"Polynomial.rootSet":{"signature":"def Polynomial.rootSet {T : Type w} [CommRing T] (p : Polynomial T) (S : Type u_1) [CommRing S] [IsDomain S] [Algebra T S] :Set S","doc":"The set of distinct roots of p in S. If you have a non-separable polynomial, use Polynomial.aroots for the multiset where multiple roots have the appropriate multiplicity.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L532-L538"},"Polynomial.IsSplittingField":{"signature":"class Polynomial.IsSplittingField (K : Type v) (L : Type w) [Field K] [Field L] [Algebra K L] (f : Polynomial K) :Prop","doc":"Typeclass characterising splitting fields.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/SplittingField/IsSplittingField.lean#L45-L49"},"Polynomial.map_multiset_prod":{"signature":"theorem Polynomial.map_multiset_prod {R : Type u} {S : Type v} [CommSemiring R] [CommSemiring S] (f : R \u2192+* S) (m : Multiset (Polynomial R)) :map f m.prod = (Multiset.map (map f) m).prod","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L708-L709"},"Polynomial.count_map_roots":{"signature":"theorem Polynomial.count_map_roots {A : Type u_1} {B : Type u_2} [CommRing A] [CommRing B] [IsDomain A] [DecidableEq B] {p : Polynomial A} {f : A \u2192+* B} (hmap : map f p \u2260 0) (b : B) :Multiset.count b (Multiset.map (\u21d1f) p.roots) \u2264 rootMultiplicity b (map f p)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L881-L891"},"Polynomial.map_roots_le":{"signature":"theorem Polynomial.map_roots_le {A : Type u_1} {B : Type u_2} [CommRing A] [CommRing B] [IsDomain A] [IsDomain B] {p : Polynomial A} {f : A \u2192+* B} (h : map f p \u2260 0) :Multiset.map (\u21d1f) p.roots \u2264 (map f p).roots","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L901-L906"},"Polynomial.map_ne_zero_iff":{"signature":"theorem Polynomial.map_ne_zero_iff {R : Type u} {S : Type v} [Semiring R] {p : Polynomial R} [Semiring S] {f : R \u2192+* S} (hf : Function.Injective \u21d1f) :map f p \u2260 0 \u2194 p \u2260 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Coeff.lean#L124-L125"},"Polynomial.map_roots_le_of_injective":{"signature":"theorem Polynomial.map_roots_le_of_injective {A : Type u_1} {B : Type u_2} [CommRing A] [CommRing B] [IsDomain A] [IsDomain B] (p : Polynomial A) {f : A \u2192+* B} (hf : Function.Injective \u21d1f) :Multiset.map (\u21d1f) p.roots \u2264 (map f p).roots","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L908-L912"},"Polynomial.card_roots'":{"signature":"theorem Polynomial.card_roots' {R : Type u} [CommRing R] [IsDomain R] (p : Polynomial R) :p.roots.card \u2264 p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L79-L82"},"Polynomial.card_roots_map_le_natDegree":{"signature":"theorem Polynomial.card_roots_map_le_natDegree {A : Type u_3} {B : Type u_4} [Semiring A] [CommRing B] [IsDomain B] {f : A \u2192+* B} (p : Polynomial A) :(map f p).roots.card \u2264 p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L920-L922"},"Polynomial.roots_map_of_injective_of_card_eq_natDegree":{"signature":"theorem Polynomial.roots_map_of_injective_of_card_eq_natDegree {A : Type u_1} {B : Type u_2} [CommRing A] [CommRing B] [IsDomain A] [IsDomain B] {p : Polynomial A} {f : A \u2192+* B} (hf : Function.Injective \u21d1f) (hroots : p.roots.card = p.natDegree) :Multiset.map (\u21d1f) p.roots = (map f p).roots","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L953-L957"},"Polynomial.coeff_C_succ":{"signature":"theorem Polynomial.coeff_C_succ {R : Type u} [Semiring R] {r : R} {n : \u2115} :(C r).coeff (n + 1) = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L642-L643"},"Polynomial.eq_X_add_C_of_degree_le_one":{"signature":"theorem Polynomial.eq_X_add_C_of_degree_le_one {R : Type u} [Semiring R] {p : Polynomial R} (h : p.degree \u2264 1) :p = C (p.coeff 1) * X + C (p.coeff 0)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/SmallDegree.lean#L31-L36"},"Polynomial.eq_X_add_C_of_natDegree_le_one":{"signature":"theorem Polynomial.eq_X_add_C_of_natDegree_le_one {R : Type u} [Semiring R] {p : Polynomial R} (h : p.natDegree \u2264 1) :p = C (p.coeff 1) * X + C (p.coeff 0)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/SmallDegree.lean#L43-L45"},"Polynomial.exists_eq_X_add_C_of_natDegree_le_one":{"signature":"theorem Polynomial.exists_eq_X_add_C_of_natDegree_le_one {R : Type u} [Semiring R] {p : Polynomial R} (h : p.natDegree \u2264 1) :\u2203 (a : R) (b : R), p = C a * X + C b","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/SmallDegree.lean#L50-L51"},"Polynomial.instIsDomainOfIsCancelAdd":{"signature":"instance Polynomial.instIsDomainOfIsCancelAdd {R : Type u} [Semiring R] [IsCancelAdd R] [IsDomain R] :IsDomain (Polynomial R)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L1159-L1159"},"Polynomial.factor_dvd_of_not_isUnit":{"signature":"theorem Polynomial.factor_dvd_of_not_isUnit {K : Type v} [Field K] {f : Polynomial K} (hf1 : \u00acIsUnit f) :f.factor \u2223 f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/SplittingField/Construction.lean#L67-L70"},"Polynomial.factor_dvd_of_degree_ne_zero":{"signature":"theorem Polynomial.factor_dvd_of_degree_ne_zero {K : Type v} [Field K] {f : Polynomial K} (hf : f.degree \u2260 0) :f.factor \u2223 f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/SplittingField/Construction.lean#L72-L73"},"Polynomial.factor_dvd_of_natDegree_ne_zero":{"signature":"theorem Polynomial.factor_dvd_of_natDegree_ne_zero {K : Type v} [Field K] {f : Polynomial K} (hf : f.natDegree \u2260 0) :f.factor \u2223 f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/SplittingField/Construction.lean#L75-L76"},"Polynomial.map_id":{"signature":"theorem Polynomial.map_id {R : Type u} [Semiring R] {p : Polynomial R} :map (RingHom.id R) p = p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Coeff.lean#L89-L90"},"Polynomial.X_sub_C_mul_removeFactor":{"signature":"theorem Polynomial.X_sub_C_mul_removeFactor {K : Type v} [Field K] (f : Polynomial K) (hf : f.natDegree \u2260 0) :(X - C (AdjoinRoot.root f.factor)) * f.removeFactor = map (AdjoinRoot.of f.factor) f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/SplittingField/Construction.lean#L93-L98"},"Polynomial.zero_divByMonic":{"signature":"theorem Polynomial.zero_divByMonic {R : Type u} [Ring R] (p : Polynomial R) :0 /\u2098 p = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L184-L186"},"Polynomial.natDegree_divByMonic":{"signature":"theorem Polynomial.natDegree_divByMonic {R : Type u} [Ring R] (f : Polynomial R) {g : Polynomial R} (hg : g.Monic) :(f /\u2098 g).natDegree = f.natDegree - g.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L319-L336"},"Polynomial.natDegree_map":{"signature":"theorem Polynomial.natDegree_map {R : Type u} {S : Type v} [Ring R] [IsSimpleRing R] [Semiring S] [Nontrivial S] {p : Polynomial R} (f : R \u2192+* S) :(map f p).natDegree = p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L278-L280"},"Polynomial.natDegree_removeFactor":{"signature":"theorem Polynomial.natDegree_removeFactor {K : Type v} [Field K] (f : Polynomial K) :f.removeFactor.natDegree = f.natDegree - 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/SplittingField/Construction.lean#L100-L101"},"Polynomial.natDegree_removeFactor'":{"signature":"theorem Polynomial.natDegree_removeFactor' {K : Type v} [Field K] {f : Polynomial K} {n : \u2115} (hfn : f.natDegree = n + 1) :f.removeFactor.natDegree = n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/SplittingField/Construction.lean#L103-L104"},"Polynomial.rootSet_def":{"signature":"theorem Polynomial.rootSet_def {T : Type w} [CommRing T] (p : Polynomial T) (S : Type u_1) [CommRing S] [IsDomain S] [Algebra T S] [DecidableEq S] :p.rootSet S = \u2191(p.aroots S).toFinset","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L540-L543"},"Polynomial.aroots_def":{"signature":"theorem Polynomial.aroots_def {T : Type w} [CommRing T] (p : Polynomial T) (S : Type u_1) [CommRing S] [IsDomain S] [Algebra T S] :p.aroots S = (map (algebraMap T S) p).roots","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L442-L444"},"Polynomial.aeval_sub":{"signature":"theorem Polynomial.aeval_sub {R : Type u} {A : Type z} [CommRing R] {p q : Polynomial R} [Ring A] [Algebra R A] (x : A) :(aeval x) (p - q) = (aeval x) p - (aeval x) q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L644-L646"},"Polynomial.coe_eval\u2082RingHom":{"signature":"theorem Polynomial.coe_eval\u2082RingHom {R : Type u} {S : Type v} [Semiring R] [CommSemiring S] (f : R \u2192+* S) (x : S) :\u21d1(eval\u2082RingHom f x) = eval\u2082 f x","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L216-L218"},"Polynomial.natDegree_le_of_dvd":{"signature":"theorem Polynomial.natDegree_le_of_dvd {R : Type u} [Semiring R] [NoZeroDivisors R] {p q : Polynomial R} (h1 : p \u2223 q) (h2 : q \u2260 0) :p.natDegree \u2264 q.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Domain.lean#L61-L64"},"Polynomial.aeval_monomial":{"signature":"theorem Polynomial.aeval_monomial {R : Type u} {A : Type z} [CommSemiring R] [Semiring A] [Algebra R A] (x : A) {n : \u2115} {r : R} :(aeval x) ((monomial n) r) = (algebraMap R A) r * x ^ n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L283-L285"},"Polynomial.degree_sum_le":{"signature":"theorem Polynomial.degree_sum_le {R : Type u} [Semiring R] {\u03b9 : Type u_1} (s : Finset \u03b9) (f : \u03b9 \u2192 Polynomial R) :(\u2211 i \u2208 s, f i).degree \u2264 s.sup fun (b : \u03b9) => (f b).degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L389-L396"},"Polynomial.degree_sum_fin_lt":{"signature":"theorem Polynomial.degree_sum_fin_lt {R : Type u} [Semiring R] {n : \u2115} (f : Fin n \u2192 R) :(\u2211 i : Fin n, C (f i) * X ^ \u2191i).degree < \u2191n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L499-L503"},"Polynomial.exists_iff_exists_finsupp":{"signature":"theorem Polynomial.exists_iff_exists_finsupp {R : Type u} [Semiring R] (P : Polynomial R \u2192 Prop) :(\u2203 (p : Polynomial R), P p) \u2194 \u2203 (q : AddMonoidAlgebra R \u2115), P { toFinsupp := q }","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L94-L96"},"Polynomial.aeval_add":{"signature":"theorem Polynomial.aeval_add {R : Type u} {A : Type z} [CommSemiring R] [Semiring A] [Algebra R A] {p q : Polynomial R} (x : A) :(aeval x) (p + q) = (aeval x) p + (aeval x) q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L290-L291"},"Polynomial.aeval_algebraMap_apply":{"signature":"theorem Polynomial.aeval_algebraMap_apply {R : Type u_1} {A : Type u_2} (B : Type u_3) [CommSemiring R] [CommSemiring A] [Semiring B] [Algebra R A] [Algebra A B] [Algebra R B] [IsScalarTower R A B] (x : A) (p : Polynomial R) :(aeval ((algebraMap A B) x)) p = (algebraMap A B) ((aeval x) p)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Tower.lean#L50-L52"},"Polynomial.irreducible_of_monic":{"signature":"theorem Polynomial.irreducible_of_monic {R : Type u} [CommSemiring R] [NoZeroDivisors R] {p : Polynomial R} (hp : p.Monic) (hp1 : p \u2260 1) :Irreducible p \u2194 \u2200 (f g : Polynomial R), f.Monic \u2192 g.Monic \u2192 f * g = p \u2192 f = 1 \u2228 g = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Monic.lean#L320-L331"},"Polynomial.aeval_mul":{"signature":"theorem Polynomial.aeval_mul {R : Type u} {A : Type z} [CommSemiring R] [Semiring A] [Algebra R A] {p q : Polynomial R} (x : A) :(aeval x) (p * q) = (aeval x) p * (aeval x) q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L299-L300"},"Polynomial.natDegree_pos_iff_degree_pos":{"signature":"theorem Polynomial.natDegree_pos_iff_degree_pos {R : Type u} [Semiring R] {p : Polynomial R} :0 < p.natDegree \u2194 0 < p.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L506-L507"},"Polynomial.as_sum_range'":{"signature":"theorem Polynomial.as_sum_range' {R : Type u} [Semiring R] (p : Polynomial R) (n : \u2115) (hn : p.natDegree < n) :p = \u2211 i \u2208 Finset.range n, (monomial i) (p.coeff i)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Support.lean#L86-L88"},"Polynomial.eval\u2082_eq_sum_range'":{"signature":"theorem Polynomial.eval\u2082_eq_sum_range' {R : Type u} {S : Type v} [Semiring R] [Semiring S] (f : R \u2192+* S) {p : Polynomial R} {n : \u2115} (hn : p.natDegree < n) (x : S) :eval\u2082 f x p = \u2211 i \u2208 Finset.range n, f (p.coeff i) * x ^ i","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Degree.lean#L49-L53"},"Polynomial.aeval_eq_sum_range'":{"signature":"theorem Polynomial.aeval_eq_sum_range' {R : Type u} {S : Type v} [CommSemiring R] [Semiring S] [Algebra R S] {p : Polynomial R} {n : \u2115} (hn : p.natDegree < n) (x : S) :(aeval x) p = \u2211 i \u2208 Finset.range n, p.coeff i \u2022 x ^ i","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L497-L500"},"Polynomial.degree_sub_eq_left_of_degree_lt":{"signature":"theorem Polynomial.degree_sub_eq_left_of_degree_lt {R : Type u} [Ring R] {p q : Polynomial R} (h : q.degree < p.degree) :(p - q).degree = p.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L573-L575"},"Polynomial.mem_roots_map_of_injective":{"signature":"theorem Polynomial.mem_roots_map_of_injective {R : Type u} {S : Type v} [CommRing R] [IsDomain R] [Semiring S] {p : Polynomial S} {f : S \u2192+* R} (hf : Function.Injective \u21d1f) {x : R} (hp : p \u2260 0) :x \u2208 (map f p).roots \u2194 eval\u2082 f x p = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L128-L130"},"Polynomial.mem_roots_iff_aeval_eq_zero":{"signature":"theorem Polynomial.mem_roots_iff_aeval_eq_zero {R : Type u} [CommRing R] [IsDomain R] {p : Polynomial R} {x : R} (w : p \u2260 0) :x \u2208 p.roots \u2194 (aeval x) p = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L132-L134"},"Polynomial.map_monic_eq_zero_iff":{"signature":"theorem Polynomial.map_monic_eq_zero_iff {R : Type u} {S : Type v} [Semiring R] [Semiring S] {f : R \u2192+* S} {p : Polynomial R} (hp : p.Monic) :map f p = 0 \u2194 \u2200 (x : R), f x = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Degree.lean#L135-L141"},"Polynomial.map_monic_ne_zero":{"signature":"theorem Polynomial.map_monic_ne_zero {R : Type u} {S : Type v} [Semiring R] [Semiring S] {f : R \u2192+* S} {p : Polynomial R} (hp : p.Monic) [Nontrivial S] :map f p \u2260 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Degree.lean#L143-L144"},"Polynomial.rootOfSplits":{"signature":"noncomputable def Polynomial.rootOfSplits {R : Type u_1} [CommRing R] {f : Polynomial R} (hf : f.Splits) (hfd : f.degree \u2260 0) :R","doc":"Pick a root of a polynomial that splits.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L271-L273"},"Polynomial.mem_aroots'":{"signature":"theorem Polynomial.mem_aroots' {S : Type v} {T : Type w} [CommRing T] [CommRing S] [IsDomain S] [Algebra T S] {p : Polynomial T} {a : S} :a \u2208 p.aroots S \u2194 map (algebraMap T S) p \u2260 0 \u2227 (aeval a) p = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L446-L448"},"Polynomial.mem_aroots":{"signature":"theorem Polynomial.mem_aroots {S : Type v} {T : Type w} [CommRing T] [IsDomain T] [CommRing S] [IsDomain S] [Algebra T S] [Module.IsTorsionFree T S] {p : Polynomial T} {a : S} :a \u2208 p.aroots S \u2194 p \u2260 0 \u2227 (aeval a) p = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L450-L453"},"Polynomial.eval_rootOfSplits":{"signature":"theorem Polynomial.eval_rootOfSplits {R : Type u_1} [CommRing R] {f : Polynomial R} (hf : f.Splits) (hfd : f.degree \u2260 0) :eval (rootOfSplits hf hfd) f = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L275-L278"},"Polynomial.coeff_one_zero":{"signature":"theorem Polynomial.coeff_one_zero {R : Type u} [Semiring R] :coeff 1 0 = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L602-L604"},"Polynomial.constantCoeff":{"signature":"def Polynomial.constantCoeff {R : Type u} [Semiring R] :Polynomial R \u2192+* R","doc":"constantCoeff p returns the constant term of the polynomial p, defined as coeff p 0. This is a ring homomorphism.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L123-L131"},"Polynomial.isUnit_C":{"signature":"theorem Polynomial.isUnit_C {R : Type u} [Semiring R] {x : R} :IsUnit (C x) \u2194 IsUnit x","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L136-L137"},"Polynomial.isUnit_iff":{"signature":"theorem Polynomial.isUnit_iff {R : Type u} [Semiring R] [NoZeroDivisors R] {p : Polynomial R} :IsUnit p \u2194 \u2203 (r : R), IsUnit r \u2227 C r = p","doc":"Characterization of a unit of a polynomial ring over an integral domain R. See Polynomial.isUnit_iff_coeff_isUnit_isNilpotent when R is a commutative ring.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Units.lean#L46-L53"},"Polynomial.instNormalizationMonoid":{"signature":"instance Polynomial.instNormalizationMonoid {R : Type u} [CommRing R] [NoZeroDivisors R] [NormalizationMonoid R] :NormalizationMonoid (Polynomial R)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L205-L223"},"Polynomial.coe_normUnit":{"signature":"theorem Polynomial.coe_normUnit {R : Type u} [CommRing R] [NoZeroDivisors R] [NormalizationMonoid R] {p : Polynomial R} :\u2191(normUnit p) = C \u2191(normUnit p.leadingCoeff)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L225-L227"},"Polynomial.coe_normUnit_of_ne_zero":{"signature":"theorem Polynomial.coe_normUnit_of_ne_zero {R : Type u} [Field R] {p : Polynomial R} [DecidableEq R] (hp : p \u2260 0) :\u2191(normUnit p) = C p.leadingCoeff\u207b\u00b9","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L591-L594"},"Polynomial.map_dvd_map'":{"signature":"theorem Polynomial.map_dvd_map' {R : Type u} {k : Type y} [Field R] [Field k] (f : R \u2192+* k) {x y : Polynomial R} :map f x \u2223 map f y \u2194 x \u2223 y","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L596-L603"},"Polynomial.aroots_C":{"signature":"theorem Polynomial.aroots_C {S : Type v} {T : Type w} [CommRing T] [CommRing S] [IsDomain S] [Algebra T S] (a : T) :(C a).aroots S = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L473-L475"},"Polynomial.rootSet_C":{"signature":"theorem Polynomial.rootSet_C {S : Type v} {T : Type w} [CommRing T] [CommRing S] [IsDomain S] [Algebra T S] (a : T) :(C a).rootSet S = \u2205","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L545-L548"},"Polynomial.rootSet_zero":{"signature":"theorem Polynomial.rootSet_zero {T : Type w} [CommRing T] (S : Type u_1) [CommRing S] [IsDomain S] [Algebra T S] :rootSet 0 S = \u2205","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L550-L552"},"Polynomial.ne_zero_of_mem_rootSet":{"signature":"theorem Polynomial.ne_zero_of_mem_rootSet {S : Type v} {T : Type w} [CommRing T] {p : Polynomial T} [CommRing S] [IsDomain S] [Algebra T S] {a : S} (h : a \u2208 p.rootSet S) :p \u2260 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L619-L620"},"Polynomial.mem_rootSet'":{"signature":"theorem Polynomial.mem_rootSet' {T : Type w} [CommRing T] {p : Polynomial T} {S : Type u_1} [CommRing S] [IsDomain S] [Algebra T S] {a : S} :a \u2208 p.rootSet S \u2194 map (algebraMap T S) p \u2260 0 \u2227 (aeval a) p = 0","doc":"A version of mem_rootSet that requires the polynomial to be non-zero after mapping instead of requiring it to be non-zero and NoZeroSMulDivisors.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L588-L593"},"Polynomial.aeval_eq_zero_of_mem_rootSet":{"signature":"theorem Polynomial.aeval_eq_zero_of_mem_rootSet {S : Type v} {T : Type w} [CommRing T] {p : Polynomial T} [CommRing S] [IsDomain S] [Algebra T S] {a : S} (hx : a \u2208 p.rootSet S) :(aeval a) p = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L622-L624"},"Polynomial.mem_rootSet":{"signature":"theorem Polynomial.mem_rootSet {T : Type w} [CommRing T] {p : Polynomial T} {S : Type u_1} [IsDomain T] [CommRing S] [IsDomain S] [Algebra T S] [Module.IsTorsionFree T S] {a : S} :a \u2208 p.rootSet S \u2194 p \u2260 0 \u2227 (aeval a) p = 0","doc":"A version of mem_rootSet' that requires Module.IsTorsionFree and for the polynomial to be non-zero instead of requiring it to be non-zero after mapping.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L595-L599"},"Polynomial.mem_rootSet_of_ne":{"signature":"theorem Polynomial.mem_rootSet_of_ne {T : Type w} [CommRing T] {p : Polynomial T} {S : Type u_1} [IsDomain T] [CommRing S] [IsDomain S] [Algebra T S] [Module.IsTorsionFree T S] (hp : p \u2260 0) {a : S} :a \u2208 p.rootSet S \u2194 (aeval a) p = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L601-L603"},"Polynomial.natDegree_lt_natDegree_iff":{"signature":"theorem Polynomial.natDegree_lt_natDegree_iff {R : Type u} [Semiring R] {p q : Polynomial R} (hp : p \u2260 0) :p.natDegree < q.natDegree \u2194 p.degree < q.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L158-L161"},"Polynomial.C_ne_zero":{"signature":"theorem Polynomial.C_ne_zero {R : Type u} {a : R} [Semiring R] :C a \u2260 0 \u2194 a \u2260 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L686-L687"},"Polynomial.instCommSemiringAdjoinSingleton":{"signature":"instance Polynomial.instCommSemiringAdjoinSingleton (R : Type u) {A : Type z} [CommSemiring R] [Semiring A] [Algebra R A] (x : A) :CommSemiring \u21a5(Algebra.adjoin R {x})","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Adjoin/Polynomial.lean#L77-L83"},"Polynomial.instCommRingAdjoinSingleton":{"signature":"instance Polynomial.instCommRingAdjoinSingleton {R : Type u_3} {A : Type u_4} [CommRing R] [Ring A] [Algebra R A] (x : A) :CommRing \u21a5(Algebra.adjoin R {x})","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Adjoin/Polynomial.lean#L85-L87"},"Polynomial.coe_aeval_mk_apply":{"signature":"theorem Polynomial.coe_aeval_mk_apply {R : Type u} {A : Type z} [CommSemiring R] [Semiring A] [Algebra R A] {p : Polynomial R} (x : A) {S : Subalgebra R A} (h : x \u2208 S) :\u2191((aeval \u27e8x, h\u27e9) p) = (aeval x) p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L407-L410"},"Polynomial.lift_of_splits":{"signature":"theorem Polynomial.lift_of_splits {F : Type u_2} {K : Type u_3} {L : Type u_4} [Field F] [Field K] [Field L] [Algebra F K] [Algebra F L] (s : Finset K) :(\u2200 x \u2208 s, IsIntegral F x \u2227 (map (algebraMap F L) (minpoly F x)).Splits) \u2192 Nonempty (\u21a5(Algebra.adjoin F \u2191s) \u2192\u2090[F] L)","doc":"If K and L are field extensions of F and we have s : Finset K such that the minimal polynomial of each x \u2208 s splits in L then Algebra.adjoin F s embeds in L.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Adjoin/Field.lean#L65-L92"},"Polynomial.roots_prod":{"signature":"theorem Polynomial.roots_prod {R : Type u} [CommRing R] [IsDomain R] {\u03b9 : Type u_1} (f : \u03b9 \u2192 Polynomial R) (s : Finset \u03b9) :s.prod f \u2260 0 \u2192 (s.prod f).roots = s.val.bind fun (i : \u03b9) => (f i).roots","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L242-L245"},"Polynomial.rootSet_prod":{"signature":"theorem Polynomial.rootSet_prod {R : Type u} {S : Type v} [Field R] [CommRing S] [IsDomain S] [Algebra R S] {\u03b9 : Type u_1} (f : \u03b9 \u2192 Polynomial R) (s : Finset \u03b9) (h : s.prod f \u2260 0) :(s.prod f).rootSet S = \u22c3 i \u2208 s, (f i).rootSet S","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L515-L520"},"Polynomial.eval_map_algebraMap":{"signature":"theorem Polynomial.eval_map_algebraMap {R : Type u} {B : Type u_2} [CommSemiring R] [Semiring B] [Algebra R B] (P : Polynomial R) (b : B) :eval b (map (algebraMap R B) P) = (aeval b) P","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L262-L265"},"Polynomial.irreducible_mul_leadingCoeff_inv":{"signature":"theorem Polynomial.irreducible_mul_leadingCoeff_inv {K : Type u_1} [DivisionRing K] {p : Polynomial K} :Irreducible (p * C p.leadingCoeff\u207b\u00b9) \u2194 Irreducible p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L437-L443"},"Polynomial.degree_eq_one_of_irreducible_of_root":{"signature":"theorem Polynomial.degree_eq_one_of_irreducible_of_root {R : Type u} [CommRing R] {p : Polynomial R} [IsDomain R] (hi : Irreducible p) {x : R} (hx : p.IsRoot x) :p.degree = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L774-L783"},"Polynomial.evalRingHom":{"signature":"def Polynomial.evalRingHom {R : Type u} [CommSemiring R] :R \u2192 Polynomial R \u2192+* R","doc":"eval r, regarded as a ring homomorphism from R[X] to R.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L603-L605"},"Polynomial.eval_prod":{"signature":"theorem Polynomial.eval_prod {R : Type u} [CommSemiring R] {\u03b9 : Type u_1} (s : Finset \u03b9) (p : \u03b9 \u2192 Polynomial R) (x : R) :eval x (\u220f j \u2208 s, p j) = \u220f j \u2208 s, eval x (p j)","doc":"Polynomial evaluation commutes with Finset.prod","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L659-L663"},"Polynomial.integralNormalization":{"signature":"noncomputable def Polynomial.integralNormalization {R : Type u} [Semiring R] (p : Polynomial R) :Polynomial R","doc":"If p : R[X] is a nonzero polynomial with root z, integralNormalization p is a monic polynomial with root leadingCoeff f * z. Moreover, integralNormalization 0 = 0.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/IntegralNormalization.lean#L34-L41"},"Polynomial.support_integralNormalization_subset":{"signature":"theorem Polynomial.support_integralNormalization_subset {R : Type u} [Semiring R] {p : Polynomial R} :p.integralNormalization.support \u2286 p.support","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/IntegralNormalization.lean#L60-L63"},"Polynomial.coeff_ne_zero_of_eq_degree":{"signature":"theorem Polynomial.coeff_ne_zero_of_eq_degree {R : Type u} {n : \u2115} [Semiring R] {p : Polynomial R} (hn : p.degree = \u2191n) :p.coeff n \u2260 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L237-L238"},"Polynomial.integralNormalization_coeff":{"signature":"theorem Polynomial.integralNormalization_coeff {R : Type u} [Semiring R] {p : Polynomial R} {i : \u2115} :p.integralNormalization.coeff i = if p.degree = \u2191i then 1 else p.coeff i * p.leadingCoeff ^ (p.natDegree - 1 - i)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/IntegralNormalization.lean#L53-L58"},"Polynomial.integralNormalization_coeff_degree":{"signature":"theorem Polynomial.integralNormalization_coeff_degree {R : Type u} [Semiring R] {p : Polynomial R} {i : \u2115} (hi : p.degree = \u2191i) :p.integralNormalization.coeff i = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/IntegralNormalization.lean#L65-L66"},"Polynomial.integralNormalization_coeff_natDegree":{"signature":"theorem Polynomial.integralNormalization_coeff_natDegree {R : Type u} [Semiring R] {p : Polynomial R} (hp : p \u2260 0) :p.integralNormalization.coeff p.natDegree = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/IntegralNormalization.lean#L68-L70"},"Polynomial.monic_integralNormalization":{"signature":"theorem Polynomial.monic_integralNormalization {R : Type u} [Semiring R] {p : Polynomial R} (hp : p \u2260 0) :p.integralNormalization.Monic","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/IntegralNormalization.lean#L95-L99"},"Polynomial.integralNormalization_zero":{"signature":"theorem Polynomial.integralNormalization_zero {R : Type u} [Semiring R] :integralNormalization 0 = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/IntegralNormalization.lean#L43-L45"},"Polynomial.as_sum_range":{"signature":"theorem Polynomial.as_sum_range {R : Type u} [Semiring R] (p : Polynomial R) :p = \u2211 i \u2208 Finset.range (p.natDegree + 1), (monomial i) (p.coeff i)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Support.lean#L90-L91"},"Polynomial.eval\u2082_finset_sum":{"signature":"theorem Polynomial.eval\u2082_finset_sum {R : Type u} {S : Type v} {\u03b9 : Type y} [Semiring R] [Semiring S] (f : R \u2192+* S) (s : Finset \u03b9) (g : \u03b9 \u2192 Polynomial R) (x : S) :eval\u2082 f x (\u2211 i \u2208 s, g i) = \u2211 i \u2208 s, eval\u2082 f x (g i)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L132-L134"},"Polynomial.eval\u2082_eq_sum_range":{"signature":"theorem Polynomial.eval\u2082_eq_sum_range {R : Type u} {S : Type v} [Semiring R] {p : Polynomial R} [Semiring S] (f : R \u2192+* S) (x : S) :eval\u2082 f x p = \u2211 i \u2208 Finset.range (p.natDegree + 1), f (p.coeff i) * x ^ i","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Degree.lean#L44-L47"},"Polynomial.degree_eq_of_le_of_coeff_ne_zero":{"signature":"theorem Polynomial.degree_eq_of_le_of_coeff_ne_zero {R : Type u} {n : \u2115} [Semiring R] {p : Polynomial R} (pn : p.degree \u2264 \u2191n) (p1 : p.coeff n \u2260 0) :p.degree = \u2191n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L66-L67"},"Polynomial.degree_integralNormalization":{"signature":"theorem Polynomial.degree_integralNormalization {R : Type u} [Semiring R] {p : Polynomial R} :p.integralNormalization.degree = p.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/IntegralNormalization.lean#L80-L87"},"Polynomial.coe_lt_degree":{"signature":"theorem Polynomial.coe_lt_degree {R : Type u} [Semiring R] {p : Polynomial R} {n : \u2115} :\u2191n < p.degree \u2194 n < p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L252-L256"},"Polynomial.integralNormalization_coeff_mul_leadingCoeff_pow":{"signature":"theorem Polynomial.integralNormalization_coeff_mul_leadingCoeff_pow {R : Type u} [Semiring R] {p : Polynomial R} (i : \u2115) (hp : 1 \u2264 p.natDegree) :p.integralNormalization.coeff i * p.leadingCoeff ^ i = p.coeff i * p.leadingCoeff ^ (p.natDegree - 1)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/IntegralNormalization.lean#L101-L113"},"Polynomial.integralNormalization_eval\u2082_leadingCoeff_mul_of_commute":{"signature":"theorem Polynomial.integralNormalization_eval\u2082_leadingCoeff_mul_of_commute {R : Type u} [Semiring R] {p : Polynomial R} {A : Type u_1} [Semiring A] (h : 1 \u2264 p.natDegree) (f : R \u2192+* A) (x : A) (h\u2081 : Commute (f p.leadingCoeff) x) (h\u2082 : \u2200 {r r' : R}, Commute (f r) (f r')) :eval\u2082 f (f p.leadingCoeff * x) p.integralNormalization = f p.leadingCoeff ^ (p.natDegree - 1) * eval\u2082 f x p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/IntegralNormalization.lean#L137-L146"},"Polynomial.integralNormalization_eval\u2082_eq_zero_of_commute":{"signature":"theorem Polynomial.integralNormalization_eval\u2082_eq_zero_of_commute {R : Type u} [Semiring R] {A : Type u_1} [Semiring A] {p : Polynomial R} (f : R \u2192+* A) {z : A} (hz : eval\u2082 f z p = 0) (h\u2081 : Commute (f p.leadingCoeff) z) (h\u2082 : \u2200 {r r' : R}, Commute (f r) (f r')) (inj : \u2200 (x : R), f x = 0 \u2192 x = 0) :eval\u2082 f (f p.leadingCoeff * z) p.integralNormalization = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/IntegralNormalization.lean#L153-L163"},"Polynomial.integralNormalization_aeval_eq_zero":{"signature":"theorem Polynomial.integralNormalization_aeval_eq_zero {S : Type v} {A : Type u_1} [CommSemiring S] [Semiring A] [Algebra S A] {f : Polynomial S} {z : A} (hz : (aeval z) f = 0) (inj : \u2200 (x : S), (algebraMap S A) x = 0 \u2192 x = 0) :(aeval ((algebraMap S A) f.leadingCoeff * z)) f.integralNormalization = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/IntegralNormalization.lean#L170-L174"},"Polynomial.mem_coeffs_iff":{"signature":"theorem Polynomial.mem_coeffs_iff {R : Type u} [Semiring R] {p : Polynomial R} {c : R} :c \u2208 p.coeffs \u2194 \u2203 n \u2208 p.support, c = p.coeff n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L1022-L1023"},"Polynomial.support_smul":{"signature":"theorem Polynomial.support_smul {R : Type u} {S : Type v} [Semiring R] [SMulZeroClass S R] (r : S) (p : Polynomial R) :(r \u2022 p).support \u2286 p.support","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L52-L57"},"Polynomial.instIsTorsionFree":{"signature":"instance Polynomial.instIsTorsionFree {R : Type u} [Semiring R] {S : Type u_1} [Semiring S] [Module S R] [Module.IsTorsionFree S R] :Module.IsTorsionFree S (Polynomial R)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L312-L314"},"Polynomial.exists_eq_pow_rootMultiplicity_mul_and_not_dvd":{"signature":"theorem Polynomial.exists_eq_pow_rootMultiplicity_mul_and_not_dvd {R : Type u} [Ring R] (p : Polynomial R) (hp : p \u2260 0) (a : R) :\u2203 (q : Polynomial R), p = (X - C a) ^ rootMultiplicity a p * q \u2227 \u00acX - C a \u2223 q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L559-L563"},"Polynomial.divX":{"signature":"def Polynomial.divX {R : Type u} [Semiring R] (p : Polynomial R) :Polynomial R","doc":"divX p returns a polynomial q such that q * X + C (p.coeff 0) = p. It can be used in a semiring where the usual division algorithm is not possible","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Inductions.lean#L38-L41"},"Polynomial.coeff_divX":{"signature":"theorem Polynomial.coeff_divX {R : Type u} {n : \u2115} [Semiring R] {p : Polynomial R} :p.divX.coeff n = p.coeff (n + 1)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Inductions.lean#L43-L45"},"Polynomial.coeff_X_mul":{"signature":"theorem Polynomial.coeff_X_mul {R : Type u} [Semiring R] (p : Polynomial R) (n : \u2115) :(X * p).coeff (n + 1) = p.coeff n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L254-L256"},"Polynomial.X_mul_divX_add":{"signature":"theorem Polynomial.X_mul_divX_add {R : Type u} [Semiring R] (p : Polynomial R) :X * p.divX + C (p.coeff 0) = p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Inductions.lean#L50-L52"},"Polynomial.X_dvd_iff":{"signature":"theorem Polynomial.X_dvd_iff {R : Type u} [Semiring R] {f : Polynomial R} :X \u2223 f \u2194 f.coeff 0 = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L40-L42"},"Polynomial.aeval_X_pow":{"signature":"theorem Polynomial.aeval_X_pow {R : Type u} {A : Type z} [CommSemiring R] [Semiring A] [Algebra R A] (x : A) {n : \u2115} :(aeval x) (X ^ n) = x ^ n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L287-L288"},"Polynomial.X_pow_mul":{"signature":"theorem Polynomial.X_pow_mul {R : Type u} [Semiring R] {p : Polynomial R} {n : \u2115} :X ^ n * p = p * X ^ n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L507-L512"},"Polynomial.X_dvd_sub_C":{"signature":"theorem Polynomial.X_dvd_sub_C {R : Type u} [Ring R] {p : Polynomial R} :X \u2223 p - C (p.coeff 0)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L235-L236"},"Polynomial.comp":{"signature":"def Polynomial.comp {R : Type u} [Semiring R] (p q : Polynomial R) :Polynomial R","doc":"The composition of polynomials as a polynomial.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L368-L370"},"Polynomial.leadingCoeff_C_mul_X":{"signature":"theorem Polynomial.leadingCoeff_C_mul_X {R : Type u} [Semiring R] (a : R) :(C a * X).leadingCoeff = a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L427-L428"},"Polynomial.natDegree_pow_le":{"signature":"theorem Polynomial.natDegree_pow_le {R : Type u} [Semiring R] {p : Polynomial R} {n : \u2115} :(p ^ n).natDegree \u2264 n * p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L482-L485"},"Polynomial.coeff_pow_mul_natDegree":{"signature":"theorem Polynomial.coeff_pow_mul_natDegree {R : Type u} [Semiring R] (p : Polynomial R) (n : \u2115) :(p ^ n).coeff (n * p.natDegree) = p.leadingCoeff ^ n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L420-L440"},"Polynomial.coeff_comp_degree_mul_degree":{"signature":"theorem Polynomial.coeff_comp_degree_mul_degree {R : Type u} [Semiring R] {p q : Polynomial R} (hqd0 : q.natDegree \u2260 0) :(p.comp q).coeff (p.natDegree * q.natDegree) = p.leadingCoeff * q.leadingCoeff ^ p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Degree.lean#L90-L105"},"Polynomial.natDegree_C_mul_X":{"signature":"theorem Polynomial.natDegree_C_mul_X {R : Type u} [Semiring R] (a : R) (ha : a \u2260 0) :(C a * X).natDegree = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L216-L218"},"Polynomial.add_comp":{"signature":"theorem Polynomial.add_comp {R : Type u} [Semiring R] {p q r : Polynomial R} :(p + q).comp r = p.comp r + q.comp r","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L409-L411"},"Polynomial.monomial_comp":{"signature":"theorem Polynomial.monomial_comp {R : Type u} {a : R} [Semiring R] {p : Polynomial R} (n : \u2115) :((monomial n) a).comp p = C a * p ^ n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L413-L415"},"Polynomial.eval\u2082_mul'":{"signature":"theorem Polynomial.eval\u2082_mul' {R : Type u} {S : Type v} [CommSemiring R] [Semiring S] [Algebra R S] (x : S) (p q : Polynomial R) :eval\u2082 (algebraMap R S) x (p * q) = eval\u2082 (algebraMap R S) x p * eval\u2082 (algebraMap R S) x q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Algebra.lean#L41-L44"},"Polynomial.eval\u2082_pow'":{"signature":"theorem Polynomial.eval\u2082_pow' {R : Type u} {S : Type v} [CommSemiring R] [Semiring S] [Algebra R S] (x : S) (p : Polynomial R) (n : \u2115) :eval\u2082 (algebraMap R S) x (p ^ n) = eval\u2082 (algebraMap R S) x p ^ n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Algebra.lean#L46-L51"},"Polynomial.eval\u2082_comp'":{"signature":"theorem Polynomial.eval\u2082_comp' {R : Type u} {S : Type v} [CommSemiring R] [Semiring S] [Algebra R S] (x : S) (p q : Polynomial R) :eval\u2082 (algebraMap R S) x (p.comp q) = eval\u2082 (algebraMap R S) (eval\u2082 (algebraMap R S) x q) p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Algebra.lean#L53-L58"},"Polynomial.aeval_comp":{"signature":"theorem Polynomial.aeval_comp {R : Type u} [CommSemiring R] {p q : Polynomial R} {A : Type u_3} [Semiring A] [Algebra R A] (x : A) :(aeval x) (p.comp q) = (aeval ((aeval x) q)) p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L304-L306"},"Polynomial.notMem_support_iff":{"signature":"theorem Polynomial.notMem_support_iff {R : Type u} {n : \u2115} [Semiring R] {p : Polynomial R} :n \u2209 p.support \u2194 p.coeff n = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L629-L629"},"Polynomial.eval\u2082_smul":{"signature":"theorem Polynomial.eval\u2082_smul {R : Type u} {S : Type v} [Semiring R] [Semiring S] (g : R \u2192+* S) (p : Polynomial R) (x : S) {s : R} :eval\u2082 g x (s \u2022 p) = g s * eval\u2082 g x p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/SMul.lean#L43-L49"},"Polynomial.rootSet_maps_to'":{"signature":"theorem Polynomial.rootSet_maps_to' {T : Type w} [CommRing T] {p : Polynomial T} {S : Type u_1} {S' : Type u_2} [CommRing S] [IsDomain S] [Algebra T S] [CommRing S'] [IsDomain S'] [Algebra T S'] (hp : map (algebraMap T S') p = 0 \u2192 map (algebraMap T S) p = 0) (f : S \u2192\u2090[T] S') :Set.MapsTo (\u21d1f) (p.rootSet S) (p.rootSet S')","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L612-L617"},"Polynomial.rootSetFintype":{"signature":"instance Polynomial.rootSetFintype {T : Type w} [CommRing T] (p : Polynomial T) (S : Type u_1) [CommRing S] [IsDomain S] [Algebra T S] :Fintype \u2191(p.rootSet S)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L563-L565"},"Polynomial.derivative_add":{"signature":"theorem Polynomial.derivative_add {R : Type u} [Semiring R] {f g : Polynomial R} :derivative (f + g) = derivative f + derivative g","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L124-L126"},"Polynomial.derivative_X":{"signature":"theorem Polynomial.derivative_X {R : Type u} [Semiring R] :derivative X = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L116-L118"},"Polynomial.derivative_C":{"signature":"theorem Polynomial.derivative_C {R : Type u} [Semiring R] {a : R} :derivative (C a) = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L110-L111"},"Polynomial.separable_X_add_C":{"signature":"theorem Polynomial.separable_X_add_C {R : Type u} [CommSemiring R] (a : R) :(X + C a).Separable","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Separable.lean#L73-L75"},"Polynomial.separable_X_sub_C":{"signature":"theorem Polynomial.separable_X_sub_C {R : Type u} [CommRing R] {x : R} :(X - C x).Separable","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Separable.lean#L201-L202"},"Polynomial.natSepDegree":{"signature":"def Polynomial.natSepDegree {F : Type u} [Field F] (f : Polynomial F) :\u2115","doc":"The separable degree Polynomial.natSepDegree of a polynomial is a natural number, defined to be the number of distinct roots of it over its splitting field. This is similar to Polynomial.natDegree but not to Polynomial.degree, namely, the separable degree of 0 is 0, not negative infinity.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/SeparableDegree.lean#L295-L299"},"Polynomial.natSepDegree_eq_of_splits":{"signature":"theorem Polynomial.natSepDegree_eq_of_splits {F : Type u} {E : Type v} [Field F] [Field E] [Algebra F E] (f : Polynomial F) [DecidableEq E] (h : (map (algebraMap F E) f).Splits) :f.natSepDegree = (f.aroots E).toFinset.card","doc":"If a polynomial splits over E, then its separable degree is equal to the number of distinct roots of it over E.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/SeparableDegree.lean#L366-L373"},"Polynomial.natSepDegree_eq_of_isAlgClosed":{"signature":"theorem Polynomial.natSepDegree_eq_of_isAlgClosed {F : Type u} (E : Type v) [Field F] [Field E] [Algebra F E] (f : Polynomial F) [DecidableEq E] [IsAlgClosed E] :f.natSepDegree = (f.aroots E).toFinset.card","doc":"The separable degree of a polynomial is equal to the number of distinct roots of it over any algebraically closed field.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/SeparableDegree.lean#L376-L380"},"Polynomial.IsPrimitive":{"signature":"def Polynomial.IsPrimitive {R : Type u_1} [CommSemiring R] (p : Polynomial R) :Prop","doc":"A polynomial is primitive when the only constant polynomials dividing it are units. Note: This has nothing to do with minimal polynomials of primitive elements in finite fields.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Content.lean#L43-L46"},"Polynomial.content":{"signature":"def Polynomial.content {R : Type u_1} [CommRing R] [NormalizedGCDMonoid R] (p : Polynomial R) :R","doc":"p.content is the gcd of the coefficients of p.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Content.lean#L89-L91"},"Polynomial.C_dvd_iff_dvd_coeff":{"signature":"theorem Polynomial.C_dvd_iff_dvd_coeff {R : Type u} [Semiring R] (r : R) (\u03c6 : Polynomial R) :C r \u2223 \u03c6 \u2194 \u2200 (i : \u2115), r \u2223 \u03c6.coeff i","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L311-L328"},"Polynomial.content_dvd_coeff":{"signature":"theorem Polynomial.content_dvd_coeff {R : Type u_1} [CommRing R] [NormalizedGCDMonoid R] {p : Polynomial R} (n : \u2115) :p.content \u2223 p.coeff n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Content.lean#L93-L98"},"Polynomial.dvd_content_iff_C_dvd":{"signature":"theorem Polynomial.dvd_content_iff_C_dvd {R : Type u_1} [CommRing R] [NormalizedGCDMonoid R] {p : Polynomial R} {r : R} :r \u2223 p.content \u2194 C r \u2223 p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Content.lean#L206-L214"},"Polynomial.C_content_dvd":{"signature":"theorem Polynomial.C_content_dvd {R : Type u_1} [CommRing R] [NormalizedGCDMonoid R] (p : Polynomial R) :C p.content \u2223 p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Content.lean#L216-L217"},"Polynomial.primPart":{"signature":"noncomputable def Polynomial.primPart {R : Type u_1} [CommRing R] [NormalizedGCDMonoid R] (p : Polynomial R) :Polynomial R","doc":"The primitive part of a polynomial p is the primitive polynomial gained by dividing p by p.content. If p = 0, then p.primPart = 1.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Content.lean#L229-L233"},"Polynomial.normalize_content":{"signature":"theorem Polynomial.normalize_content {R : Type u_1} [CommRing R] [NormalizedGCDMonoid R] {p : Polynomial R} :normalize p.content = p.content","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Content.lean#L167-L168"},"Polynomial.isPrimitive_iff_content_eq_one":{"signature":"theorem Polynomial.isPrimitive_iff_content_eq_one {R : Type u_1} [CommRing R] [NormalizedGCDMonoid R] {p : Polynomial R} :p.IsPrimitive \u2194 p.content = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Content.lean#L219-L222"},"Polynomial.content_C":{"signature":"theorem Polynomial.content_C {R : Type u_1} [CommRing R] [NormalizedGCDMonoid R] {r : R} :(C r).content = normalize r","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Content.lean#L100-L106"},"Polynomial.content_zero":{"signature":"theorem Polynomial.content_zero {R : Type u_1} [CommRing R] [NormalizedGCDMonoid R] :content 0 = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Content.lean#L108-L109"},"Polynomial.eq_C_content_mul_primPart":{"signature":"theorem Polynomial.eq_C_content_mul_primPart {R : Type u_1} [CommRing R] [NormalizedGCDMonoid R] (p : Polynomial R) :p = C p.content * p.primPart","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Content.lean#L235-L237"},"Polynomial.eraseLead":{"signature":"def Polynomial.eraseLead {R : Type u_1} [Semiring R] (f : Polynomial R) :Polynomial R","doc":"eraseLead f for a polynomial f is the polynomial obtained by subtracting from f the leading term of f.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/EraseLead.lean#L37-L40"},"Polynomial.erase_zero":{"signature":"theorem Polynomial.erase_zero {R : Type u} [Semiring R] (n : \u2115) :erase n 0 = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L955-L957"},"Polynomial.eraseLead_zero":{"signature":"theorem Polynomial.eraseLead_zero {R : Type u_1} [Semiring R] :eraseLead 0 = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/EraseLead.lean#L57-L58"},"Polynomial.eraseLead_support":{"signature":"theorem Polynomial.eraseLead_support {R : Type u_1} [Semiring R] (f : Polynomial R) :f.eraseLead.support = f.support.erase f.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/EraseLead.lean#L44-L45"},"Polynomial.coeff_erase":{"signature":"theorem Polynomial.coeff_erase {R : Type u} [Semiring R] (p : Polynomial R) (n i : \u2115) :(erase n p).coeff i = if i = n then 0 else p.coeff i","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L949-L953"},"Polynomial.eraseLead_coeff":{"signature":"theorem Polynomial.eraseLead_coeff {R : Type u_1} [Semiring R] {f : Polynomial R} (i : \u2115) :f.eraseLead.coeff i = if i = f.natDegree then 0 else f.coeff i","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/EraseLead.lean#L47-L49"},"Polynomial.content_eq_gcd_leadingCoeff_content_eraseLead":{"signature":"theorem Polynomial.content_eq_gcd_leadingCoeff_content_eraseLead {R : Type u_1} [CommRing R] [NormalizedGCDMonoid R] (p : Polynomial R) :p.content = gcd p.leadingCoeff p.eraseLead.content","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Content.lean#L195-L204"},"Polynomial.content_eq_gcd_range_of_lt":{"signature":"theorem Polynomial.content_eq_gcd_range_of_lt {R : Type u_1} [CommRing R] [NormalizedGCDMonoid R] (p : Polynomial R) (n : \u2115) (h : p.natDegree < n) :p.content = (Finset.range n).gcd p.coeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Content.lean#L178-L189"},"Polynomial.gcd_content_eq_of_dvd_sub":{"signature":"theorem Polynomial.gcd_content_eq_of_dvd_sub {R : Type u_1} [CommRing R] [NormalizedGCDMonoid R] {a : R} {p q : Polynomial R} (h : C a \u2223 p - q) :gcd a p.content = gcd a q.content","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Content.lean#L303-L313"},"Polynomial.eraseLead_add_monomial_natDegree_leadingCoeff":{"signature":"theorem Polynomial.eraseLead_add_monomial_natDegree_leadingCoeff {R : Type u_1} [Semiring R] (f : Polynomial R) :f.eraseLead + (monomial f.natDegree) f.leadingCoeff = f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/EraseLead.lean#L60-L63"},"Polynomial.self_sub_monomial_natDegree_leadingCoeff":{"signature":"theorem Polynomial.self_sub_monomial_natDegree_leadingCoeff {R : Type u_2} [Ring R] (f : Polynomial R) :f - (monomial f.natDegree) f.leadingCoeff = f.eraseLead","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/EraseLead.lean#L70-L73"},"Polynomial.self_sub_C_mul_X_pow":{"signature":"theorem Polynomial.self_sub_C_mul_X_pow {R : Type u_2} [Ring R] (f : Polynomial R) :f - C f.leadingCoeff * X ^ f.natDegree = f.eraseLead","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/EraseLead.lean#L75-L78"},"Polynomial.content_mul_aux":{"signature":"theorem Polynomial.content_mul_aux {R : Type u_1} [CommRing R] [NormalizedGCDMonoid R] {p q : Polynomial R} :gcd (p * q).eraseLead.content p.leadingCoeff = gcd (p.eraseLead * q).content p.leadingCoeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Content.lean#L315-L322"},"Polynomial.primPart_zero":{"signature":"theorem Polynomial.primPart_zero {R : Type u_1} [CommRing R] [NormalizedGCDMonoid R] :primPart 0 = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Content.lean#L239-L241"},"Polynomial.isPrimitive_one":{"signature":"theorem Polynomial.isPrimitive_one {R : Type u_1} [CommSemiring R] :IsPrimitive 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Content.lean#L51-L53"},"Polynomial.content_eq_zero_iff":{"signature":"theorem Polynomial.content_eq_zero_iff {R : Type u_1} [CommRing R] [NormalizedGCDMonoid R] {p : Polynomial R} :p.content = 0 \u2194 p = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Content.lean#L154-L164"},"Polynomial.content_C_mul":{"signature":"theorem Polynomial.content_C_mul {R : Type u_1} [CommRing R] [NormalizedGCDMonoid R] (r : R) (p : Polynomial R) :(C r * p).content = normalize r * p.content","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Content.lean#L145-L148"},"Polynomial.isPrimitive_primPart":{"signature":"theorem Polynomial.isPrimitive_primPart {R : Type u_1} [CommRing R] [NormalizedGCDMonoid R] (p : Polynomial R) :p.primPart.IsPrimitive","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Content.lean#L243-L248"},"Polynomial.primPart_ne_zero":{"signature":"theorem Polynomial.primPart_ne_zero {R : Type u_1} [CommRing R] [NormalizedGCDMonoid R] [Nontrivial R] (p : Polynomial R) :p.primPart \u2260 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Content.lean#L253-L254"},"Polynomial.natDegree_primPart":{"signature":"theorem Polynomial.natDegree_primPart {R : Type u_1} [CommRing R] [NormalizedGCDMonoid R] (p : Polynomial R) :p.primPart.natDegree = p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Content.lean#L256-L262"},"Polynomial.content_primPart":{"signature":"theorem Polynomial.content_primPart {R : Type u_1} [CommRing R] [NormalizedGCDMonoid R] (p : Polynomial R) :p.primPart.content = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Content.lean#L250-L251"},"Polynomial.content_mul":{"signature":"theorem Polynomial.content_mul {R : Type u_1} [CommRing R] [NormalizedGCDMonoid R] {p q : Polynomial R} :(p * q).content = p.content * q.content","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Content.lean#L324-L368"},"Polynomial.primPart_mul":{"signature":"theorem Polynomial.primPart_mul {R : Type u_1} [CommRing R] [NormalizedGCDMonoid R] {p q : Polynomial R} (h0 : p * q \u2260 0) :(p * q).primPart = p.primPart * q.primPart","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Content.lean#L374-L383"},"Polynomial.cancelLeads":{"signature":"def Polynomial.cancelLeads {R : Type u_1} [Ring R] (p q : Polynomial R) :Polynomial R","doc":"cancelLeads p q is formed by multiplying p and q by monomials so that they have the same leading term, and then subtracting.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/CancelLeads.lean#L39-L43"},"Polynomial.natDegree_add_le":{"signature":"theorem Polynomial.natDegree_add_le {R : Type u} [Semiring R] (p q : Polynomial R) :(p + q).natDegree \u2264 max p.natDegree q.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L341-L342"},"Polynomial.natDegree_add_le_of_le":{"signature":"theorem Polynomial.natDegree_add_le_of_le {R : Type u} {n m : \u2115} [Semiring R] {p q : Polynomial R} (hp : p.natDegree \u2264 m) (hq : q.natDegree \u2264 n) :(p + q).natDegree \u2264 max m n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L348-L350"},"Polynomial.natDegree_mul_le_of_le":{"signature":"theorem Polynomial.natDegree_mul_le_of_le {R : Type u} {n m : \u2115} [Semiring R] {p q : Polynomial R} (hp : p.natDegree \u2264 m) (hg : q.natDegree \u2264 n) :(p * q).natDegree \u2264 m + n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L478-L480"},"Polynomial.natDegree_neg_le_of_le":{"signature":"theorem Polynomial.natDegree_neg_le_of_le {R : Type u} {m : \u2115} [Ring R] {p : Polynomial R} (hp : p.natDegree \u2264 m) :(-p).natDegree \u2264 m","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L286-L287"},"Polynomial.natDegree_pow_le_of_le":{"signature":"theorem Polynomial.natDegree_pow_le_of_le {R : Type u} {m : \u2115} [Semiring R] {p : Polynomial R} (n : \u2115) (hp : p.natDegree \u2264 m) :(p ^ n).natDegree \u2264 n * m","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L487-L489"},"Polynomial.natDegree_cancelLeads_lt_of_natDegree_le_natDegree_of_comm":{"signature":"theorem Polynomial.natDegree_cancelLeads_lt_of_natDegree_le_natDegree_of_comm {R : Type u_1} [Ring R] {p q : Polynomial R} (comm : p.leadingCoeff * q.leadingCoeff = q.leadingCoeff * p.leadingCoeff) (h : p.natDegree \u2264 q.natDegree) (hq : 0 < q.natDegree) :(p.cancelLeads q).natDegree < q.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/CancelLeads.lean#L51-L70"},"Polynomial.natDegree_cancelLeads_lt_of_natDegree_le_natDegree":{"signature":"theorem Polynomial.natDegree_cancelLeads_lt_of_natDegree_le_natDegree {R : Type u_1} [CommRing R] {p q : Polynomial R} (h : p.natDegree \u2264 q.natDegree) (hq : 0 < q.natDegree) :(p.cancelLeads q).natDegree < q.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/CancelLeads.lean#L81-L83"},"Polynomial.dvd_cancelLeads_of_dvd_of_dvd":{"signature":"theorem Polynomial.dvd_cancelLeads_of_dvd_of_dvd {R : Type u_1} [CommRing R] {p q r : Polynomial R} (pq : p \u2223 q) (pr : p \u2223 r) :p \u2223 q.cancelLeads r","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/CancelLeads.lean#L78-L79"},"Polynomial.isUnit_primPart_C":{"signature":"theorem Polynomial.isUnit_primPart_C {R : Type u_1} [CommRing R] [NormalizedGCDMonoid R] (r : R) :IsUnit (C r).primPart","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Content.lean#L268-L280"},"Polynomial.exists_primitive_lcm_of_isPrimitive":{"signature":"theorem Polynomial.exists_primitive_lcm_of_isPrimitive {R : Type u_1} [CommRing R] [NormalizedGCDMonoid R] {p q : Polynomial R} (hp : p.IsPrimitive) (hq : q.IsPrimitive) :\u2203 (r : Polynomial R), r.IsPrimitive \u2227 \u2200 (s : Polynomial R), p \u2223 s \u2227 q \u2223 s \u2194 r \u2223 s","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Content.lean#L392-L433"},"Polynomial.primPart_dvd":{"signature":"theorem Polynomial.primPart_dvd {R : Type u_1} [CommRing R] [NormalizedGCDMonoid R] (p : Polynomial R) :p.primPart \u2223 p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Content.lean#L282-L283"},"Polynomial.dvd_iff_content_dvd_content_and_primPart_dvd_primPart":{"signature":"theorem Polynomial.dvd_iff_content_dvd_content_and_primPart_dvd_primPart {R : Type u_1} [CommRing R] [NormalizedGCDMonoid R] {p q : Polynomial R} (hq : q \u2260 0) :p \u2223 q \u2194 p.content \u2223 q.content \u2227 p.primPart \u2223 q.primPart","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Content.lean#L435-L443"},"Polynomial.normalizedGcdMonoid":{"signature":"noncomputable instance Polynomial.normalizedGcdMonoid {R : Type u_1} [CommRing R] [NormalizedGCDMonoid R] :NormalizedGCDMonoid (Polynomial R)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Content.lean#L445-L462"},"Polynomial.derivative_one":{"signature":"theorem Polynomial.derivative_one {R : Type u} [Semiring R] :derivative 1 = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L120-L122"},"Polynomial.derivative_pow_succ":{"signature":"theorem Polynomial.derivative_pow_succ {R : Type u} [CommSemiring R] (p : Polynomial R) (n : \u2115) :derivative (p ^ (n + 1)) = C (\u2191n + 1) * p ^ n * derivative p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L449-L453"},"Polynomial.derivative_pow":{"signature":"theorem Polynomial.derivative_pow {R : Type u} [CommSemiring R] (p : Polynomial R) (n : \u2115) :derivative (p ^ n) = C \u2191n * p ^ (n - 1) * derivative p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L455-L458"},"Polynomial.pow_sub_one_dvd_derivative_of_pow_dvd":{"signature":"theorem Polynomial.pow_sub_one_dvd_derivative_of_pow_dvd {R : Type u} [CommSemiring R] {p q : Polynomial R} {n : \u2115} (dvd : q ^ n \u2223 p) :q ^ (n - 1) \u2223 derivative p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L463-L467"},"Polynomial.pow_sub_dvd_iterate_derivative_of_pow_dvd":{"signature":"theorem Polynomial.pow_sub_dvd_iterate_derivative_of_pow_dvd {R : Type u} [CommSemiring R] {p q : Polynomial R} {n : \u2115} (m : \u2115) (dvd : q ^ n \u2223 p) :q ^ (n - m) \u2223 (\u21d1derivative)^[m] p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L469-L475"},"Polynomial.isRoot_iterate_derivative_of_lt_rootMultiplicity":{"signature":"theorem Polynomial.isRoot_iterate_derivative_of_lt_rootMultiplicity {R : Type u} [CommRing R] {p : Polynomial R} {t : R} {n : \u2115} (hn : n < rootMultiplicity t p) :((\u21d1derivative)^[n] p).IsRoot t","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L66-L69"},"Polynomial.pow_mul_divByMonic_rootMultiplicity_eq":{"signature":"theorem Polynomial.pow_mul_divByMonic_rootMultiplicity_eq {R : Type u} [Ring R] (p : Polynomial R) (a : R) :(X - C a) ^ rootMultiplicity a p * (p /\u2098 (X - C a) ^ rootMultiplicity a p) = p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L551-L557"},"Polynomial.eval_divByMonic_pow_rootMultiplicity_ne_zero":{"signature":"theorem Polynomial.eval_divByMonic_pow_rootMultiplicity_ne_zero {R : Type u} [CommRing R] {p : Polynomial R} (a : R) (hp : p \u2260 0) :eval a (p /\u2098 (X - C a) ^ rootMultiplicity a p) \u2260 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L643-L655"},"Polynomial.derivative_sum":{"signature":"theorem Polynomial.derivative_sum {R : Type u} {\u03b9 : Type y} [Semiring R] {s : Finset \u03b9} {f : \u03b9 \u2192 Polynomial R} :derivative (\u2211 b \u2208 s, f b) = \u2211 b \u2208 s, derivative (f b)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L131-L133"},"Polynomial.derivative_smul":{"signature":"theorem Polynomial.derivative_smul {R : Type u} [Semiring R] {S : Type u_1} [SMulZeroClass S R] [IsScalarTower S R R] (s : S) (p : Polynomial R) :derivative (s \u2022 p) = s \u2022 derivative p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L139-L141"},"Polynomial.iterate_derivative_mul":{"signature":"theorem Polynomial.iterate_derivative_mul {R : Type u} [Semiring R] {n : \u2115} (p q : Polynomial R) :(\u21d1derivative)^[n] (p * q) = \u2211 k \u2208 Finset.range n.succ, n.choose k \u2022 ((\u21d1derivative)^[n - k] p * (\u21d1derivative)^[k] q)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L350-L392"},"Polynomial.eval_finset_sum":{"signature":"theorem Polynomial.eval_finset_sum {R : Type u} {\u03b9 : Type y} [Semiring R] (s : Finset \u03b9) (g : \u03b9 \u2192 Polynomial R) (x : R) :eval x (\u2211 i \u2208 s, g i) = \u2211 i \u2208 s, eval x (g i)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L338-L340"},"Polynomial.derivative_X_add_C":{"signature":"theorem Polynomial.derivative_X_add_C {R : Type u} [Semiring R] (c : R) :derivative (X + C c) = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L128-L129"},"Polynomial.derivative_X_add_C_pow":{"signature":"theorem Polynomial.derivative_X_add_C_pow {R : Type u} [CommSemiring R] (c : R) (m : \u2115) :derivative ((X + C c) ^ m) = C \u2191m * (X + C c) ^ (m - 1)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L505-L507"},"Polynomial.iterate_derivative_X_add_pow":{"signature":"theorem Polynomial.iterate_derivative_X_add_pow {R : Type u} [CommSemiring R] (n k : \u2115) (c : R) :(\u21d1derivative)^[k] ((X + C c) ^ n) = n.descFactorial k \u2022 (X + C c) ^ (n - k)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L512-L520"},"Polynomial.iterate_derivative_X_sub_pow":{"signature":"theorem Polynomial.iterate_derivative_X_sub_pow {R : Type u} [CommRing R] (n k : \u2115) (c : R) :(\u21d1derivative)^[k] ((X - C c) ^ n) = n.descFactorial k \u2022 (X - C c) ^ (n - k)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L671-L673"},"Polynomial.eval\u2082_id":{"signature":"theorem Polynomial.eval\u2082_id {R : Type u} [Semiring R] {p : Polynomial R} {x : R} :eval\u2082 (RingHom.id R) x p = eval x p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L246-L247"},"Polynomial.eval_smul":{"signature":"theorem Polynomial.eval_smul {R : Type u} {S : Type v} [Semiring R] [SMulZeroClass S R] [IsScalarTower S R R] (s : S) (p : Polynomial R) (x : R) :eval x (s \u2022 p) = s \u2022 eval x p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/SMul.lean#L57-L60"},"Polynomial.iterate_derivative_X_sub_pow_self":{"signature":"theorem Polynomial.iterate_derivative_X_sub_pow_self {R : Type u} [CommRing R] (n : \u2115) (c : R) :(\u21d1derivative)^[n] ((X - C c) ^ n) = \u2191n.factorial","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L675-L677"},"Polynomial.C_eq_natCast":{"signature":"theorem Polynomial.C_eq_natCast {R : Type u} [Semiring R] (n : \u2115) :C \u2191n = \u2191n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L469-L470"},"Polynomial.eval_natCast":{"signature":"theorem Polynomial.eval_natCast {R : Type u} [Semiring R] {x : R} {n : \u2115} :eval x \u2191n = \u2191n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L279-L280"},"Polynomial.eval_iterate_derivative_rootMultiplicity":{"signature":"theorem Polynomial.eval_iterate_derivative_rootMultiplicity {R : Type u} [CommRing R] {p : Polynomial R} {t : R} :eval t ((\u21d1derivative)^[rootMultiplicity t p] p) = (rootMultiplicity t p).factorial \u2022 eval t (p /\u2098 (X - C t) ^ rootMultiplicity t p)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L72-L83"},"Polynomial.lt_rootMultiplicity_of_isRoot_iterate_derivative_of_mem_nonZeroDivisors":{"signature":"theorem Polynomial.lt_rootMultiplicity_of_isRoot_iterate_derivative_of_mem_nonZeroDivisors {R : Type u} [CommRing R] {p : Polynomial R} {t : R} {n : \u2115} (h : p \u2260 0) (hroot : \u2200 m \u2264 n, ((\u21d1derivative)^[m] p).IsRoot t) (hnzd : \u2191n.factorial \u2208 nonZeroDivisors R) :n < rootMultiplicity t p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L85-L96"},"Polynomial.lt_rootMultiplicity_iff_isRoot_iterate_derivative_of_mem_nonZeroDivisors":{"signature":"theorem Polynomial.lt_rootMultiplicity_iff_isRoot_iterate_derivative_of_mem_nonZeroDivisors {R : Type u} [CommRing R] {p : Polynomial R} {t : R} {n : \u2115} (h : p \u2260 0) (hnzd : \u2191n.factorial \u2208 nonZeroDivisors R) :n < rootMultiplicity t p \u2194 \u2200 m \u2264 n, ((\u21d1derivative)^[m] p).IsRoot t","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L113-L118"},"Polynomial.one_lt_rootMultiplicity_iff_isRoot_iterate_derivative":{"signature":"theorem Polynomial.one_lt_rootMultiplicity_iff_isRoot_iterate_derivative {R : Type u} [CommRing R] {p : Polynomial R} {t : R} (h : p \u2260 0) :1 < rootMultiplicity t p \u2194 \u2200 m \u2264 1, ((\u21d1derivative)^[m] p).IsRoot t","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L127-L131"},"Polynomial.one_lt_rootMultiplicity_iff_isRoot":{"signature":"theorem Polynomial.one_lt_rootMultiplicity_iff_isRoot {R : Type u} [CommRing R] {p : Polynomial R} {t : R} (h : p \u2260 0) :1 < rootMultiplicity t p \u2194 p.IsRoot t \u2227 (derivative p).IsRoot t","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L133-L139"},"Polynomial.one_lt_rootMultiplicity_iff_isRoot_gcd":{"signature":"theorem Polynomial.one_lt_rootMultiplicity_iff_isRoot_gcd {R : Type u} [CommRing R] [GCDMonoid (Polynomial R)] {p : Polynomial R} {t : R} (h : p \u2260 0) :1 < rootMultiplicity t p \u2194 (gcd p (derivative p)).IsRoot t","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L147-L150"},"Polynomial.isUnit_of_self_mul_dvd_separable":{"signature":"theorem Polynomial.isUnit_of_self_mul_dvd_separable {R : Type u} [CommSemiring R] {p q : Polynomial R} (hp : p.Separable) (hq : q * q \u2223 p) :IsUnit q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Separable.lean#L165-L176"},"Polynomial.emultiplicity_le_one_of_separable":{"signature":"theorem Polynomial.emultiplicity_le_one_of_separable {R : Type u} [CommSemiring R] {p q : Polynomial R} (hq : \u00acIsUnit q) (hsep : p.Separable) :emultiplicity q p \u2264 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Separable.lean#L178-L184"},"Polynomial.rootMultiplicity_le_one_of_separable":{"signature":"theorem Polynomial.rootMultiplicity_le_one_of_separable {R : Type u} [CommRing R] [Nontrivial R] {p : Polynomial R} (hsep : p.Separable) (x : R) :rootMultiplicity x p \u2264 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Separable.lean#L284-L291"},"Polynomial.count_roots_le_one":{"signature":"theorem Polynomial.count_roots_le_one {R : Type u} [CommRing R] [IsDomain R] [DecidableEq R] {p : Polynomial R} (hsep : p.Separable) (x : R) :Multiset.count x p.roots \u2264 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Separable.lean#L299-L302"},"Polynomial.nodup_roots":{"signature":"theorem Polynomial.nodup_roots {R : Type u} [CommRing R] [IsDomain R] {p : Polynomial R} (hsep : p.Separable) :p.roots.Nodup","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Separable.lean#L304-L306"},"Polynomial.nodup_roots_iff_of_splits":{"signature":"theorem Polynomial.nodup_roots_iff_of_splits {F : Type u} [Field F] {f : Polynomial F} (hf : f \u2260 0) (h : f.Splits) :f.roots.Nodup \u2194 f.Separable","doc":"If a non-zero polynomial splits, then it has no repeated roots on that field if and only if it is separable.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Separable.lean#L464-L473"},"Polynomial.nodup_aroots_iff_of_splits":{"signature":"theorem Polynomial.nodup_aroots_iff_of_splits {F : Type u} [Field F] {K : Type v} [Field K] [Algebra F K] {f : Polynomial F} (hf : f \u2260 0) (h : (map (algebraMap F K) f).Splits) :(f.aroots K).Nodup \u2194 f.Separable","doc":"If a non-zero polynomial over F splits in K, then it has no repeated roots on K if and only if it is separable.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Separable.lean#L475-L480"},"Polynomial.card_rootSet_eq_natDegree_iff_of_splits":{"signature":"theorem Polynomial.card_rootSet_eq_natDegree_iff_of_splits {F : Type u} [Field F] {K : Type v} [Field K] [Algebra F K] {f : Polynomial F} (hf : f \u2260 0) (h : (map (algebraMap F K) f).Splits) :Fintype.card \u2191(f.rootSet K) = f.natDegree \u2194 f.Separable","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Separable.lean#L482-L488"},"Polynomial.natSepDegree_eq_natDegree_iff":{"signature":"theorem Polynomial.natSepDegree_eq_natDegree_iff {F : Type u} [Field F] (f : Polynomial F) (hf : f \u2260 0) :f.natSepDegree = f.natDegree \u2194 f.Separable","doc":"The separable degree of a non-zero polynomial is equal to its degree if and only if it is separable.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/SeparableDegree.lean#L347-L354"},"Polynomial.expand":{"signature":"noncomputable def Polynomial.expand (R : Type u) [CommSemiring R] (p : \u2115) :Polynomial R \u2192\u2090[R] Polynomial R","doc":"Expand the polynomial by a factor of p, so \u2211 a\u2099 x\u207f becomes \u2211 a\u2099 x\u207f\u1d56.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Expand.lean#L39-L41"},"Polynomial.IsSeparableContraction":{"signature":"def Polynomial.IsSeparableContraction {F : Type u_1} [CommSemiring F] (q : \u2115) (f g : Polynomial F) :Prop","doc":"A separable contraction of a polynomial f is a separable polynomial g such that g(x^(q^m)) = f(x) for some m : \u2115.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/SeparableDegree.lean#L50-L53"},"Polynomial.HasSeparableContraction":{"signature":"def Polynomial.HasSeparableContraction {F : Type u_1} [CommSemiring F] (q : \u2115) (f : Polynomial F) :Prop","doc":"The condition of having a separable contraction.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/SeparableDegree.lean#L55-L57"},"Polynomial.coeff_derivative":{"signature":"theorem Polynomial.coeff_derivative {R : Type u} [Semiring R] (p : Polynomial R) (n : \u2115) :(derivative p).coeff n = p.coeff (n + 1) * (\u2191n + 1)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L58-L70"},"Polynomial.of_mem_support_derivative":{"signature":"theorem Polynomial.of_mem_support_derivative {R : Type u} [Semiring R] {p : Polynomial R} {n : \u2115} (h : n \u2208 (derivative p).support) :n + 1 \u2208 p.support","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L158-L161"},"Polynomial.degree_derivative_lt":{"signature":"theorem Polynomial.degree_derivative_lt {R : Type u} [Semiring R] {p : Polynomial R} (hp : p \u2260 0) :(derivative p).degree < p.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L163-L166"},"Polynomial.natDegree_derivative_lt":{"signature":"theorem Polynomial.natDegree_derivative_lt {R : Type u} [Semiring R] {p : Polynomial R} (hp : p.natDegree \u2260 0) :(derivative p).natDegree < p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L172-L178"},"Polynomial.derivative_of_natDegree_zero":{"signature":"theorem Polynomial.derivative_of_natDegree_zero {R : Type u} [Semiring R] {p : Polynomial R} (hp : p.natDegree = 0) :derivative p = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L113-L114"},"Polynomial.separable_iff_derivative_ne_zero":{"signature":"theorem Polynomial.separable_iff_derivative_ne_zero {F : Type u} [Field F] {f : Polynomial F} (hf : Irreducible f) :f.Separable \u2194 derivative f \u2260 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Separable.lean#L314-L323"},"Polynomial.mem_support_derivative":{"signature":"theorem Polynomial.mem_support_derivative {R : Type u} [Semiring R] [IsAddTorsionFree R] (p : Polynomial R) (n : \u2115) :n \u2208 (derivative p).support \u2194 n + 1 \u2208 p.support","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L295-L300"},"Polynomial.degree_derivative_eq":{"signature":"theorem Polynomial.degree_derivative_eq {R : Type u} [Semiring R] [IsAddTorsionFree R] (p : Polynomial R) (hp : 0 < p.natDegree) :(derivative p).degree = \u2191(p.natDegree - 1)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L302-L317"},"Polynomial.expand_C":{"signature":"theorem Polynomial.expand_C {R : Type u} [CommSemiring R] (p : \u2115) (r : R) :(expand R p) (C r) = C r","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Expand.lean#L53-L55"},"Polynomial.expand_X":{"signature":"theorem Polynomial.expand_X {R : Type u} [CommSemiring R] (p : \u2115) :(expand R p) X = X ^ p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Expand.lean#L57-L59"},"Polynomial.expand_one":{"signature":"theorem Polynomial.expand_one {R : Type u} [CommSemiring R] (f : Polynomial R) :(expand R 1) f = f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Expand.lean#L76-L80"},"Polynomial.natDegree_eq_zero_of_derivative_eq_zero":{"signature":"theorem Polynomial.natDegree_eq_zero_of_derivative_eq_zero {R : Type u} [Semiring R] [IsAddTorsionFree R] {f : Polynomial R} (h : derivative f = 0) :f.natDegree = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L226-L240"},"Polynomial.not_irreducible_C":{"signature":"theorem Polynomial.not_irreducible_C {R : Type u} [Field R] (x : R) :\u00acIrreducible (C x)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L619-L623"},"Polynomial.degree_pos_of_irreducible":{"signature":"theorem Polynomial.degree_pos_of_irreducible {R : Type u} [Field R] {p : Polynomial R} (hp : Irreducible p) :0 < p.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L625-L628"},"Polynomial.contract":{"signature":"noncomputable def Polynomial.contract {R : Type u} [CommSemiring R] (p : \u2115) (f : Polynomial R) :Polynomial R","doc":"The opposite of expand: sends \u2211 a\u2099 x\u207f\u1d56 to \u2211 a\u2099 x\u207f.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Expand.lean#L182-L184"},"Polynomial.eq_C_of_degree_eq_zero":{"signature":"theorem Polynomial.eq_C_of_degree_eq_zero {R : Type u} [Semiring R] {p : Polynomial R} (h : p.degree = 0) :p = C (p.coeff 0)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L169-L170"},"Polynomial.expand_eq_sum":{"signature":"theorem Polynomial.expand_eq_sum {R : Type u} [CommSemiring R] (p : \u2115) {f : Polynomial R} :(expand R p) f = f.sum fun (e : \u2115) (a : R) => C a * (X ^ p) ^ e","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Expand.lean#L50-L51"},"Polynomial.coeff_expand":{"signature":"theorem Polynomial.coeff_expand {R : Type u} [CommSemiring R] {p : \u2115} (hp : 0 < p) (f : Polynomial R) (n : \u2115) :((expand R p) f).coeff n = if p \u2223 n then f.coeff (n / p) else 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Expand.lean#L90-L107"},"Polynomial.coeff_expand_mul":{"signature":"theorem Polynomial.coeff_expand_mul {R : Type u} [CommSemiring R] {p : \u2115} (hp : 0 < p) (f : Polynomial R) (n : \u2115) :((expand R p) f).coeff (n * p) = f.coeff n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Expand.lean#L109-L112"},"Polynomial.expand_injective":{"signature":"theorem Polynomial.expand_injective {R : Type u} [CommSemiring R] {n : \u2115} (hn : 0 < n) :Function.Injective \u21d1(expand R n)","doc":"Expansion is injective.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Expand.lean#L118-L120"},"Polynomial.expand_inj":{"signature":"theorem Polynomial.expand_inj {R : Type u} [CommSemiring R] {p : \u2115} (hp : 0 < p) {f g : Polynomial R} :(expand R p) f = (expand R p) g \u2194 f = g","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Expand.lean#L122-L123"},"Polynomial.expand_eq_C":{"signature":"theorem Polynomial.expand_eq_C {R : Type u} [CommSemiring R] {p : \u2115} (hp : 0 < p) {f : Polynomial R} {r : R} :(expand R p) f = C r \u2194 f = C r","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Expand.lean#L131-L132"},"Polynomial.isLocalHom_expand":{"signature":"theorem Polynomial.isLocalHom_expand (R : Type u) [CommRing R] [IsDomain R] {p : \u2115} (hp : 0 < p) :IsLocalHom (expand R p)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Expand.lean#L312-L316"},"Polynomial.coeff_contract":{"signature":"theorem Polynomial.coeff_contract {R : Type u} [CommSemiring R] {p : \u2115} (hp : p \u2260 0) (f : Polynomial R) (n : \u2115) :(contract p f).coeff n = f.coeff (n * p)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Expand.lean#L186-L195"},"Polynomial.expand_contract":{"signature":"theorem Polynomial.expand_contract {R : Type u} [CommSemiring R] (p : \u2115) [CharP R p] [NoZeroDivisors R] {f : Polynomial R} (hf : derivative f = 0) (hp : p \u2260 0) :(expand R p) (contract p f) = f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Expand.lean#L238-L252"},"Polynomial.separable_or":{"signature":"theorem Polynomial.separable_or {F : Type u} [Field F] (p : \u2115) [HF : CharP F p] {f : Polynomial F} (hf : Irreducible f) :f.Separable \u2228 \u00acf.Separable \u2227 \u2203 (g : Polynomial F), Irreducible g \u2227 (expand F p) g = f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Separable.lean#L352-L367"},"Polynomial.separable_C":{"signature":"theorem Polynomial.separable_C {R : Type u} [CommSemiring R] (r : R) :(C r).Separable \u2194 IsUnit r","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Separable.lean#L81-L82"},"Polynomial.degree_le_zero_iff":{"signature":"theorem Polynomial.degree_le_zero_iff {R : Type u} [Semiring R] {p : Polynomial R} :p.degree \u2264 0 \u2194 p = C (p.coeff 0)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L172-L173"},"Polynomial.coe_expand":{"signature":"theorem Polynomial.coe_expand (R : Type u) [CommSemiring R] (p : \u2115) :\u21d1(expand R p) = eval\u2082 C (X ^ p)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Expand.lean#L43-L44"},"Polynomial.eval\u2082_hom":{"signature":"theorem Polynomial.eval\u2082_hom {R : Type u} {S : Type v} [Semiring R] {p : Polynomial R} [Semiring S] (f : R \u2192+* S) (x : R) :eval\u2082 f (f x) p = f (eval x p)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Coeff.lean#L187-L188"},"Polynomial.expand_eq_zero":{"signature":"theorem Polynomial.expand_eq_zero {R : Type u} [CommSemiring R] {p : \u2115} (hp : 0 < p) {f : Polynomial R} :(expand R p) f = 0 \u2194 f = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Expand.lean#L125-L126"},"Polynomial.natDegree_expand":{"signature":"theorem Polynomial.natDegree_expand {R : Type u} [CommSemiring R] (p : \u2115) (f : Polynomial R) :((expand R p) f).natDegree = f.natDegree * p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Expand.lean#L134-L152"},"Polynomial.expand_expand":{"signature":"theorem Polynomial.expand_expand {R : Type u} [CommSemiring R] (p q : \u2115) (f : Polynomial R) :(expand R p) ((expand R q) f) = (expand R (p * q)) f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Expand.lean#L65-L68"},"Polynomial.exists_separable_of_irreducible":{"signature":"theorem Polynomial.exists_separable_of_irreducible {F : Type u} [Field F] (p : \u2115) [HF : CharP F p] {f : Polynomial F} (hf : Irreducible f) (hp : p \u2260 0) :\u2203 (n : \u2115) (g : Polynomial F), g.Separable \u2227 (expand F (p ^ n)) g = f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Separable.lean#L369-L392"},"Polynomial.eval\u2082_at_one":{"signature":"theorem Polynomial.eval\u2082_at_one {R : Type u} [Semiring R] {p : Polynomial R} {S : Type u_1} [Semiring S] (f : R \u2192+* S) :eval\u2082 f 1 p = f (eval 1 p)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L259-L262"},"Polynomial.expand_zero":{"signature":"theorem Polynomial.expand_zero {R : Type u} [CommSemiring R] (f : Polynomial R) :(expand R 0) f = C (eval 1 f)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Expand.lean#L73-L74"},"Polynomial.eval_one_map":{"signature":"theorem Polynomial.eval_one_map {R : Type u} {S : Type v} [Semiring R] [Semiring S] (f : R \u2192+* S) (p : Polynomial R) :eval 1 (map f p) = f (eval 1 p)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Coeff.lean#L146-L150"},"Polynomial.map_expand":{"signature":"theorem Polynomial.map_expand {R : Type u} [CommSemiring R] {S : Type v} [CommSemiring S] {p : \u2115} {f : R \u2192+* S} {q : Polynomial R} :map f ((expand R p) q) = (expand S p) (map f q)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Expand.lean#L163-L169"},"Polynomial.coeff_natCast_ite":{"signature":"theorem Polynomial.coeff_natCast_ite {R : Type u} {m n : \u2115} [Semiring R] :(\u2191m).coeff n = \u2191(if n = 0 then m else 0)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L645-L647"},"Polynomial.natCast_inj":{"signature":"theorem Polynomial.natCast_inj {m n : \u2115} {R : Type u_1} [Semiring R] [CharZero R] :\u2191m = \u2191n \u2194 m = n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L345-L353"},"Polynomial.charZero":{"signature":"instance Polynomial.charZero {R : Type u} [Semiring R] [CharZero R] :CharZero (Polynomial R)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L370-L370"},"Polynomial.instCharP":{"signature":"instance Polynomial.instCharP {R : Type u} [Semiring R] (p : \u2115) [h : CharP R p] :CharP (Polynomial R) p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Basic.lean#L43-L45"},"Polynomial.instExpChar":{"signature":"instance Polynomial.instExpChar {R : Type u} [Semiring R] (p : \u2115) [h : ExpChar R p] :ExpChar (Polynomial R) p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Basic.lean#L47-L48"},"Polynomial.expand_ne_zero":{"signature":"theorem Polynomial.expand_ne_zero {R : Type u} [CommSemiring R] {p : \u2115} (hp : 0 < p) {f : Polynomial R} :(expand R p) f \u2260 0 \u2194 f \u2260 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Expand.lean#L128-L129"},"Polynomial.expand_eq_comp_X_pow":{"signature":"theorem Polynomial.expand_eq_comp_X_pow {R : Type u} [CommSemiring R] (p : \u2115) {f : Polynomial R} :(expand R p) f = f.comp (X ^ p)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Expand.lean#L48-L48"},"Polynomial.eval_comp":{"signature":"theorem Polynomial.eval_comp {R : Type u} [CommSemiring R] {p q : Polynomial R} {x : R} :eval x (p.comp q) = eval (eval x q) p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L615-L619"},"Polynomial.isRoot_comp":{"signature":"theorem Polynomial.isRoot_comp {R : Type u_1} [CommSemiring R] {p q : Polynomial R} {r : R} :(p.comp q).IsRoot r \u2194 p.IsRoot (eval r q)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L621-L622"},"Polynomial.rootMultiplicity_expand_pow":{"signature":"theorem Polynomial.rootMultiplicity_expand_pow {R : Type u} [CommRing R] {p n : \u2115} [ExpChar R p] {f : Polynomial R} {r : R} :rootMultiplicity r ((expand R (p ^ n)) f) = p ^ n * rootMultiplicity (r ^ p ^ n) f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Expand.lean#L293-L300"},"Polynomial.roots_expand_pow":{"signature":"theorem Polynomial.roots_expand_pow {R : Type u_1} [CommRing R] [IsDomain R] {p n : \u2115} [ExpChar R p] {f : Polynomial R} [PerfectRing R p] :((expand R (p ^ n)) f).roots = p ^ n \u2022 Multiset.map (\u21d1(iterateFrobeniusEquiv R p n).symm) f.roots","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Perfect.lean#L342-L348"},"Polynomial.roots_expand_pow_map_iterateFrobenius":{"signature":"theorem Polynomial.roots_expand_pow_map_iterateFrobenius {R : Type u_1} [CommRing R] [IsDomain R] {p n : \u2115} [ExpChar R p] {f : Polynomial R} [PerfectRing R p] :Multiset.map (\u21d1(iterateFrobenius R p n)) ((expand R (p ^ n)) f).roots = p ^ n \u2022 f.roots","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Perfect.lean#L373-L376"},"Polynomial.roots_expand_image_iterateFrobenius":{"signature":"theorem Polynomial.roots_expand_image_iterateFrobenius {R : Type u_1} [CommRing R] [IsDomain R] {p n : \u2115} [ExpChar R p] {f : Polynomial R} [PerfectRing R p] [DecidableEq R] :Finset.image (\u21d1(iterateFrobenius R p n)) ((expand R (p ^ n)) f).roots.toFinset = f.roots.toFinset","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Perfect.lean#L381-L384"},"Polynomial.rootsExpandPowEquivRoots":{"signature":"noncomputable def Polynomial.rootsExpandPowEquivRoots {R : Type u_1} [CommRing R] [IsDomain R] (p : \u2115) [ExpChar R p] (f : Polynomial R) [DecidableEq R] [PerfectRing R p] (n : \u2115) :\u21a5((expand R (p ^ n)) f).roots.toFinset \u2243 \u21a5f.roots.toFinset","doc":"If f is a polynomial over a perfect integral domain R of characteristic p, then there is a bijection from the set of roots of Polynomial.expand R (p ^ n) f to the set of roots of f. It's given by x \u21a6 x ^ (p ^ n), see rootsExpandPowEquivRoots_apply.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Perfect.lean#L432-L441"},"Polynomial.eval\u2082_dvd":{"signature":"theorem Polynomial.eval\u2082_dvd {R : Type u} {S : Type v} [Semiring R] {p q : Polynomial R} [CommSemiring S] (f : R \u2192+* S) (x : S) :p \u2223 q \u2192 eval\u2082 f x p \u2223 eval\u2082 f x q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L223-L225"},"Polynomial.eval\u2082_eq_zero_of_dvd_of_eval\u2082_eq_zero":{"signature":"theorem Polynomial.eval\u2082_eq_zero_of_dvd_of_eval\u2082_eq_zero {R : Type u} {S : Type v} [Semiring R] {p q : Polynomial R} [CommSemiring S] (f : R \u2192+* S) (x : S) (h : p \u2223 q) (h0 : eval\u2082 f x p = 0) :eval\u2082 f x q = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L227-L229"},"Polynomial.natDegree_X_pow_sub_C":{"signature":"theorem Polynomial.natDegree_X_pow_sub_C {R : Type u} [Ring R] [Nontrivial R] {n : \u2115} {r : R} :(X ^ n - C r).natDegree = n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L782-L783"},"Polynomial.eq_of_dvd_of_natDegree_le_of_leadingCoeff":{"signature":"theorem Polynomial.eq_of_dvd_of_natDegree_le_of_leadingCoeff {R : Type u} [CommRing R] [IsDomain R] {p q : Polynomial R} (hpq : p \u2223 q) (h\u2081 : q.natDegree \u2264 p.natDegree) (h\u2082 : p.leadingCoeff = q.leadingCoeff) :p = q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L806-L817"},"Polynomial.natDegree_mul_C":{"signature":"theorem Polynomial.natDegree_mul_C {R : Type u} [Semiring R] {p : Polynomial R} {a : R} [NoZeroDivisors R] (a0 : a \u2260 0) :(p * C a).natDegree = p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L377-L378"},"Polynomial.associated_of_dvd_of_natDegree_le_of_leadingCoeff":{"signature":"theorem Polynomial.associated_of_dvd_of_natDegree_le_of_leadingCoeff {R : Type u} [CommRing R] [IsDomain R] {p q : Polynomial R} (hpq : p \u2223 q) (h\u2081 : q.natDegree \u2264 p.natDegree) (h\u2082 : q.leadingCoeff \u2223 p.leadingCoeff) :Associated p q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L819-L825"},"Polynomial.associated_of_dvd_of_natDegree_le":{"signature":"theorem Polynomial.associated_of_dvd_of_natDegree_le {K : Type u_1} [Field K] {p q : Polynomial K} (hpq : p \u2223 q) (hq : q \u2260 0) (h\u2081 : q.natDegree \u2264 p.natDegree) :Associated p q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L827-L830"},"Polynomial.derivative_sub":{"signature":"theorem Polynomial.derivative_sub {R : Type u} [Ring R] {f g : Polynomial R} :derivative (f - g) = derivative f - derivative g","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L615-L617"},"Polynomial.derivative_C_mul_X_pow":{"signature":"theorem Polynomial.derivative_C_mul_X_pow {R : Type u} [Semiring R] (a : R) (n : \u2115) :derivative (C a * X ^ n) = C (a * \u2191n) * X ^ (n - 1)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L92-L94"},"Polynomial.derivative_X_pow":{"signature":"theorem Polynomial.derivative_X_pow {R : Type u} [Semiring R] (n : \u2115) :derivative (X ^ n) = C \u2191n * X ^ (n - 1)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L99-L100"},"Polynomial.expand_monomial":{"signature":"theorem Polynomial.expand_monomial {R : Type u} [CommSemiring R] (p q : \u2115) (r : R) :(expand R p) ((monomial q) r) = (monomial (q * p)) r","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Expand.lean#L61-L63"},"Polynomial.map_frobenius_expand":{"signature":"theorem Polynomial.map_frobenius_expand {R : Type u} [CommSemiring R] (p : \u2115) [ExpChar R p] (f : Polynomial R) :map (frobenius R p) ((expand R p) f) = f ^ p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Expand.lean#L262-L267"},"Polynomial.natSepDegree_expand":{"signature":"theorem Polynomial.natSepDegree_expand {F : Type u} [Field F] (f : Polynomial F) (q : \u2115) [hF : ExpChar F q] {n : \u2115} :((expand F (q ^ n)) f).natSepDegree = f.natSepDegree","doc":"If a field F is of exponential characteristic q, then Polynomial.expand F (q ^ n) f and f have the same separable degree.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/SeparableDegree.lean#L474-L484"},"Polynomial.natSepDegree_eq_natDegree_of_separable":{"signature":"theorem Polynomial.natSepDegree_eq_natDegree_of_separable {F : Type u} [Field F] (f : Polynomial F) (h : f.Separable) :f.natSepDegree = f.natDegree","doc":"If a polynomial is separable, then its separable degree is equal to its degree.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/SeparableDegree.lean#L356-L358"},"Polynomial.mem_roots_map":{"signature":"theorem Polynomial.mem_roots_map {R : Type u} {k : Type y} [Field R] {p : Polynomial R} [CommRing k] [IsDomain k] {f : R \u2192+* k} {x : k} (hp : p \u2260 0) :x \u2208 (map f p).roots \u2194 eval\u2082 f x p = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L496-L498"},"Polynomial.aeval_algebraMap_eq_zero_iff":{"signature":"theorem Polynomial.aeval_algebraMap_eq_zero_iff {R : Type u_1} {A : Type u_2} (B : Type u_3) [CommSemiring R] [CommSemiring A] [Semiring B] [Algebra R A] [Algebra A B] [Algebra R B] [IsScalarTower R A B] [IsDomain A] [Module.IsTorsionFree A B] [Nontrivial B] (x : A) (p : Polynomial R) :(aeval ((algebraMap A B) x)) p = 0 \u2194 (aeval x) p = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Tower.lean#L54-L58"},"Polynomial.not_isUnit_of_degree_pos":{"signature":"theorem Polynomial.not_isUnit_of_degree_pos {R : Type u} [Semiring R] [NoZeroDivisors R] (p : Polynomial R) (hpl : 0 < p.degree) :\u00acIsUnit p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Units.lean#L55-L59"},"Polynomial.not_isUnit_of_natDegree_pos":{"signature":"theorem Polynomial.not_isUnit_of_natDegree_pos {R : Type u} [Semiring R] [NoZeroDivisors R] (p : Polynomial R) (hpl : 0 < p.natDegree) :\u00acIsUnit p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Units.lean#L61-L62"},"Polynomial.leadingCoeffHom":{"signature":"def Polynomial.leadingCoeffHom {R : Type u} [Semiring R] [NoZeroDivisors R] :Polynomial R \u2192* R","doc":"Polynomial.leadingCoeff bundled as a MonoidHom when R has NoZeroDivisors, and thus leadingCoeff is multiplicative","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L701-L706"},"Polynomial.leadingCoeff_pow":{"signature":"theorem Polynomial.leadingCoeff_pow {R : Type u} [Semiring R] [NoZeroDivisors R] (p : Polynomial R) (n : \u2115) :(p ^ n).leadingCoeff = p.leadingCoeff ^ n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L712-L714"},"Polynomial.natDegree_pow":{"signature":"theorem Polynomial.natDegree_pow {R : Type u} [Semiring R] [NoZeroDivisors R] (p : Polynomial R) (n : \u2115) :(p ^ n).natDegree = n * p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Domain.lean#L53-L59"},"Polynomial.separable_of_subsingleton":{"signature":"theorem Polynomial.separable_of_subsingleton {R : Type u} [CommSemiring R] [Subsingleton R] (f : Polynomial R) :f.Separable","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Separable.lean#L69-L71"},"Polynomial.separable_def'":{"signature":"theorem Polynomial.separable_def' {R : Type u} [CommSemiring R] (f : Polynomial R) :f.Separable \u2194 \u2203 (a : Polynomial R) (b : Polynomial R), a * f + b * derivative f = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Separable.lean#L55-L56"},"Polynomial.separable_X_pow_sub_C_unit":{"signature":"theorem Polynomial.separable_X_pow_sub_C_unit {R : Type u} [CommRing R] {n : \u2115} (u : R\u02e3) (hn : IsUnit \u2191n) :(X ^ n - C \u2191u).Separable","doc":"If IsUnit n in a CommRing R, then X ^ n - u is separable for any unit u.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Separable.lean#L246-L263"},"Polynomial.separable_X_pow_sub_C":{"signature":"theorem Polynomial.separable_X_pow_sub_C {F : Type u} [Field F] {n : \u2115} (a : F) (hn : \u2191n \u2260 0) (ha : a \u2260 0) :(X ^ n - C a).Separable","doc":"If n \u2260 0 in F, then X ^ n - a is separable for any a \u2260 0.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Separable.lean#L428-L431"},"Polynomial.X_pow_sub_C_separable_iff":{"signature":"theorem Polynomial.X_pow_sub_C_separable_iff {F : Type u} [Field F] {n : \u2115} {x : F} (hn : 0 < n) (hx : x \u2260 0) :(X ^ n - C x).Separable \u2194 \u2191n \u2260 0","doc":"In a field F, for any t \u2208 F and n > 0, the polynomial X ^ n - t is separable iff \u2191n \u2260 0. The assumption n > 0 is needed, since for n = 0 the polynomial X ^ n - t is separable iff t \u2260 1.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Separable.lean#L438-L445"},"Polynomial.X_pow_sub_one_separable_iff":{"signature":"theorem Polynomial.X_pow_sub_one_separable_iff {F : Type u} [Field F] {n : \u2115} :(X ^ n - 1).Separable \u2194 \u2191n \u2260 0","doc":"In a field F, X ^ n - 1 is separable iff \u2191n \u2260 0.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Separable.lean#L449-L453"},"Polynomial.eq_X_add_C_of_degree_eq_one":{"signature":"theorem Polynomial.eq_X_add_C_of_degree_eq_one {R : Type u} [Semiring R] {p : Polynomial R} (h : p.degree = 1) :p = C p.leadingCoeff * X + C (p.coeff 0)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/SmallDegree.lean#L38-L41"},"Polynomial.natDegree_X_add_C":{"signature":"theorem Polynomial.natDegree_X_add_C {R : Type u} [Nontrivial R] [Semiring R] (x : R) :(X + C x).natDegree = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L608-L609"},"Polynomial.not_isUnit_X_add_C":{"signature":"theorem Polynomial.not_isUnit_X_add_C {R : Type u} [Semiring R] [Nontrivial R] (a : R) :\u00acIsUnit (X + C a)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Monic.lean#L165-L169"},"Polynomial.aeval_eq_sum_range":{"signature":"theorem Polynomial.aeval_eq_sum_range {R : Type u} {S : Type v} [CommSemiring R] [Semiring S] [Algebra R S] {p : Polynomial R} (x : S) :(aeval x) p = \u2211 i \u2208 Finset.range (p.natDegree + 1), p.coeff i \u2022 x ^ i","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L492-L495"},"Polynomial.lifts":{"signature":"def Polynomial.lifts {R : Type u} [Semiring R] {S : Type v} [Semiring S] (f : R \u2192+* S) :Subsemiring (Polynomial S)","doc":"We define the subsemiring of polynomials that lifts as the image of RingHom.of (map f).","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Lifts.lean#L59-L61"},"Polynomial.lifts_iff_ringHom_rangeS":{"signature":"theorem Polynomial.lifts_iff_ringHom_rangeS {R : Type u} [Semiring R] {S : Type v} [Semiring S] {f : R \u2192+* S} (p : Polynomial S) :p \u2208 lifts f \u2194 p \u2208 (mapRingHom f).rangeS","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Lifts.lean#L69-L70"},"Polynomial.coe_mapRingHom":{"signature":"theorem Polynomial.coe_mapRingHom {R : Type u} {S : Type v} [Semiring R] [Semiring S] (f : R \u2192+* S) :\u21d1(mapRingHom f) = map f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L529-L531"},"Polynomial.as_sum_range_C_mul_X_pow'":{"signature":"theorem Polynomial.as_sum_range_C_mul_X_pow' {R : Type u} [Semiring R] (p : Polynomial R) {n : \u2115} (hn : p.natDegree < n) :p = \u2211 i \u2208 Finset.range n, C (p.coeff i) * X ^ i","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Support.lean#L93-L95"},"Polynomial.as_sum_range_C_mul_X_pow":{"signature":"theorem Polynomial.as_sum_range_C_mul_X_pow {R : Type u} [Semiring R] (p : Polynomial R) :p = \u2211 i \u2208 Finset.range (p.natDegree + 1), C (p.coeff i) * X ^ i","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Support.lean#L97-L99"},"Polynomial.mem_map_rangeS":{"signature":"theorem Polynomial.mem_map_rangeS {R : Type u} {S : Type v} [Semiring R] [Semiring S] (f : R \u2192+* S) {p : Polynomial S} :p \u2208 (mapRingHom f).rangeS \u2194 \u2200 (n : \u2115), p.coeff n \u2208 f.rangeS","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Subring.lean#L33-L44"},"Polynomial.lifts_iff_coeff_lifts":{"signature":"theorem Polynomial.lifts_iff_coeff_lifts {R : Type u} [Semiring R] {S : Type v} [Semiring S] {f : R \u2192+* S} (p : Polynomial S) :p \u2208 lifts f \u2194 \u2200 (n : \u2115), p.coeff n \u2208 Set.range \u21d1f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Lifts.lean#L72-L74"},"Polynomial.lifts_and_degree_eq_and_monic":{"signature":"theorem Polynomial.lifts_and_degree_eq_and_monic {R : Type u} [Semiring R] {S : Type v} [Semiring S] {f : R \u2192+* S} [Nontrivial S] {p : Polynomial S} (hlifts : p \u2208 lifts f) (hp : p.Monic) :\u2203 (q : Polynomial R), map f q = p \u2227 q.degree = p.degree \u2227 q.Monic","doc":"A monic polynomial lifts if and only if it can be lifted to a monic polynomial of the same degree.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Lifts.lean#L162-L174"},"Polynomial.lifts_and_natDegree_eq_and_monic":{"signature":"theorem Polynomial.lifts_and_natDegree_eq_and_monic {R : Type u} [Semiring R] {S : Type v} [Semiring S] {f : R \u2192+* S} {p : Polynomial S} (hlifts : p \u2208 lifts f) (hp : p.Monic) :\u2203 (q : Polynomial R), map f q = p \u2227 q.natDegree = p.natDegree \u2227 q.Monic","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Lifts.lean#L176-L182"},"Polynomial.smul_eq_map":{"signature":"theorem Polynomial.smul_eq_map {M : Type u_1} [Monoid M] (R : Type u_2) [Semiring R] [MulSemiringAction M R] (m : M) :HSMul.hSMul m = map (MulSemiringAction.toRingHom M R m)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/GroupRingAction.lean#L35-L38"},"Polynomial.smul_X":{"signature":"theorem Polynomial.smul_X {M : Type u_1} [Monoid M] {R : Type u_2} [Semiring R] [MulSemiringAction M R] (m : M) :m \u2022 X = X","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/GroupRingAction.lean#L50-L52"},"Polynomial.instMulSemiringAction":{"signature":"noncomputable instance Polynomial.instMulSemiringAction (M : Type u_1) [Monoid M] (R : Type u_2) [Semiring R] [MulSemiringAction M R] :MulSemiringAction M (Polynomial R)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/GroupRingAction.lean#L40-L45"},"Polynomial.algebraMap_apply":{"signature":"theorem Polynomial.algebraMap_apply {R : Type u} {A : Type z} [CommSemiring R] [Semiring A] [Algebra R A] (r : R) :(algebraMap R (Polynomial A)) r = C ((algebraMap R A) r)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L68-L70"},"Polynomial.coeff_monomial_same":{"signature":"theorem Polynomial.coeff_monomial_same {R : Type u} [Semiring R] (n : \u2115) (c : R) :((monomial n) c).coeff n = c","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L586-L588"},"Polynomial.as_sum_support":{"signature":"theorem Polynomial.as_sum_support {R : Type u} [Semiring R] (p : Polynomial R) :p = \u2211 i \u2208 p.support, (monomial i) (p.coeff i)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Support.lean#L55-L56"},"Polynomial.scaleRoots":{"signature":"noncomputable def Polynomial.scaleRoots {R : Type u_1} [Semiring R] (p : Polynomial R) (s : R) :Polynomial R","doc":"scaleRoots p s is a polynomial with root r * s for each root r of p.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ScaleRoots.lean#L28-L30"},"Polynomial.coeff_scaleRoots":{"signature":"theorem Polynomial.coeff_scaleRoots {R : Type u_1} [Semiring R] (p : Polynomial R) (s : R) (i : \u2115) :(p.scaleRoots s).coeff i = p.coeff i * s ^ (p.natDegree - i)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ScaleRoots.lean#L32-L35"},"Polynomial.coeff_scaleRoots_natDegree":{"signature":"theorem Polynomial.coeff_scaleRoots_natDegree {R : Type u_1} [Semiring R] (p : Polynomial R) (s : R) :(p.scaleRoots s).coeff p.natDegree = p.leadingCoeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ScaleRoots.lean#L37-L39"},"Polynomial.dvd_term_of_dvd_eval_of_dvd_terms":{"signature":"theorem Polynomial.dvd_term_of_dvd_eval_of_dvd_terms {S : Type v} [CommRing S] {z p : S} {f : Polynomial S} (i : \u2115) (dvd_eval : p \u2223 eval z f) (dvd_terms : \u2200 (j : \u2115), j \u2260 i \u2192 p \u2223 f.coeff j * z ^ j) :p \u2223 f.coeff i * z ^ i","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L583-L594"},"Polynomial.dvd_term_of_isRoot_of_dvd_terms":{"signature":"theorem Polynomial.dvd_term_of_isRoot_of_dvd_terms {S : Type v} [CommRing S] {r p : S} {f : Polynomial S} (i : \u2115) (hr : f.IsRoot r) (h : \u2200 (j : \u2115), j \u2260 i \u2192 p \u2223 f.coeff j * r ^ j) :p \u2223 f.coeff i * r ^ i","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L596-L598"},"Polynomial.isRoot_of_eval\u2082_map_eq_zero":{"signature":"theorem Polynomial.isRoot_of_eval\u2082_map_eq_zero {R : Type u} {S : Type v} [CommSemiring R] {p : Polynomial R} [Semiring S] {f : R \u2192+* S} (hf : Function.Injective \u21d1f) {r : R} :eval\u2082 f (f r) p = 0 \u2192 p.IsRoot r","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L502-L506"},"Polynomial.support_scaleRoots_le":{"signature":"theorem Polynomial.support_scaleRoots_le {R : Type u_1} [Semiring R] (p : Polynomial R) (s : R) :(p.scaleRoots s).support \u2264 p.support","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ScaleRoots.lean#L54-L56"},"Polynomial.scaleRoots_eval\u2082_mul_of_commute":{"signature":"theorem Polynomial.scaleRoots_eval\u2082_mul_of_commute {S : Type u_2} {A : Type u_3} [Semiring S] [Semiring A] {p : Polynomial S} (f : S \u2192+* A) (a : A) (s : S) (hsa : Commute (f s) a) (hf : \u2200 (s\u2081 s\u2082 : S), Commute (f s\u2081) (f s\u2082)) :eval\u2082 f (f s * a) (p.scaleRoots s) = f s ^ p.natDegree * eval\u2082 f a p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ScaleRoots.lean#L130-L148"},"Polynomial.scaleRoots_eval\u2082_mul":{"signature":"theorem Polynomial.scaleRoots_eval\u2082_mul {R : Type u_1} {S : Type u_2} [Semiring S] [CommSemiring R] {p : Polynomial S} (f : S \u2192+* R) (r : R) (s : S) :eval\u2082 f (f s * r) (p.scaleRoots s) = f s ^ p.natDegree * eval\u2082 f r p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ScaleRoots.lean#L150-L152"},"Polynomial.scaleRoots_eval\u2082_eq_zero":{"signature":"theorem Polynomial.scaleRoots_eval\u2082_eq_zero {R : Type u_1} {S : Type u_2} [Semiring S] [CommSemiring R] {p : Polynomial S} (f : S \u2192+* R) {r : R} {s : S} (hr : eval\u2082 f r p = 0) :eval\u2082 f (f s * r) (p.scaleRoots s) = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ScaleRoots.lean#L154-L155"},"Polynomial.zero_scaleRoots":{"signature":"theorem Polynomial.zero_scaleRoots {R : Type u_1} [Semiring R] (s : R) :scaleRoots 0 s = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ScaleRoots.lean#L41-L44"},"Polynomial.degree_scaleRoots":{"signature":"theorem Polynomial.degree_scaleRoots {R : Type u_1} [Semiring R] (p : Polynomial R) {s : R} :(p.scaleRoots s).degree = p.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ScaleRoots.lean#L67-L76"},"Polynomial.natDegree_scaleRoots":{"signature":"theorem Polynomial.natDegree_scaleRoots {R : Type u_1} [Semiring R] (p : Polynomial R) (s : R) :(p.scaleRoots s).natDegree = p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ScaleRoots.lean#L78-L80"},"Polynomial.leadingCoeff_X":{"signature":"theorem Polynomial.leadingCoeff_X {R : Type u} [Semiring R] :X.leadingCoeff = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L437-L438"},"Polynomial.monic_X":{"signature":"theorem Polynomial.monic_X {R : Type u} [Semiring R] :X.Monic","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L444-L446"},"Polynomial.leadingCoeff_scaleRoots":{"signature":"theorem Polynomial.leadingCoeff_scaleRoots {R : Type u_1} [Semiring R] (p : Polynomial R) (r : R) :(p.scaleRoots r).leadingCoeff = p.leadingCoeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ScaleRoots.lean#L82-L85"},"Polynomial.monic_scaleRoots_iff":{"signature":"theorem Polynomial.monic_scaleRoots_iff {R : Type u_1} [Semiring R] {p : Polynomial R} (s : R) :(p.scaleRoots s).Monic \u2194 p.Monic","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ScaleRoots.lean#L87-L88"},"Polynomial.coeff_X_one":{"signature":"theorem Polynomial.coeff_X_one {R : Type u} [Semiring R] :X.coeff 1 = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L606-L608"},"Polynomial.X_ne_zero":{"signature":"theorem Polynomial.X_ne_zero {R : Type u} [Semiring R] [Nontrivial R] :X \u2260 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L1144-L1146"},"Polynomial.separable_gcd_right":{"signature":"theorem Polynomial.separable_gcd_right {F : Type u_1} [Field F] [DecidableEq (Polynomial F)] {g : Polynomial F} (f : Polynomial F) (hg : g.Separable) :(EuclideanDomain.gcd f g).Separable","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Separable.lean#L101-L104"},"Polynomial.eval\u2082_gcd_eq_zero":{"signature":"theorem Polynomial.eval\u2082_gcd_eq_zero {R : Type u} {k : Type y} [Field R] [CommSemiring k] [DecidableEq R] {\u03d5 : R \u2192+* k} {f g : Polynomial R} {\u03b1 : k} (hf : eval\u2082 \u03d5 \u03b1 f = 0) (hg : eval\u2082 \u03d5 \u03b1 g = 0) :eval\u2082 \u03d5 \u03b1 (EuclideanDomain.gcd f g) = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L463-L467"},"Polynomial.eval_gcd_eq_zero":{"signature":"theorem Polynomial.eval_gcd_eq_zero {R : Type u} [Field R] [DecidableEq R] {f g : Polynomial R} {\u03b1 : R} (hf : eval \u03b1 f = 0) (hg : eval \u03b1 g = 0) :eval \u03b1 (EuclideanDomain.gcd f g) = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L469-L471"},"Polynomial.root_left_of_root_gcd":{"signature":"theorem Polynomial.root_left_of_root_gcd {R : Type u} {k : Type y} [Field R] [CommSemiring k] [DecidableEq R] {\u03d5 : R \u2192+* k} {f g : Polynomial R} {\u03b1 : k} (h\u03b1 : eval\u2082 \u03d5 \u03b1 (EuclideanDomain.gcd f g) = 0) :eval\u2082 \u03d5 \u03b1 f = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L473-L476"},"Polynomial.eval\u2082_comp":{"signature":"theorem Polynomial.eval\u2082_comp {R : Type u} {S : Type v} [Semiring R] {p q : Polynomial R} [CommSemiring S] (f : R \u2192+* S) {x : S} :eval\u2082 f x (p.comp q) = eval\u2082 f (eval\u2082 f x q) p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Degree.lean#L205-L206"},"Polynomial.root_right_of_root_gcd":{"signature":"theorem Polynomial.root_right_of_root_gcd {R : Type u} {k : Type y} [Field R] [CommSemiring k] [DecidableEq R] {\u03d5 : R \u2192+* k} {f g : Polynomial R} {\u03b1 : k} (h\u03b1 : eval\u2082 \u03d5 \u03b1 (EuclideanDomain.gcd f g) = 0) :eval\u2082 \u03d5 \u03b1 g = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L478-L481"},"Polynomial.eq_X_sub_C_of_separable_of_root_eq":{"signature":"theorem Polynomial.eq_X_sub_C_of_separable_of_root_eq {F : Type u} [Field F] {K : Type v} [Field K] {i : F \u2192+* K} {x : F} {h : Polynomial F} (h_sep : h.Separable) (h_root : eval x h = 0) (h_splits : (map i h).Splits) (h_roots : \u2200 y \u2208 (map i h).roots, y = i x) :h = C h.leadingCoeff * (X - C x)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Separable.lean#L492-L509"},"Polynomial.C_comp":{"signature":"theorem Polynomial.C_comp {R : Type u} {a : R} [Semiring R] {p : Polynomial R} :(C a).comp p = C a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L386-L388"},"Polynomial.coeff_X_of_ne_one":{"signature":"theorem Polynomial.coeff_X_of_ne_one {R : Type u} [Semiring R] {n : \u2115} (hn : n \u2260 1) :X.coeff n = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L621-L622"},"Polynomial.eval\u2082_mul_X":{"signature":"theorem Polynomial.eval\u2082_mul_X {R : Type u} {S : Type v} [Semiring R] {p : Polynomial R} [Semiring S] (f : R \u2192+* S) (x : S) :eval\u2082 f x (p * X) = eval\u2082 f x p * x","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L148-L153"},"Polynomial.map_comp":{"signature":"theorem Polynomial.map_comp {R : Type u} {S : Type v} [Semiring R] [Semiring S] (f : R \u2192+* S) (p q : Polynomial R) :map f (p.comp q) = (map f p).comp (map f q)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L575-L582"},"Polynomial.nextCoeff":{"signature":"def Polynomial.nextCoeff {R : Type u} [Semiring R] (p : Polynomial R) :R","doc":"The second-highest coefficient, or 0 for constants","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L305-L307"},"Polynomial.nextCoeff_C_eq_zero":{"signature":"theorem Polynomial.nextCoeff_C_eq_zero {R : Type u} [Semiring R] (c : R) :(C c).nextCoeff = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L316-L319"},"Polynomial.revAtFun":{"signature":"def Polynomial.revAtFun (N i : \u2115) :\u2115","doc":"If i \u2264 N, then revAtFun N i returns N - i, otherwise it returns i. This is the map used by the embedding revAt.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Reverse.lean#L34-L38"},"Polynomial.revAtFun_invol":{"signature":"theorem Polynomial.revAtFun_invol {N i : \u2115} :revAtFun N (revAtFun N i) = i","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Reverse.lean#L40-L42"},"Polynomial.revAtFun_inj":{"signature":"theorem Polynomial.revAtFun_inj {N : \u2115} :Function.Injective (revAtFun N)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Reverse.lean#L44-L46"},"Polynomial.revAt":{"signature":"def Polynomial.revAt (N : \u2115) :\u2115 \u21aa \u2115","doc":"If i \u2264 N, then revAt N i returns N - i, otherwise it returns i. Essentially, this embedding is only used for i \u2264 N. The advantage of revAt N i over N - i is that revAt is an involution.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Reverse.lean#L48-L54"},"Polynomial.reflect":{"signature":"noncomputable def Polynomial.reflect {R : Type u_1} [Semiring R] (N : \u2115) :Polynomial R \u2192 Polynomial R","doc":"reflect N f is the polynomial such that (reflect N f).coeff i = f.coeff (revAt N i). In other words, the terms with exponent [0, ..., N] now have exponent [N, ..., 0]. In practice, reflect is only used when N is at least as large as the degree of f. Eventually, it will be used with N exactly equal to the degree of f.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Reverse.lean#L81-L88"},"Polynomial.reverse":{"signature":"noncomputable def Polynomial.reverse {R : Type u_1} [Semiring R] (f : Polynomial R) :Polynomial R","doc":"The reverse of a polynomial f is the polynomial obtained by \"reading f backwards\". Even though this is not the actual definition, reverse f = f (1/X) * X ^ f.natDegree.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Reverse.lean#L217-L220"},"Polynomial.revAt_invol":{"signature":"theorem Polynomial.revAt_invol {N i : \u2115} :(revAt N) ((revAt N) i) = i","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Reverse.lean#L61-L63"},"Polynomial.coeff_reflect":{"signature":"theorem Polynomial.coeff_reflect {R : Type u_1} [Semiring R] (N : \u2115) (f : Polynomial R) (i : \u2115) :(reflect N f).coeff i = f.coeff ((revAt N) i)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Reverse.lean#L96-L103"},"Polynomial.coeff_reverse":{"signature":"theorem Polynomial.coeff_reverse {R : Type u_1} [Semiring R] (f : Polynomial R) (n : \u2115) :f.reverse.coeff n = f.coeff ((revAt f.natDegree) n)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Reverse.lean#L222-L223"},"Polynomial.revAt_le":{"signature":"theorem Polynomial.revAt_le {N i : \u2115} (H : i \u2264 N) :(revAt N) i = N - i","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Reverse.lean#L65-L67"},"Polynomial.coeff_one_reverse":{"signature":"theorem Polynomial.coeff_one_reverse {R : Type u_1} [Semiring R] (f : Polynomial R) :f.reverse.coeff 1 = f.nextCoeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Reverse.lean#L290-L297"},"Polynomial.support_monomial'":{"signature":"theorem Polynomial.support_monomial' {R : Type u} [Semiring R] (n : \u2115) (a : R) :((monomial n) a).support \u2286 {n}","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L744-L746"},"Polynomial.card_support_le_one_iff_monomial":{"signature":"theorem Polynomial.card_support_le_one_iff_monomial {R : Type u} [Semiring R] {f : Polynomial R} :f.support.card \u2264 1 \u2194 \u2203 (n : \u2115) (a : R), f = (monomial n) a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Monomial.lean#L35-L52"},"Polynomial.monomial_natDegree_leadingCoeff_eq_self":{"signature":"theorem Polynomial.monomial_natDegree_leadingCoeff_eq_self {R : Type u} [Semiring R] {p : Polynomial R} (h : p.support.card \u2264 1) :(monomial p.natDegree) p.leadingCoeff = p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Monomial.lean#L39-L43"},"Polynomial.C_mul_X_pow_eq_self":{"signature":"theorem Polynomial.C_mul_X_pow_eq_self {R : Type u} [Semiring R] {p : Polynomial R} (h : p.support.card \u2264 1) :C p.leadingCoeff * X ^ p.natDegree = p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Monomial.lean#L45-L46"},"Polynomial.reflect_C_mul":{"signature":"theorem Polynomial.reflect_C_mul {R : Type u_1} [Semiring R] (f : Polynomial R) (r : R) (N : \u2115) :reflect N (C r * f) = C r * reflect N f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Reverse.lean#L120-L123"},"Polynomial.reflect_C_mul_X_pow":{"signature":"theorem Polynomial.reflect_C_mul_X_pow {R : Type u_1} [Semiring R] (N n : \u2115) {c : R} :reflect N (C c * X ^ n) = C c * X ^ (revAt N) n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Reverse.lean#L125-L127"},"Polynomial.reflect_monomial":{"signature":"theorem Polynomial.reflect_monomial {R : Type u_1} [Semiring R] (N n : \u2115) :reflect N (X ^ n) = X ^ (revAt N) n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Reverse.lean#L133-L135"},"Polynomial.revAt_add":{"signature":"theorem Polynomial.revAt_add {N O n o : \u2115} (hn : n \u2264 N) (ho : o \u2264 O) :(revAt (N + O)) (n + o) = (revAt N) n + (revAt O) o","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Reverse.lean#L71-L77"},"Polynomial.reflect_zero":{"signature":"theorem Polynomial.reflect_zero {R : Type u_1} [Semiring R] {N : \u2115} :reflect N 0 = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Reverse.lean#L107-L109"},"Polynomial.eraseLead_add_C_mul_X_pow":{"signature":"theorem Polynomial.eraseLead_add_C_mul_X_pow {R : Type u_1} [Semiring R] (f : Polynomial R) :f.eraseLead + C f.leadingCoeff * X ^ f.natDegree = f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/EraseLead.lean#L65-L68"},"Polynomial.reflect_add":{"signature":"theorem Polynomial.reflect_add {R : Type u_1} [Semiring R] (f g : Polynomial R) (N : \u2115) :reflect N (f + g) = reflect N f + reflect N g","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Reverse.lean#L115-L118"},"Polynomial.natDegree_mem_support_of_nonzero":{"signature":"theorem Polynomial.natDegree_mem_support_of_nonzero {R : Type u} [Semiring R] {p : Polynomial R} (H : p \u2260 0) :p.natDegree \u2208 p.support","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Support.lean#L124-L126"},"Polynomial.eraseLead_support_card_lt":{"signature":"theorem Polynomial.eraseLead_support_card_lt {R : Type u_1} [Semiring R] {f : Polynomial R} (h : f \u2260 0) :f.eraseLead.support.card < f.support.card","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/EraseLead.lean#L97-L99"},"Polynomial.eraseLead_degree_le":{"signature":"theorem Polynomial.eraseLead_degree_le {R : Type u_1} [Semiring R] {f : Polynomial R} :f.eraseLead.degree \u2264 f.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/EraseLead.lean#L185-L186"},"Polynomial.eraseLead_natDegree_le_aux":{"signature":"theorem Polynomial.eraseLead_natDegree_le_aux {R : Type u_1} [Semiring R] {f : Polynomial R} :f.eraseLead.natDegree \u2264 f.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/EraseLead.lean#L191-L192"},"Polynomial.support_C_mul_X_pow'":{"signature":"theorem Polynomial.support_C_mul_X_pow' {R : Type u} [Semiring R] (n : \u2115) (c : R) :(C c * X ^ n).support \u2286 {n}","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L764-L765"},"Polynomial.card_support_C_mul_X_pow_le_one":{"signature":"theorem Polynomial.card_support_C_mul_X_pow_le_one {R : Type u} [Semiring R] {c : R} {n : \u2115} :(C c * X ^ n).support.card \u2264 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Support.lean#L104-L106"},"Polynomial.natDegree_C_mul_X_pow_le":{"signature":"theorem Polynomial.natDegree_C_mul_X_pow_le {R : Type u} [Semiring R] (a : R) (n : \u2115) :(C a * X ^ n).natDegree \u2264 n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L368-L369"},"Polynomial.reflect_mul_induction":{"signature":"theorem Polynomial.reflect_mul_induction {R : Type u_1} [Semiring R] (cf cg N O : \u2115) (f g : Polynomial R) (Cf : f.support.card \u2264 cf.succ) (Cg : g.support.card \u2264 cg.succ) (Nf : f.natDegree \u2264 N) (Og : g.natDegree \u2264 O) :reflect (N + O) (f * g) = reflect N f * reflect O g","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Reverse.lean#L148-L176"},"Polynomial.reflect_mul":{"signature":"theorem Polynomial.reflect_mul {R : Type u_1} [Semiring R] (f g : Polynomial R) {F G : \u2115} (Ff : f.natDegree \u2264 F) (Gg : g.natDegree \u2264 G) :reflect (F + G) (f * g) = reflect F f * reflect G g","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Reverse.lean#L178-L181"},"Polynomial.reverse_mul":{"signature":"theorem Polynomial.reverse_mul {R : Type u_1} [Semiring R] {f g : Polynomial R} (fg : f.leadingCoeff * g.leadingCoeff \u2260 0) :(f * g).reverse = f.reverse * g.reverse","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Reverse.lean#L271-L274"},"Polynomial.mul_coeff_one":{"signature":"theorem Polynomial.mul_coeff_one {R : Type u} [Semiring R] (p q : Polynomial R) :(p * q).coeff 1 = p.coeff 0 * q.coeff 1 + p.coeff 1 * q.coeff 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L118-L121"},"Polynomial.coeff_zero_reverse":{"signature":"theorem Polynomial.coeff_zero_reverse {R : Type u_1} [Semiring R] (f : Polynomial R) :f.reverse.coeff 0 = f.leadingCoeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Reverse.lean#L225-L227"},"Polynomial.nextCoeff_of_natDegree_pos":{"signature":"theorem Polynomial.nextCoeff_of_natDegree_pos {R : Type u} [Semiring R] {p : Polynomial R} (hp : 0 < p.natDegree) :p.nextCoeff = p.coeff (p.natDegree - 1)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L321-L325"},"Polynomial.nextCoeff_X_add_C":{"signature":"theorem Polynomial.nextCoeff_X_add_C {S : Type v} [Semiring S] (c : S) :(X + C c).nextCoeff = c","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L611-L614"},"Polynomial.nextCoeff_X_sub_C":{"signature":"theorem Polynomial.nextCoeff_X_sub_C {S : Type v} [Ring S] (c : S) :(X - C c).nextCoeff = -c","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L763-L765"},"Polynomial.multiset_prod_X_sub_C_nextCoeff":{"signature":"theorem Polynomial.multiset_prod_X_sub_C_nextCoeff {R : Type u} [CommRing R] (t : Multiset R) :(Multiset.map (fun (x : R) => X - C x) t).prod.nextCoeff = -t.sum","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/BigOperators.lean#L252-L258"},"Polynomial.multiset_prod_X_sub_C_coeff_card_pred":{"signature":"theorem Polynomial.multiset_prod_X_sub_C_coeff_card_pred {R : Type u} [CommRing R] (t : Multiset R) (ht : 0 < t.card) :(Multiset.map (fun (x : R) => X - C x) t).prod.coeff (t.card - 1) = -t.sum","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/BigOperators.lean#L264-L278"},"Polynomial.prod_X_sub_C_coeff_card_pred":{"signature":"theorem Polynomial.prod_X_sub_C_coeff_card_pred {R : Type u} {\u03b9 : Type w} [CommRing R] (s : Finset \u03b9) (f : \u03b9 \u2192 R) (hs : 0 < s.card) :(\u220f i \u2208 s, (X - C (f i))).coeff (s.card - 1) = -\u2211 i \u2208 s, f i","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/BigOperators.lean#L280-L282"},"Polynomial.nextCoeff_map":{"signature":"theorem Polynomial.nextCoeff_map {R : Type u} {S : Type v} [Semiring R] [Semiring S] {f : R \u2192+* S} (hf : Function.Injective \u21d1f) (p : Polynomial R) :(map f p).nextCoeff = f p.nextCoeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L305-L308"},"Polynomial.nextCoeff_map_eq":{"signature":"theorem Polynomial.nextCoeff_map_eq {R : Type u} {S : Type v} [Ring R] [IsSimpleRing R] [Semiring S] [Nontrivial S] (p : Polynomial R) (f : R \u2192+* S) :(map f p).nextCoeff = f p.nextCoeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L286-L287"},"Polynomial.nextCoeff_C_mul":{"signature":"theorem Polynomial.nextCoeff_C_mul {R : Type u} [Semiring R] {p : Polynomial R} {a : R} [NoZeroDivisors R] :(C a * p).nextCoeff = a * p.nextCoeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L403-L405"},"Polynomial.divX_C":{"signature":"theorem Polynomial.divX_C {R : Type u} [Semiring R] (a : R) :(C a).divX = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Inductions.lean#L54-L56"},"Polynomial.divX_mul_X_add":{"signature":"theorem Polynomial.divX_mul_X_add {R : Type u} [Semiring R] (p : Polynomial R) :p.divX * X + C (p.coeff 0) = p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Inductions.lean#L47-L48"},"Polynomial.divX_eq_zero_iff":{"signature":"theorem Polynomial.divX_eq_zero_iff {R : Type u} [Semiring R] {p : Polynomial R} :p.divX = 0 \u2194 p = C (p.coeff 0)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Inductions.lean#L58-L59"},"Polynomial.degree_lt_degree_mul_X":{"signature":"theorem Polynomial.degree_lt_degree_mul_X {R : Type u} [Semiring R] {p : Polynomial R} (hp : p \u2260 0) :p.degree < (p * X).degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L472-L477"},"Polynomial.degree_divX_lt":{"signature":"theorem Polynomial.degree_divX_lt {R : Type u} [Semiring R] {p : Polynomial R} (hp0 : p \u2260 0) :p.divX.degree < p.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Inductions.lean#L108-L132"},"Polynomial.coeff_mul_X_zero":{"signature":"theorem Polynomial.coeff_mul_X_zero {R : Type u} [Semiring R] (p : Polynomial R) :(p * X).coeff 0 = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L139-L139"},"Polynomial.recOnHorner":{"signature":"noncomputable def Polynomial.recOnHorner {R : Type u} [Semiring R] {M : Polynomial R \u2192 Sort u_1} (p : Polynomial R) (M0 : M 0) (MC : (p : Polynomial R) \u2192 (a : R) \u2192 p.coeff 0 = 0 \u2192 a \u2260 0 \u2192 M p \u2192 M (p + C a)) (MX : (p : Polynomial R) \u2192 p \u2260 0 \u2192 M p \u2192 M (p * X)) :M p","doc":"An induction principle for polynomials, valued in Sort* instead of Prop.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Inductions.lean#L134-L153"},"Polynomial.eq_of_monic_of_associated":{"signature":"theorem Polynomial.eq_of_monic_of_associated {R : Type u} [Semiring R] {p q : Polynomial R} (hp : p.Monic) (hq : q.Monic) (hpq : Associated p q) :p = q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Monic.lean#L224-L226"},"Polynomial.eval_X_pow":{"signature":"theorem Polynomial.eval_X_pow {R : Type u} [Semiring R] {x : R} (n : \u2115) :eval x (X ^ n) = x ^ n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L584-L585"},"Polynomial.smul_eval_smul":{"signature":"theorem Polynomial.smul_eval_smul {M : Type u_1} [Monoid M] (S : Type u_3) [CommSemiring S] [MulSemiringAction M S] (m : M) (f : Polynomial S) (x : S) :eval (m \u2022 x) (m \u2022 f) = m \u2022 eval x f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/GroupRingAction.lean#L56-L60"},"Polynomial.eval_smul'":{"signature":"theorem Polynomial.eval_smul' (S : Type u_3) [CommSemiring S] (G : Type u_4) [Group G] [MulSemiringAction G S] (g : G) (f : Polynomial S) (x : S) :eval (g \u2022 x) f = g \u2022 eval x (g\u207b\u00b9 \u2022 f)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/GroupRingAction.lean#L64-L66"},"Polynomial.separable_one":{"signature":"theorem Polynomial.separable_one {R : Type u} [CommSemiring R] :Separable 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Separable.lean#L65-L67"},"Polynomial.separable_prod'":{"signature":"theorem Polynomial.separable_prod' {R : Type u} [CommRing R] {\u03b9 : Type u_1} {f : \u03b9 \u2192 Polynomial R} {s : Finset \u03b9} :(\u2200 x \u2208 s, \u2200 y \u2208 s, x \u2260 y \u2192 IsCoprime (f x) (f y)) \u2192 (\u2200 x \u2208 s, (f x).Separable) \u2192 (\u220f x \u2208 s, f x).Separable","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Separable.lean#L210-L218"},"Polynomial.separable_prod_X_sub_C_iff'":{"signature":"theorem Polynomial.separable_prod_X_sub_C_iff' {F : Type u} [Field F] {\u03b9 : Type u_1} {f : \u03b9 \u2192 F} {s : Finset \u03b9} :(\u220f i \u2208 s, (X - C (f i))).Separable \u2194 \u2200 x \u2208 s, \u2200 y \u2208 s, f x = f y \u2192 x = y","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Separable.lean#L333-L342"},"Polynomial.separable_prod_X_sub_C_iff":{"signature":"theorem Polynomial.separable_prod_X_sub_C_iff {F : Type u} [Field F] {\u03b9 : Type u_1} [Fintype \u03b9] {f : \u03b9 \u2192 F} :(\u220f i : \u03b9, (X - C (f i))).Separable \u2194 Function.Injective f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Separable.lean#L344-L346"},"Polynomial.separable_X":{"signature":"theorem Polynomial.separable_X {R : Type u} [CommSemiring R] :X.Separable","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Separable.lean#L77-L79"},"Polynomial.aroots_zero":{"signature":"theorem Polynomial.aroots_zero {T : Type w} [CommRing T] (S : Type u_1) [CommRing S] [IsDomain S] [Algebra T S] :aroots 0 S = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L477-L479"},"Polynomial.cyclotomic'":{"signature":"def Polynomial.cyclotomic' (n : \u2115) (R : Type u_2) [CommRing R] [IsDomain R] :Polynomial R","doc":"The modified n-th cyclotomic polynomial with coefficients in R, it is the usual cyclotomic polynomial if there is a primitive n-th root of unity in R.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean#L62-L65"},"Polynomial.nthRoots_zero":{"signature":"theorem Polynomial.nthRoots_zero {R : Type u} [CommRing R] [IsDomain R] (r : R) :nthRoots 0 r = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L328-L330"},"Polynomial.cyclotomic'_zero":{"signature":"theorem Polynomial.cyclotomic'_zero (R : Type u_2) [CommRing R] [IsDomain R] :cyclotomic' 0 R = 1","doc":"The zeroth modified cyclotomic polynomial is 1.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean#L67-L70"},"Polynomial.nthRootsFinset":{"signature":"def Polynomial.nthRootsFinset (n : \u2115) {R : Type u_1} (a : R) [CommRing R] [IsDomain R] :Finset R","doc":"The multiset nthRoots \u2191n a as a Finset. Previously nthRootsFinset n was defined to be nthRoots n (1 : R) as a Finset. That situation can be recovered by setting a to be (1 : R)","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L358-L362"},"Polynomial.roots_pow":{"signature":"theorem Polynomial.roots_pow {R : Type u} [CommRing R] [IsDomain R] (p : Polynomial R) (n : \u2115) :(p ^ n).roots = n \u2022 p.roots","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L247-L254"},"Polynomial.roots_X":{"signature":"theorem Polynomial.roots_X {R : Type u} [CommRing R] [IsDomain R] :X.roots = {0}","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L185-L186"},"Polynomial.nthRoots_zero_right":{"signature":"theorem Polynomial.nthRoots_zero_right {R : Type u_1} [CommRing R] [IsDomain R] (n : \u2115) :nthRoots n 0 = Multiset.replicate n 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L332-L335"},"Polynomial.prod_multiset_X_sub_C_of_monic_of_roots_card_eq":{"signature":"theorem Polynomial.prod_multiset_X_sub_C_of_monic_of_roots_card_eq {R : Type u} [CommRing R] [IsDomain R] {p : Polynomial R} (hp : p.Monic) (hroots : p.roots.card = p.natDegree) :(Multiset.map (fun (a : R) => X - C a) p.roots).prod = p","doc":"A monic polynomial p that has as many roots as its degree can be written p = \u220f(X - a), for a in p.roots.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L815-L820"},"Polynomial.X_pow_sub_one_eq_prod":{"signature":"theorem Polynomial.X_pow_sub_one_eq_prod {R : Type u_1} [CommRing R] [IsDomain R] {\u03b6 : R} {n : \u2115} (hpos : 0 < n) (h : IsPrimitiveRoot \u03b6 n) :X ^ n - 1 = \u220f \u03b6 \u2208 nthRootsFinset n 1, (X - C \u03b6)","doc":"If there is a primitive nth root of unity in K, then X ^ n - 1 = \u220f (X - \u03bc), where \u03bc varies over the n-th roots of unity.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean#L117-L129"},"Polynomial.nthRootsFinset_def":{"signature":"theorem Polynomial.nthRootsFinset_def (n : \u2115) {R : Type u_1} (a : R) [CommRing R] [IsDomain R] [DecidableEq R] :nthRootsFinset n a = (nthRoots n a).toFinset","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L364-L367"},"Polynomial.nthRootsFinset_zero":{"signature":"theorem Polynomial.nthRootsFinset_zero {R : Type u} [CommRing R] [IsDomain R] (a : R) :nthRootsFinset 0 a = \u2205","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L375-L377"},"Polynomial.mem_nthRootsFinset":{"signature":"theorem Polynomial.mem_nthRootsFinset {R : Type u} [CommRing R] [IsDomain R] {n : \u2115} (h : 0 < n) (a : R) {x : R} :x \u2208 nthRootsFinset n a \u2194 x ^ n = a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L369-L373"},"Polynomial.prod_cyclotomic'_eq_X_pow_sub_one":{"signature":"theorem Polynomial.prod_cyclotomic'_eq_X_pow_sub_one {K : Type u_2} [CommRing K] [IsDomain K] {\u03b6 : K} {n : \u2115} (hpos : 0 < n) (h : IsPrimitiveRoot \u03b6 n) :\u220f i \u2208 n.divisors, cyclotomic' i K = X ^ n - 1","doc":"If there is a primitive n-th root of unity in K, then \u220f i \u2208 Nat.divisors n, cyclotomic' i K = X ^ n - 1.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean#L148-L157"},"Polynomial.map_divByMonic":{"signature":"theorem Polynomial.map_divByMonic {R : Type u} {S : Type v} [Ring R] {p q : Polynomial R} [Ring S] (f : R \u2192+* S) (hq : q.Monic) :map f (p /\u2098 q) = map f p /\u2098 map f q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L378-L380"},"Polynomial.int_coeff_of_cyclotomic'":{"signature":"theorem Polynomial.int_coeff_of_cyclotomic' {K : Type u_2} [CommRing K] [IsDomain K] {\u03b6 : K} {n : \u2115} (h : IsPrimitiveRoot \u03b6 n) :\u2203 (P : Polynomial \u2124), map (Int.castRingHom K) P = cyclotomic' n K \u2227 P.degree = (cyclotomic' n K).degree \u2227 P.Monic","doc":"If there is a primitive n-th root of unity in K, then cyclotomic' n K comes from a monic polynomial with integer coefficients.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean#L177-L211"},"Polynomial.cyclotomic":{"signature":"def Polynomial.cyclotomic (n : \u2115) (R : Type u_1) [Ring R] :Polynomial R","doc":"The n-th cyclotomic polynomial with coefficients in R.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean#L229-L232"},"Polynomial.int_cyclotomic_rw":{"signature":"theorem Polynomial.int_cyclotomic_rw {n : \u2115} (h : n \u2260 0) :cyclotomic n \u2124 = \u22ef.choose","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean#L234-L238"},"Polynomial.int_cyclotomic_spec":{"signature":"theorem Polynomial.int_cyclotomic_spec (n : \u2115) :map (Int.castRingHom \u2102) (cyclotomic n \u2124) = cyclotomic' n \u2102 \u2227 (cyclotomic n \u2124).degree = (cyclotomic' n \u2102).degree \u2227 (cyclotomic n \u2124).Monic","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean#L247-L254"},"Polynomial.map_cyclotomic_int":{"signature":"theorem Polynomial.map_cyclotomic_int (n : \u2115) (R : Type u_1) [Ring R] :map (Int.castRingHom R) (cyclotomic n \u2124) = cyclotomic n R","doc":"cyclotomic n R comes from cyclotomic n \u2124.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean#L240-L245"},"Polynomial.prod_cyclotomic_eq_X_pow_sub_one":{"signature":"theorem Polynomial.prod_cyclotomic_eq_X_pow_sub_one {n : \u2115} (hpos : 0 < n) (R : Type u_1) [CommRing R] :\u220f i \u2208 n.divisors, cyclotomic i R = X ^ n - 1","doc":"\u220f i \u2208 Nat.divisors n, cyclotomic i R = X ^ n - 1.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean#L337-L346"},"Polynomial.isRoot_prod":{"signature":"theorem Polynomial.isRoot_prod {R : Type u_2} [CommSemiring R] [IsDomain R] {\u03b9 : Type u_1} (s : Finset \u03b9) (p : \u03b9 \u2192 Polynomial R) (x : R) :(\u220f j \u2208 s, p j).IsRoot x \u2194 \u2203 i \u2208 s, (p i).IsRoot x","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L677-L679"},"Polynomial.map_cyclotomic":{"signature":"theorem Polynomial.map_cyclotomic (n : \u2115) {R : Type u_1} {S : Type u_2} [Ring R] [Ring S] (f : R \u2192+* S) :map f (cyclotomic n R) = cyclotomic n S","doc":"The definition of cyclotomic n R commutes with any ring homomorphism.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean#L261-L267"},"Polynomial.cyclotomic_ne_zero":{"signature":"theorem Polynomial.cyclotomic_ne_zero (n : \u2115) (R : Type u_1) [Ring R] [Nontrivial R] :cyclotomic n R \u2260 0","doc":"cyclotomic n R is different from 0.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean#L300-L302"},"Polynomial.cyclotomic_zero":{"signature":"theorem Polynomial.cyclotomic_zero (R : Type u_1) [Ring R] :cyclotomic 0 R = 1","doc":"The zeroth cyclotomic polynomial is 1.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean#L277-L280"},"Polynomial.cyclotomic_eq_X_pow_sub_one_div":{"signature":"theorem Polynomial.cyclotomic_eq_X_pow_sub_one_div {R : Type u_1} [CommRing R] {n : \u2115} (hpos : 0 < n) :cyclotomic n R = (X ^ n - 1) /\u2098 \u220f i \u2208 n.properDivisors, cyclotomic i R","doc":"We have cyclotomic n R = (X ^ k - 1) /\u2098 (\u220f i \u2208 Nat.properDivisors k, cyclotomic i K).","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean#L442-L459"},"Polynomial.cyclotomic'_eq_X_pow_sub_one_div":{"signature":"theorem Polynomial.cyclotomic'_eq_X_pow_sub_one_div {K : Type u_2} [CommRing K] [IsDomain K] {\u03b6 : K} {n : \u2115} (hpos : 0 < n) (h : IsPrimitiveRoot \u03b6 n) :cyclotomic' n K = (X ^ n - 1) /\u2098 \u220f i \u2208 n.properDivisors, cyclotomic' i K","doc":"If there is a primitive n-th root of unity in K, then cyclotomic' n K = (X ^ k - 1) /\u2098 (\u220f i \u2208 Nat.properDivisors k, cyclotomic' i K).","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean#L159-L175"},"Polynomial.cyclotomic_eq_prod_X_sub_primitiveRoots":{"signature":"theorem Polynomial.cyclotomic_eq_prod_X_sub_primitiveRoots {K : Type u_1} [CommRing K] [IsDomain K] {\u03b6 : K} {n : \u2115} (hz : IsPrimitiveRoot \u03b6 n) :cyclotomic n K = \u220f \u03bc \u2208 primitiveRoots n K, (X - C \u03bc)","doc":"If there is a primitive n-th root of unity in K, then cyclotomic n K = \u220f \u03bc \u2208 primitiveRoots n K, (X - C \u03bc). \u2208 particular, cyclotomic n K = cyclotomic' n K","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean#L473-L488"},"Polynomial.roots_prod_X_sub_C":{"signature":"theorem Polynomial.roots_prod_X_sub_C {R : Type u} [CommRing R] [IsDomain R] (s : Finset R) :(\u220f a \u2208 s, (X - C a)).roots = s.val","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L267-L271"},"Polynomial.isRoot_map_iff":{"signature":"theorem Polynomial.isRoot_map_iff {S : Type v} [CommSemiring S] {R : Type u_1} [CommRing R] {f : R \u2192+* S} {x : R} {p : Polynomial R} (hf : Function.Injective \u21d1f) :(map f p).IsRoot (f x) \u2194 p.IsRoot x","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Coeff.lean#L225-L227"},"Polynomial.isRoot_cyclotomic_iff":{"signature":"theorem Polynomial.isRoot_cyclotomic_iff {R : Type u_1} [CommRing R] {n : \u2115} [IsDomain R] [NeZero \u2191n] {\u03bc : R} :(cyclotomic n R).IsRoot \u03bc \u2194 IsPrimitiveRoot \u03bc n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Roots.lean#L98-L103"},"Polynomial.cyclotomic'_ne_zero":{"signature":"theorem Polynomial.cyclotomic'_ne_zero (n : \u2115) (R : Type u_2) [CommRing R] [IsDomain R] :cyclotomic' n R \u2260 0","doc":"cyclotomic' n R is different from 0.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean#L92-L94"},"Polynomial.natDegree_prod":{"signature":"theorem Polynomial.natDegree_prod {R : Type u} {\u03b9 : Type w} (s : Finset \u03b9) [CommSemiring R] [NoZeroDivisors R] (f : \u03b9 \u2192 Polynomial R) (h : \u2200 i \u2208 s, f i \u2260 0) :(\u220f i \u2208 s, f i).natDegree = \u2211 i \u2208 s, (f i).natDegree","doc":"The degree of a product of polynomials is equal to the sum of the degrees. See Polynomial.natDegree_prod' (with a ') for a version for commutative semirings, where additionally, the product of the leading coefficients must be nonzero.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/BigOperators.lean#L320-L331"},"Polynomial.natDegree_cyclotomic'":{"signature":"theorem Polynomial.natDegree_cyclotomic' {R : Type u_1} [CommRing R] [IsDomain R] {\u03b6 : R} {n : \u2115} (h : IsPrimitiveRoot \u03b6 n) :(cyclotomic' n R).natDegree = n.totient","doc":"The natural degree of cyclotomic' n R is totient n if there is a primitive root of unity in R.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean#L96-L105"},"Polynomial.degree_cyclotomic'":{"signature":"theorem Polynomial.degree_cyclotomic' {R : Type u_1} [CommRing R] [IsDomain R] {\u03b6 : R} {n : \u2115} (h : IsPrimitiveRoot \u03b6 n) :(cyclotomic' n R).degree = \u2191n.totient","doc":"The degree of cyclotomic' n R is totient n if there is a primitive root of unity in R.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean#L107-L110"},"Polynomial.degree_cyclotomic":{"signature":"theorem Polynomial.degree_cyclotomic (n : \u2115) (R : Type u_1) [Ring R] [Nontrivial R] :(cyclotomic n R).degree = \u2191n.totient","doc":"The degree of cyclotomic n is totient n.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean#L304-L314"},"Polynomial.natDegree_cyclotomic":{"signature":"theorem Polynomial.natDegree_cyclotomic (n : \u2115) (R : Type u_1) [Ring R] [Nontrivial R] :(cyclotomic n R).natDegree = n.totient","doc":"The natural degree of cyclotomic n is totient n.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean#L316-L319"},"Polynomial.isPrimitive_iff_isUnit_of_C_dvd":{"signature":"theorem Polynomial.isPrimitive_iff_isUnit_of_C_dvd {R : Type u_1} [CommSemiring R] {p : Polynomial R} :p.IsPrimitive \u2194 \u2200 (r : R), C r \u2223 p \u2192 IsUnit r","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Content.lean#L48-L49"},"Polynomial.isUnit_or_eq_zero_of_isUnit_integerNormalization_primPart":{"signature":"theorem Polynomial.isUnit_or_eq_zero_of_isUnit_integerNormalization_primPart {R : Type u_1} [CommRing R] {K : Type u_2} [Field K] [Algebra R K] [IsFractionRing R K] [IsDomain R] [NormalizedGCDMonoid R] {p : Polynomial K} (h0 : p \u2260 0) (h : IsUnit (IsLocalization.integerNormalization (nonZeroDivisors R) p).primPart) :IsUnit p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/GaussLemma.lean#L220-L234"},"Polynomial.isPrimitive_of_dvd":{"signature":"theorem Polynomial.isPrimitive_of_dvd {R : Type u_1} [CommSemiring R] {p q : Polynomial R} (hp : p.IsPrimitive) (hq : q \u2223 p) :q.IsPrimitive","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Content.lean#L63-L64"},"Polynomial.eq_of_monic_of_dvd_of_natDegree_le":{"signature":"theorem Polynomial.eq_of_monic_of_dvd_of_natDegree_le {R : Type u} [Semiring R] {p q : Polynomial R} (hp : p.Monic) (hq : q.Monic) (hdvd : p \u2223 q) (hdeg : q.natDegree \u2264 p.natDegree) :q = p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L95-L98"},"Polynomial.map_aeval_eq_aeval_map":{"signature":"theorem Polynomial.map_aeval_eq_aeval_map {R : Type u} [CommSemiring R] {S : Type u_3} {T : Type u_4} {U : Type u_5} [Semiring S] [CommSemiring T] [Semiring U] [Algebra R S] [Algebra T U] {\u03c6 : R \u2192+* T} {\u03c8 : S \u2192+* U} (h : (algebraMap T U).comp \u03c6 = \u03c8.comp (algebraMap R S)) (p : Polynomial R) (a : S) :\u03c8 ((aeval a) p) = (aeval (\u03c8 a)) (map \u03c6 p)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L476-L481"},"Polynomial.mem_lifts":{"signature":"theorem Polynomial.mem_lifts {R : Type u} [Semiring R] {S : Type v} [Semiring S] {f : R \u2192+* S} (p : Polynomial S) :p \u2208 lifts f \u2194 \u2203 (q : Polynomial R), map f q = p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Lifts.lean#L63-L64"},"Polynomial.liftsRing":{"signature":"def Polynomial.liftsRing {R : Type u} [Ring R] {S : Type v} [Ring S] (f : R \u2192+* S) :Subring (Polynomial S)","doc":"The subring of polynomials that lift.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Lifts.lean#L192-L194"},"Polynomial.lifts_iff_liftsRing":{"signature":"theorem Polynomial.lifts_iff_liftsRing {R : Type u} [Ring R] {S : Type v} [Ring S] (f : R \u2192+* S) (p : Polynomial S) :p \u2208 lifts f \u2194 p \u2208 liftsRing f","doc":"If R and S are rings, p is in the subring of polynomials that lift if and only if it is in the subsemiring of polynomials that lift.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Lifts.lean#L196-L199"},"Polynomial.X_mem_lifts":{"signature":"theorem Polynomial.X_mem_lifts {R : Type u} [Semiring R] {S : Type v} [Semiring S] (f : R \u2192+* S) :X \u2208 lifts f","doc":"The polynomial X lifts.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Lifts.lean#L99-L102"},"Polynomial.C'_mem_lifts":{"signature":"theorem Polynomial.C'_mem_lifts {R : Type u} [Semiring R] {S : Type v} [Semiring S] {f : R \u2192+* S} {s : S} (h : s \u2208 Set.range \u21d1f) :C s \u2208 lifts f","doc":"If (s : S) is in the image of f, then C s lifts.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Lifts.lean#L93-L97"},"Polynomial.degree_le_of_dvd":{"signature":"theorem Polynomial.degree_le_of_dvd {R : Type u} [Semiring R] [NoZeroDivisors R] {p q : Polynomial R} (h1 : p \u2223 q) (h2 : q \u2260 0) :p.degree \u2264 q.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Domain.lean#L66-L68"},"Polynomial.eq_zero_of_dvd_of_degree_lt":{"signature":"theorem Polynomial.eq_zero_of_dvd_of_degree_lt {R : Type u} [Semiring R] [NoZeroDivisors R] {p q : Polynomial R} (h\u2081 : p \u2223 q) (h\u2082 : q.degree < p.degree) :q = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Domain.lean#L70-L72"},"Polynomial.IsWeaklyEisensteinAt":{"signature":"structure Polynomial.IsWeaklyEisensteinAt {R : Type u} [CommSemiring R] (f : Polynomial R) (\ud835\udcdf : Ideal R) :Prop","doc":"Given an ideal \ud835\udcdf of a commutative semiring R, we say that a polynomial f : R[X] is weakly Eisenstein at \ud835\udcdf if \u2200 n, n < f.natDegree \u2192 f.coeff n \u2208 \ud835\udcdf.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Eisenstein/Basic.lean#L45-L49"},"Polynomial.sum_over_range":{"signature":"theorem Polynomial.sum_over_range {R : Type u} {S : Type v} [Semiring R] [AddCommMonoid S] (p : Polynomial R) {f : \u2115 \u2192 R \u2192 S} (h : \u2200 (n : \u2115), f n 0 = 0) :p.sum f = \u2211 a \u2208 Finset.range (p.natDegree + 1), f a (p.coeff a)","doc":"We can reexpress a sum over p.support as a sum over range (p.natDegree + 1).","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Support.lean#L70-L74"},"Polynomial.eval_eq_sum_range":{"signature":"theorem Polynomial.eval_eq_sum_range {R : Type u} [Semiring R] {p : Polynomial R} (x : R) :eval x p = \u2211 i \u2208 Finset.range (p.natDegree + 1), p.coeff i * x ^ i","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Degree.lean#L63-L65"},"Polynomial.dvd_pow_natDegree_of_eval\u2082_eq_zero":{"signature":"theorem Polynomial.dvd_pow_natDegree_of_eval\u2082_eq_zero {R : Type u} {A : Type u_1} [CommRing R] [CommRing A] {f : R \u2192+* A} (hf : Function.Injective \u21d1f) {p : Polynomial R} (hp : p.Monic) (x y : R) (z : A) (h : eval\u2082 f z p = 0) (hz : f x * z = f y) :x \u2223 y ^ p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Eisenstein/Basic.lean#L177-L187"},"Polynomial.expand_mul":{"signature":"theorem Polynomial.expand_mul {R : Type u} [CommSemiring R] (p q : \u2115) (f : Polynomial R) :(expand R (p * q)) f = (expand R p) ((expand R q) f)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Expand.lean#L70-L71"},"Polynomial.map_iterateFrobenius_expand":{"signature":"theorem Polynomial.map_iterateFrobenius_expand {R : Type u} [CommSemiring R] (p : \u2115) [ExpChar R p] (f : Polynomial R) (n : \u2115) :map (iterateFrobenius R p n) ((expand R (p ^ n)) f) = f ^ p ^ n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Expand.lean#L272-L280"},"Polynomial.cyclotomic_eq_minpoly":{"signature":"theorem Polynomial.cyclotomic_eq_minpoly {n : \u2115} {K : Type u_2} [Field K] {\u03bc : K} (h : IsPrimitiveRoot \u03bc n) (hpos : 0 < n) [CharZero K] :cyclotomic n \u2124 = minpoly \u2124 \u03bc","doc":"cyclotomic n \u2124 is the minimal polynomial of a primitive n-th root of unity \u03bc.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Roots.lean#L170-L175"},"Polynomial.eq_cyclotomic_iff":{"signature":"theorem Polynomial.eq_cyclotomic_iff {R : Type u_1} [CommRing R] {n : \u2115} (hpos : 0 < n) (P : Polynomial R) :P = cyclotomic n R \u2194 P * \u220f i \u2208 n.properDivisors, cyclotomic i R = X ^ n - 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean#L490-L505"},"Polynomial.cyclotomic'_one":{"signature":"theorem Polynomial.cyclotomic'_one (R : Type u_2) [CommRing R] [IsDomain R] :cyclotomic' 1 R = X - 1","doc":"The first modified cyclotomic polynomial is X - 1.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean#L72-L75"},"Polynomial.int_cyclotomic_unique":{"signature":"theorem Polynomial.int_cyclotomic_unique {n : \u2115} {P : Polynomial \u2124} (h : map (Int.castRingHom \u2102) P = cyclotomic' n \u2102) :P = cyclotomic n \u2124","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean#L256-L259"},"Polynomial.cyclotomic_one":{"signature":"theorem Polynomial.cyclotomic_one (R : Type u_1) [Ring R] :cyclotomic 1 R = X - 1","doc":"The first cyclotomic polynomial is X - 1.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean#L282-L289"},"Polynomial.prod_cyclotomic_eq_geom_sum":{"signature":"theorem Polynomial.prod_cyclotomic_eq_geom_sum {n : \u2115} (h : 0 < n) (R : Type u_1) [CommRing R] :\u220f i \u2208 n.divisors.erase 1, cyclotomic i R = \u2211 i \u2208 Finset.range n, X ^ i","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean#L358-L364"},"Polynomial.cyclotomic_prime":{"signature":"theorem Polynomial.cyclotomic_prime (R : Type u_1) [Ring R] (p : \u2115) [hp : Fact (Nat.Prime p)] :cyclotomic p R = \u2211 i \u2208 Finset.range p, X ^ i","doc":"If p is prime, then cyclotomic p R = \u2211 i \u2208 range p, X ^ i.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean#L366-L373"},"Polynomial.cyclotomic_prime_pow_eq_geom_sum":{"signature":"theorem Polynomial.cyclotomic_prime_pow_eq_geom_sum {R : Type u_1} [CommRing R] {p n : \u2115} (hp : Nat.Prime p) :cyclotomic (p ^ (n + 1)) R = \u2211 i \u2208 Finset.range p, (X ^ p ^ n) ^ i","doc":"If p ^ k is a prime power, then cyclotomic (p ^ (n + 1)) R = \u2211 i \u2208 range p, (X ^ (p ^ n)) ^ i.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean#L507-L526"},"Polynomial.eval_one_cyclotomic_prime_pow":{"signature":"theorem Polynomial.eval_one_cyclotomic_prime_pow {R : Type u_1} [CommRing R] {p : \u2115} (k : \u2115) [hn : Fact (Nat.Prime p)] :eval 1 (cyclotomic (p ^ (k + 1)) R) = \u2191p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Eval.lean#L39-L43"},"Polynomial.degree_cyclotomic_pos":{"signature":"theorem Polynomial.degree_cyclotomic_pos (n : \u2115) (R : Type u_1) (hpos : 0 < n) [Ring R] [Nontrivial R] :0 < (cyclotomic n R).degree","doc":"The degree of cyclotomic n R is positive.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean#L330-L333"},"Polynomial.coe_evalRingHom":{"signature":"theorem Polynomial.coe_evalRingHom {R : Type u} [CommSemiring R] (r : R) :\u21d1(evalRingHom r) = eval r","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L607-L609"},"Polynomial.coeff_zero_eq_eval_zero":{"signature":"theorem Polynomial.coeff_zero_eq_eval_zero {R : Type u} [Semiring R] (p : Polynomial R) :p.coeff 0 = eval 0 p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Coeff.lean#L57-L63"},"Polynomial.eval_multiset_prod":{"signature":"theorem Polynomial.eval_multiset_prod {R : Type u} [CommSemiring R] (s : Multiset (Polynomial R)) (x : R) :eval x s.prod = (Multiset.map (eval x) s).prod","doc":"Polynomial evaluation commutes with Multiset.prod","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L654-L657"},"Polynomial.eval_intCast_map":{"signature":"theorem Polynomial.eval_intCast_map {R : Type u_1} {S : Type u_2} [Ring R] [Ring S] (f : R \u2192+* S) (p : Polynomial R) (i : \u2124) :eval (\u2191i) (map f p) = f (eval (\u2191i) p)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Coeff.lean#L159-L164"},"Polynomial.irreducible_X_sub_C":{"signature":"theorem Polynomial.irreducible_X_sub_C {R : Type u} [CommRing R] [IsDomain R] (r : R) :Irreducible (X - C r)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/RingDivision.lean#L216-L217"},"Polynomial.of_irreducible_expand":{"signature":"theorem Polynomial.of_irreducible_expand {R : Type u} [CommRing R] [IsDomain R] {p : \u2115} (hp : p \u2260 0) {f : Polynomial R} (hf : Irreducible ((expand R p) f)) :Irreducible f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Expand.lean#L320-L323"},"Polynomial.leadingCoeff_expand":{"signature":"theorem Polynomial.leadingCoeff_expand {R : Type u} [CommSemiring R] {p : \u2115} {f : Polynomial R} (hp : 0 < p) :((expand R p) f).leadingCoeff = f.leadingCoeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Expand.lean#L154-L156"},"Polynomial.monic_expand_iff":{"signature":"theorem Polynomial.monic_expand_iff {R : Type u} [CommSemiring R] {p : \u2115} {f : Polynomial R} (hp : 0 < p) :((expand R p) f).Monic \u2194 f.Monic","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Expand.lean#L158-L159"},"Polynomial.expand_eval":{"signature":"theorem Polynomial.expand_eval {R : Type u} [CommSemiring R] (p : \u2115) (P : Polynomial R) (r : R) :eval r ((expand R p) P) = eval (r ^ p) P","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Expand.lean#L171-L174"},"Polynomial.cyclotomic_expand_eq_cyclotomic":{"signature":"theorem Polynomial.cyclotomic_expand_eq_cyclotomic {p n : \u2115} (hp : Nat.Prime p) (hdiv : p \u2223 n) (R : Type u_1) [CommRing R] :(expand R p) (cyclotomic n R) = cyclotomic (n * p) R","doc":"If p is a prime such that p \u2223 n, then expand R p (cyclotomic n R) = cyclotomic (p * n) R.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Expand.lean#L85-L106"},"Polynomial.cyclotomic_irreducible_pow_of_irreducible_pow":{"signature":"theorem Polynomial.cyclotomic_irreducible_pow_of_irreducible_pow {p : \u2115} (hp : Nat.Prime p) {R : Type u_1} [CommRing R] [IsDomain R] {n m : \u2115} (hmn : m \u2264 n) (h : Irreducible (cyclotomic (p ^ n) R)) :Irreducible (cyclotomic (p ^ m) R)","doc":"If the p ^ nth cyclotomic polynomial is irreducible, so is the p ^ mth, for m \u2264 n.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Expand.lean#L108-L120"},"Polynomial.mapRingHom_comp":{"signature":"theorem Polynomial.mapRingHom_comp {R : Type u} {S : Type v} {T : Type w} [Semiring R] [Semiring S] [Semiring T] (f : S \u2192+* T) (g : R \u2192+* S) :(mapRingHom f).comp (mapRingHom g) = mapRingHom (f.comp g)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Coeff.lean#L133-L136"},"Polynomial.mapRingHom_id":{"signature":"theorem Polynomial.mapRingHom_id {R : Type u} [Semiring R] :mapRingHom (RingHom.id R) = RingHom.id (Polynomial R)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Coeff.lean#L129-L131"},"Polynomial.mapEquiv":{"signature":"def Polynomial.mapEquiv {R : Type u} {S : Type v} [Semiring R] [Semiring S] (e : R \u2243+* S) :Polynomial R \u2243+* Polynomial S","doc":"If R and S are isomorphic, then so are their polynomial rings.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Degree.lean#L129-L133"},"Polynomial.mapEquiv_apply":{"signature":"theorem Polynomial.mapEquiv_apply {R : Type u} {S : Type v} [Semiring R] [Semiring S] (e : R \u2243+* S) (a : Polynomial R) :(mapEquiv e) a = map (\u2191e) a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Degree.lean#L130-L130"},"Polynomial.reflect_eq_zero_iff":{"signature":"theorem Polynomial.reflect_eq_zero_iff {R : Type u_1} [Semiring R] {N : \u2115} {f : Polynomial R} :reflect N f = 0 \u2194 f = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Reverse.lean#L111-L113"},"Polynomial.reverse_eq_zero":{"signature":"theorem Polynomial.reverse_eq_zero {R : Type u_1} [Semiring R] {f : Polynomial R} :f.reverse = 0 \u2194 f = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Reverse.lean#L233-L234"},"Polynomial.card_support_eq_zero":{"signature":"theorem Polynomial.card_support_eq_zero {R : Type u} [Semiring R] {p : Polynomial R} :p.support.card = 0 \u2194 p = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L375-L375"},"Polynomial.support_zero":{"signature":"theorem Polynomial.support_zero {R : Type u} [Semiring R] :support 0 = \u2205","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L363-L365"},"Polynomial.card_support_eraseLead_add_one":{"signature":"theorem Polynomial.card_support_eraseLead_add_one {R : Type u_1} [Semiring R] {f : Polynomial R} (h : f \u2260 0) :f.eraseLead.support.card + 1 = f.support.card","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/EraseLead.lean#L101-L109"},"Polynomial.card_support_eraseLead":{"signature":"theorem Polynomial.card_support_eraseLead {R : Type u_1} [Semiring R] {f : Polynomial R} :f.eraseLead.support.card = f.support.card - 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/EraseLead.lean#L111-L115"},"Polynomial.card_support_eraseLead'":{"signature":"theorem Polynomial.card_support_eraseLead' {R : Type u_1} [Semiring R] {f : Polynomial R} {c : \u2115} (fc : f.support.card = c + 1) :f.eraseLead.support.card = c","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/EraseLead.lean#L117-L119"},"Polynomial.lt_natDegree_of_mem_eraseLead_support":{"signature":"theorem Polynomial.lt_natDegree_of_mem_eraseLead_support {R : Type u_1} [Semiring R] {f : Polynomial R} {a : \u2115} (h : a \u2208 f.eraseLead.support) :a < f.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/EraseLead.lean#L85-L88"},"Polynomial.ne_natDegree_of_mem_eraseLead_support":{"signature":"theorem Polynomial.ne_natDegree_of_mem_eraseLead_support {R : Type u_1} [Semiring R] {f : Polynomial R} {a : \u2115} (h : a \u2208 f.eraseLead.support) :a \u2260 f.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/EraseLead.lean#L90-L92"},"Polynomial.eraseLead_ne_zero":{"signature":"theorem Polynomial.eraseLead_ne_zero {R : Type u_1} [Semiring R] {f : Polynomial R} (f0 : 2 \u2264 f.support.card) :f.eraseLead \u2260 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/EraseLead.lean#L80-L83"},"Polynomial.eraseLead_natDegree_lt":{"signature":"theorem Polynomial.eraseLead_natDegree_lt {R : Type u_1} [Semiring R] {f : Polynomial R} (f0 : 2 \u2264 f.support.card) :f.eraseLead.natDegree < f.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/EraseLead.lean#L194-L197"},"Polynomial.induction_with_natDegree_le":{"signature":"theorem Polynomial.induction_with_natDegree_le {R : Type u_1} [Semiring R] (motive : Polynomial R \u2192 Prop) (N : \u2115) (zero : motive 0) (C_mul_pow : \u2200 (n : \u2115) (r : R), r \u2260 0 \u2192 n \u2264 N \u2192 motive (C r * X ^ n)) (add : \u2200 (f g : Polynomial R), f.natDegree < g.natDegree \u2192 g.natDegree \u2264 N \u2192 motive f \u2192 motive g \u2192 motive (f + g)) (f : Polynomial R) (df : f.natDegree \u2264 N) :motive f","doc":"An induction lemma for polynomials. It takes a natural number N as a parameter, that is required to be at least as big as the natDegree of the polynomial. This is useful to prove results where you want to change each term in a polynomial to something else depending on the natDegree of the polynomial itself and not on the specific natDegree of each term.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/EraseLead.lean#L315-L345"},"Polynomial.eval\u2082_reflect_mul_pow":{"signature":"theorem Polynomial.eval\u2082_reflect_mul_pow {R : Type u_1} [Semiring R] {S : Type u_2} [CommSemiring S] (i : R \u2192+* S) (x : S) [Invertible x] (N : \u2115) (f : Polynomial R) (hf : f.natDegree \u2264 N) :eval\u2082 i (\u215fx) (reflect N f) * x ^ N = eval\u2082 i x f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Reverse.lean#L192-L203"},"Polynomial.eval\u2082_reflect_eq_zero_iff":{"signature":"theorem Polynomial.eval\u2082_reflect_eq_zero_iff {R : Type u_1} [Semiring R] {S : Type u_2} [CommSemiring S] (i : R \u2192+* S) (x : S) [Invertible x] (N : \u2115) (f : Polynomial R) (hf : f.natDegree \u2264 N) :eval\u2082 i (\u215fx) (reflect N f) = 0 \u2194 eval\u2082 i x f = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Reverse.lean#L205-L213"},"Polynomial.eval\u2082_reverse_eq_zero_iff":{"signature":"theorem Polynomial.eval\u2082_reverse_eq_zero_iff {R : Type u_1} [Semiring R] {S : Type u_2} [CommSemiring S] (i : R \u2192+* S) (x : S) [Invertible x] (f : Polynomial R) :eval\u2082 i (\u215fx) f.reverse = 0 \u2194 eval\u2082 i x f = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Reverse.lean#L336-L339"},"Polynomial.aeval_X_left":{"signature":"theorem Polynomial.aeval_X_left {R : Type u} [CommSemiring R] :aeval X = AlgHom.id R (Polynomial R)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L382-L384"},"Polynomial.eval_unique":{"signature":"theorem Polynomial.eval_unique {R : Type u} {A : Type z} [CommSemiring R] [Semiring A] [Algebra R A] (\u03c6 : Polynomial R \u2192\u2090[R] A) (p : Polynomial R) :\u03c6 p = eval\u2082 (algebraMap R A) (\u03c6 X) p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L393-L394"},"Polynomial.support_nonempty":{"signature":"theorem Polynomial.support_nonempty {R : Type u} [Semiring R] {p : Polynomial R} :p.support.Nonempty \u2194 p \u2260 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L372-L373"},"Polynomial.coeff_C_mul_X":{"signature":"theorem Polynomial.coeff_C_mul_X {R : Type u} [Semiring R] (x : R) (n : \u2115) :(C x * X).coeff n = if n = 1 then x else 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L148-L149"},"Polynomial.degree_add_le_of_degree_le":{"signature":"theorem Polynomial.degree_add_le_of_degree_le {R : Type u} [Semiring R] {p q : Polynomial R} {n : \u2115} (hp : p.degree \u2264 \u2191n) (hq : q.degree \u2264 \u2191n) :(p + q).degree \u2264 \u2191n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L333-L335"},"Polynomial.degree_C_mul_X_le":{"signature":"theorem Polynomial.degree_C_mul_X_le {R : Type u} [Semiring R] (a : R) :(C a * X).degree \u2264 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L209-L210"},"Polynomial.degree_linear_le":{"signature":"theorem Polynomial.degree_linear_le {R : Type u} {a b : R} [Semiring R] :(C a * X + C b).degree \u2264 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/SmallDegree.lean#L69-L70"},"Polynomial.degree_quadratic_le":{"signature":"theorem Polynomial.degree_quadratic_le {R : Type u} {a b c : R} [Semiring R] :(C a * X ^ 2 + C b * X + C c).degree \u2264 2","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/SmallDegree.lean#L90-L93"},"Polynomial.degree_quadratic_lt":{"signature":"theorem Polynomial.degree_quadratic_lt {R : Type u} {a b c : R} [Semiring R] :(C a * X ^ 2 + C b * X + C c).degree < 3","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/SmallDegree.lean#L95-L96"},"Polynomial.degree_quadratic_lt_degree_C_mul_X_cb":{"signature":"theorem Polynomial.degree_quadratic_lt_degree_C_mul_X_cb {R : Type u} {a b c d : R} [Semiring R] (ha : a \u2260 0) :(C b * X ^ 2 + C c * X + C d).degree < (C a * X ^ 3).degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/SmallDegree.lean#L127-L129"},"Polynomial.degree_cubic":{"signature":"theorem Polynomial.degree_cubic {R : Type u} {a b c d : R} [Semiring R] (ha : a \u2260 0) :(C a * X ^ 3 + C b * X ^ 2 + C c * X + C d).degree = 3","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/SmallDegree.lean#L131-L136"},"Polynomial.leadingCoeff_cubic":{"signature":"theorem Polynomial.leadingCoeff_cubic {R : Type u} {a b c d : R} [Semiring R] (ha : a \u2260 0) :(C a * X ^ 3 + C b * X ^ 2 + C c * X + C d).leadingCoeff = a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/SmallDegree.lean#L144-L149"},"Polynomial.degree_modByMonic_le":{"signature":"theorem Polynomial.degree_modByMonic_le {R : Type u} [Ring R] (p : Polynomial R) {q : Polynomial R} (hq : q.Monic) :(p %\u2098 q).degree \u2264 q.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L216-L218"},"Polynomial.mem_image_comap_C_basicOpen":{"signature":"theorem Polynomial.mem_image_comap_C_basicOpen {R : Type u_1} [CommRing R] (f : Polynomial R) (x : PrimeSpectrum R) :x \u2208 PrimeSpectrum.comap C '' \u2191(PrimeSpectrum.basicOpen f) \u2194 \u2203 (i : \u2115), f.coeff i \u2209 x.asIdeal","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Spectrum/Prime/Polynomial.lean#L138-L150"},"Polynomial.image_comap_C_basicOpen":{"signature":"theorem Polynomial.image_comap_C_basicOpen {R : Type u_1} [CommRing R] (f : Polynomial R) :PrimeSpectrum.comap C '' \u2191(PrimeSpectrum.basicOpen f) = (PrimeSpectrum.zeroLocus (Set.range f.coeff))\u1d9c","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Spectrum/Prime/Polynomial.lean#L152-L156"},"Polynomial.modByMonic_eq_of_not_monic":{"signature":"theorem Polynomial.modByMonic_eq_of_not_monic {R : Type u} [Ring R] {q : Polynomial R} (p : Polynomial R) (hq : \u00acq.Monic) :p %\u2098 q = p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L207-L208"},"Polynomial.add_modByMonic":{"signature":"theorem Polynomial.add_modByMonic {R : Type u} [CommRing R] {q : Polynomial R} (p\u2081 p\u2082 : Polynomial R) :(p\u2081 + p\u2082) %\u2098 q = p\u2081 %\u2098 q + p\u2082 %\u2098 q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L670-L681"},"Polynomial.smul_modByMonic":{"signature":"theorem Polynomial.smul_modByMonic {R : Type u} [CommRing R] {q : Polynomial R} (c : R) (p : Polynomial R) :c \u2022 p %\u2098 q = c \u2022 (p %\u2098 q)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/RingDivision.lean#L54-L62"},"Polynomial.modByMonicHom":{"signature":"def Polynomial.modByMonicHom {R : Type u} [CommRing R] (q : Polynomial R) :Polynomial R \u2192\u2097[R] Polynomial R","doc":"_ %\u2098 q as an R-linear map.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/RingDivision.lean#L64-L69"},"Polynomial.mem_ker_modByMonic":{"signature":"theorem Polynomial.mem_ker_modByMonic {R : Type u} [CommRing R] {q : Polynomial R} (hq : q.Monic) {p : Polynomial R} :p \u2208 q.modByMonicHom.ker \u2194 q \u2223 p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/RingDivision.lean#L71-L73"},"Polynomial.natDegree_lt_iff_degree_lt":{"signature":"theorem Polynomial.natDegree_lt_iff_degree_lt {R : Type u} {n : \u2115} [Semiring R] {p : Polynomial R} (hp : p \u2260 0) :p.natDegree < n \u2194 p.degree < \u2191n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L142-L143"},"Polynomial.sum_fin":{"signature":"theorem Polynomial.sum_fin {R : Type u} {S : Type v} [Semiring R] [AddCommMonoid S] (f : \u2115 \u2192 R \u2192 S) (hf : \u2200 (i : \u2115), f i 0 = 0) {n : \u2115} {p : Polynomial R} (hn : p.degree < \u2191n) :\u2211 i : Fin n, f (\u2191i) (p.coeff \u2191i) = p.sum f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Support.lean#L77-L84"},"Polynomial.sum_modByMonic_coeff":{"signature":"theorem Polynomial.sum_modByMonic_coeff {R : Type u} [Ring R] {p q : Polynomial R} (hq : q.Monic) {n : \u2115} (hn : q.degree \u2264 \u2191n) :\u2211 i : Fin n, (monomial \u2191i) ((p %\u2098 q).coeff \u2191i) = p %\u2098 q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L426-L431"},"Polynomial.algebraMap_eq":{"signature":"theorem Polynomial.algebraMap_eq {R : Type u} [CommSemiring R] :algebraMap R (Polynomial R) = C","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L89-L91"},"Polynomial.modByMonic_X":{"signature":"theorem Polynomial.modByMonic_X {R : Type u} [CommRing R] (p : Polynomial R) :p %\u2098 X = C (eval 0 p)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L608-L609"},"Polynomial.X_mul_C":{"signature":"theorem Polynomial.X_mul_C {R : Type u} [Semiring R] (r : R) :X * C r = C r * X","doc":"Prefer putting constants to the left of X. This lemma is the loop-avoiding simp version of Polynomial.X_mul.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L514-L519"},"Polynomial.eval_neg":{"signature":"theorem Polynomial.eval_neg {R : Type u} [Ring R] (p : Polynomial R) (x : R) :eval x (-p) = -eval x p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L747-L749"},"Polynomial.coeff_isUnit_isNilpotent_of_isUnit":{"signature":"theorem Polynomial.coeff_isUnit_isNilpotent_of_isUnit {R : Type u_1} [CommRing R] {P : Polynomial R} (hunit : IsUnit P) :IsUnit (P.coeff 0) \u2227 \u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (P.coeff i)","doc":"Let P be a polynomial over R. If P is a unit, then all its coefficients are nilpotent, except its constant term which is a unit. See also Polynomial.isUnit_iff_coeff_isUnit_isNilpotent.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Nilpotent.lean#L131-L153"},"Polynomial.erase_monomial":{"signature":"theorem Polynomial.erase_monomial {R : Type u} [Semiring R] {n : \u2115} {a : R} :erase n ((monomial n) a) = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L959-L961"},"Polynomial.eraseLead_monomial":{"signature":"theorem Polynomial.eraseLead_monomial {R : Type u_1} [Semiring R] (i : \u2115) (r : R) :((monomial i) r).eraseLead = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/EraseLead.lean#L130-L136"},"Polynomial.eraseLead_C_mul_X_pow":{"signature":"theorem Polynomial.eraseLead_C_mul_X_pow {R : Type u_1} [Semiring R] (r : R) (n : \u2115) :(C r * X ^ n).eraseLead = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/EraseLead.lean#L150-L152"},"Polynomial.eraseLead_natDegree_lt_or_eraseLead_eq_zero":{"signature":"theorem Polynomial.eraseLead_natDegree_lt_or_eraseLead_eq_zero {R : Type u_1} [Semiring R] (f : Polynomial R) :f.eraseLead.natDegree < f.natDegree \u2228 f.eraseLead = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/EraseLead.lean#L204-L211"},"Polynomial.eraseLead_natDegree_le":{"signature":"theorem Polynomial.eraseLead_natDegree_le {R : Type u_1} [Semiring R] (f : Polynomial R) :f.eraseLead.natDegree \u2264 f.natDegree - 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/EraseLead.lean#L213-L216"},"Polynomial.eraseLead_coeff_of_ne":{"signature":"theorem Polynomial.eraseLead_coeff_of_ne {R : Type u_1} [Semiring R] {f : Polynomial R} (i : \u2115) (hi : i \u2260 f.natDegree) :f.eraseLead.coeff i = f.coeff i","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/EraseLead.lean#L54-L55"},"Polynomial.commute_X_pow":{"signature":"theorem Polynomial.commute_X_pow {R : Type u} [Semiring R] (p : Polynomial R) (n : \u2115) :Commute (X ^ n) p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L541-L542"},"Polynomial.C_pow":{"signature":"theorem Polynomial.C_pow {R : Type u} {a : R} {n : \u2115} [Semiring R] :C (a ^ n) = C a ^ n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L466-L467"},"Polynomial.isNilpotent_C_mul_pow_X_of_isNilpotent":{"signature":"theorem Polynomial.isNilpotent_C_mul_pow_X_of_isNilpotent {R : Type u_1} {r : R} [Semiring R] (n : \u2115) (hnil : IsNilpotent r) :IsNilpotent (C r * X ^ n)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Nilpotent.lean#L36-L41"},"Polynomial.isUnit_of_coeff_isUnit_isNilpotent":{"signature":"theorem Polynomial.isUnit_of_coeff_isUnit_isNilpotent {R : Type u_1} [CommRing R] {P : Polynomial R} (hunit : IsUnit (P.coeff 0)) (hnil : \u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (P.coeff i)) :IsUnit P","doc":"Let P be a polynomial over R. If its constant term is a unit and its other coefficients are nilpotent, then P is a unit. See also Polynomial.isUnit_iff_coeff_isUnit_isNilpotent.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Nilpotent.lean#L107-L129"},"Polynomial.isUnit_iff_coeff_isUnit_isNilpotent":{"signature":"theorem Polynomial.isUnit_iff_coeff_isUnit_isNilpotent {R : Type u_1} [CommRing R] {P : Polynomial R} :IsUnit P \u2194 IsUnit (P.coeff 0) \u2227 \u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (P.coeff i)","doc":"Let P be a polynomial over R. P is a unit if and only if all its coefficients are nilpotent, except its constant term which is a unit. See also Polynomial.isUnit_iff'.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Nilpotent.lean#L155-L161"},"Polynomial.coeff_zero_eq_aeval_zero":{"signature":"theorem Polynomial.coeff_zero_eq_aeval_zero {R : Type u} [CommSemiring R] (p : Polynomial R) :p.coeff 0 = (aeval 0) p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L470-L471"},"Polynomial.isNilpotent_C_iff":{"signature":"theorem Polynomial.isNilpotent_C_iff {R : Type u_1} {r : R} [Semiring R] :IsNilpotent (C r) \u2194 IsNilpotent r","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Nilpotent.lean#L52-L54"},"Polynomial.coeff_X_pow_mul":{"signature":"theorem Polynomial.coeff_X_pow_mul {R : Type u} [Semiring R] (p : Polynomial R) (n d : \u2115) :(X ^ n * p).coeff (d + n) = p.coeff d","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L233-L236"},"Polynomial.isRegular_X_pow":{"signature":"theorem Polynomial.isRegular_X_pow {R : Type u} [Semiring R] (n : \u2115) :IsRegular (X ^ n)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L279-L284"},"Polynomial.isRegular_X":{"signature":"theorem Polynomial.isRegular_X {R : Type u} [Semiring R] :IsRegular X","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L286-L286"},"Polynomial.isNilpotent_X_mul_iff":{"signature":"theorem Polynomial.isNilpotent_X_mul_iff {R : Type u_1} [Semiring R] {P : Polynomial R} :IsNilpotent (X * P) \u2194 IsNilpotent P","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Nilpotent.lean#L56-L61"},"Polynomial.isNilpotent_mul_X_iff":{"signature":"theorem Polynomial.isNilpotent_mul_X_iff {R : Type u_1} [Semiring R] {P : Polynomial R} :IsNilpotent (P * X) \u2194 IsNilpotent P","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Nilpotent.lean#L63-L66"},"Polynomial.monomial_pow":{"signature":"theorem Polynomial.monomial_pow {R : Type u} [Semiring R] (n : \u2115) (r : R) (k : \u2115) :(monomial n) r ^ k = (monomial (n * k)) (r ^ k)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L407-L411"},"Polynomial.monomial_eq_zero_iff":{"signature":"theorem Polynomial.monomial_eq_zero_iff {R : Type u} [Semiring R] (t : R) (n : \u2115) :(monomial n) t = 0 \u2194 t = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L420-L422"},"Polynomial.isNilpotent_monomial_iff":{"signature":"theorem Polynomial.isNilpotent_monomial_iff {R : Type u_1} {r : R} [Semiring R] {n : \u2115} :IsNilpotent ((monomial n) r) \u2194 IsNilpotent r","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Nilpotent.lean#L48-L50"},"Polynomial.isNilpotent_iff":{"signature":"theorem Polynomial.isNilpotent_iff {R : Type u_1} [CommRing R] {P : Polynomial R} :IsNilpotent P \u2194 \u2200 (i : \u2115), IsNilpotent (P.coeff i)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Nilpotent.lean#L74-L92"},"Polynomial.isUnit_C_add_X_mul_iff":{"signature":"theorem Polynomial.isUnit_C_add_X_mul_iff {R : Type u_1} {r : R} [CommRing R] {P : Polynomial R} :IsUnit (C r + X * P) \u2194 IsUnit r \u2227 IsNilpotent P","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Nilpotent.lean#L163-L168"},"Polynomial.isUnit_iff'":{"signature":"theorem Polynomial.isUnit_iff' {R : Type u_1} [CommRing R] {P : Polynomial R} :IsUnit P \u2194 IsUnit (eval 0 P) \u2227 IsNilpotent (P /\u2098 X)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Nilpotent.lean#L170-L175"},"Polynomial.natDegree_sub_le":{"signature":"theorem Polynomial.natDegree_sub_le {R : Type u} [Ring R] (p q : Polynomial R) :(p - q).natDegree \u2264 max p.natDegree q.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L536-L537"},"Polynomial.revAt_eq_self_of_lt":{"signature":"theorem Polynomial.revAt_eq_self_of_lt {N i : \u2115} (h : N < i) :(revAt N) i = i","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Reverse.lean#L69-L69"},"Polynomial.isNilpotent_reflect_iff":{"signature":"theorem Polynomial.isNilpotent_reflect_iff {R : Type u_1} [CommRing R] {P : Polynomial R} {N : \u2115} (hN : P.natDegree \u2264 N) :IsNilpotent (reflect N P) \u2194 IsNilpotent P","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Nilpotent.lean#L94-L101"},"Polynomial.reflect_neg":{"signature":"theorem Polynomial.reflect_neg {R : Type u_1} [Ring R] (f : Polynomial R) (N : \u2115) :reflect N (-f) = -reflect N f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Reverse.lean#L349-L351"},"Polynomial.reflect_sub":{"signature":"theorem Polynomial.reflect_sub {R : Type u_1} [Ring R] (f g : Polynomial R) (N : \u2115) :reflect N (f - g) = reflect N f - reflect N g","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Reverse.lean#L353-L355"},"Polynomial.toFinsuppIsoAlg":{"signature":"def Polynomial.toFinsuppIsoAlg (R : Type u) [CommSemiring R] :Polynomial R \u2243\u2090[R] AddMonoidAlgebra R \u2115","doc":"Algebra isomorphism between R[X] and R[\u2115]. This is just an implementation detail, but it can be useful to transfer results from Finsupp to polynomials.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L109-L115"},"Polynomial.toLaurentAlg":{"signature":"def Polynomial.toLaurentAlg {R : Type u_1} [CommSemiring R] :Polynomial R \u2192\u2090[R] LaurentPolynomial R","doc":"The R-algebra map, taking a polynomial with coefficients in R to a Laurent polynomial with coefficients in R.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Laurent.lean#L107-L110"},"Polynomial.toLaurent":{"signature":"def Polynomial.toLaurent {R : Type u_1} [Semiring R] :Polynomial R \u2192+* LaurentPolynomial R","doc":"The ring homomorphism, taking a polynomial with coefficients in R to a Laurent polynomial with coefficients in R.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Laurent.lean#L96-L99"},"Polynomial.toLaurent_C_mul_T":{"signature":"theorem Polynomial.toLaurent_C_mul_T {R : Type u_1} [Semiring R] (n : \u2115) (r : R) :toLaurent ((monomial n) r) = LaurentPolynomial.C r * LaurentPolynomial.T \u2191n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Laurent.lean#L186-L188"},"Polynomial.toLaurent_X":{"signature":"theorem Polynomial.toLaurent_X {R : Type u_1} [Semiring R] :toLaurent X = LaurentPolynomial.T 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Laurent.lean#L199-L202"},"Polynomial.toLaurent_C":{"signature":"theorem Polynomial.toLaurent_C {R : Type u_1} [Semiring R] (r : R) :toLaurent (C r) = LaurentPolynomial.C r","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Laurent.lean#L190-L193"},"Polynomial.toLaurent_comp_C":{"signature":"theorem Polynomial.toLaurent_comp_C {R : Type u_1} [Semiring R] :\u21d1toLaurent \u2218 \u21d1C = \u21d1LaurentPolynomial.C","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Laurent.lean#L195-L197"},"Polynomial.toLaurent_one":{"signature":"theorem Polynomial.toLaurent_one {R : Type u_1} [Semiring R] :toLaurent 1 = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Laurent.lean#L204-L206"},"Polynomial.toFinsupp_inj":{"signature":"theorem Polynomial.toFinsupp_inj {R : Type u} [Semiring R] {a b : Polynomial R} :a.toFinsupp = b.toFinsupp \u2194 a = b","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L227-L229"},"Polynomial.toLaurent_injective":{"signature":"theorem Polynomial.toLaurent_injective {R : Type u_1} [Semiring R] :Function.Injective \u21d1toLaurent","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Laurent.lean#L330-L332"},"Polynomial.revAt_zero":{"signature":"theorem Polynomial.revAt_zero (N : \u2115) :(revAt N) 0 = N","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Reverse.lean#L79-L79"},"Polynomial.reflect_C":{"signature":"theorem Polynomial.reflect_C {R : Type u_1} [Semiring R] (r : R) (N : \u2115) :reflect N (C r) = C r * X ^ N","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Reverse.lean#L129-L131"},"Polynomial.reverse_add_C":{"signature":"theorem Polynomial.reverse_add_C {R : Type u_1} [Semiring R] (p : Polynomial R) (t : R) :(p + C t).reverse = p.reverse + C t * X ^ p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Reverse.lean#L320-L322"},"Polynomial.toLaurent_C_mul_eq":{"signature":"theorem Polynomial.toLaurent_C_mul_eq {R : Type u_1} [Semiring R] (r : R) (f : Polynomial R) :toLaurent (C r * f) = LaurentPolynomial.C r * toLaurent f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Laurent.lean#L208-L211"},"Polynomial.toLaurent_X_pow":{"signature":"theorem Polynomial.toLaurent_X_pow {R : Type u_1} [Semiring R] (n : \u2115) :toLaurent (X ^ n) = LaurentPolynomial.T \u2191n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Laurent.lean#L213-L215"},"Polynomial.reverse_zero":{"signature":"theorem Polynomial.reverse_zero {R : Type u_1} [Semiring R] :reverse 0 = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Reverse.lean#L229-L231"},"Polynomial.reverse_mul_of_domain":{"signature":"theorem Polynomial.reverse_mul_of_domain {R : Type u_2} [Semiring R] [NoZeroDivisors R] (f g : Polynomial R) :(f * g).reverse = f.reverse * g.reverse","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Reverse.lean#L276-L283"},"Polynomial.natDegree_mul_X":{"signature":"theorem Polynomial.natDegree_mul_X {R : Type u} [Semiring R] [Nontrivial R] {p : Polynomial R} (hp : p \u2260 0) :(p * X).natDegree = p.natDegree + 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L514-L515"},"Polynomial.reflect_one_X":{"signature":"theorem Polynomial.reflect_one_X {R : Type u_1} [Semiring R] :reflect 1 X = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Reverse.lean#L137-L138"},"Polynomial.reverse_mul_X":{"signature":"theorem Polynomial.reverse_mul_X {R : Type u_1} [Semiring R] (p : Polynomial R) :(p * X).reverse = p.reverse","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Reverse.lean#L303-L307"},"Polynomial.coe_toLaurentAlg":{"signature":"theorem Polynomial.coe_toLaurentAlg {R : Type u_1} [CommSemiring R] :\u21d1toLaurentAlg = \u21d1toLaurent","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Laurent.lean#L112-L114"},"Polynomial.map_neg":{"signature":"theorem Polynomial.map_neg {R : Type u} [Ring R] {p : Polynomial R} {S : Type u_1} [Ring S] (f : R \u2192+* S) :map f (-p) = -map f p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L727-L729"},"Polynomial.idealSpan_range_update_divByMonic":{"signature":"theorem Polynomial.idealSpan_range_update_divByMonic {\u03b9 : Type u_1} {R : Type u_2} [CommRing R] [DecidableEq \u03b9] {i j : \u03b9} (hij : i \u2260 j) (v : \u03b9 \u2192 Polynomial R) (hi : (v i).Monic) :Ideal.span (Set.range (Function.update v j (v j %\u2098 v i))) = Ideal.span (Set.range v)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/CoeffMem.lean#L114-L117"},"Polynomial.not_monic_zero_iff":{"signature":"theorem Polynomial.not_monic_zero_iff {R : Type u} [Semiring R] :\u00acMonic 0 \u2194 0 \u2260 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Monic.lean#L40-L41"},"Polynomial.not_monic_zero":{"signature":"theorem Polynomial.not_monic_zero {R : Type u} [Semiring R] [Nontrivial R] :\u00acMonic 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Monic.lean#L476-L478"},"Polynomial.coeff_mul_X_pow'":{"signature":"theorem Polynomial.coeff_mul_X_pow' {R : Type u} [Semiring R] (p : Polynomial R) (n d : \u2115) :(p * X ^ n).coeff d = if n \u2264 d then p.coeff (d - n) else 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L238-L244"},"Polynomial.coeff_divModByMonicAux_mem_span_pow_mul_span":{"signature":"theorem Polynomial.coeff_divModByMonicAux_mem_span_pow_mul_span {R : Type u_2} {S : Type u_3} [CommRing R] [Ring S] [Algebra R S] (p q : Polynomial S) (hq : q.Monic) (i : \u2115) :(p.divModByMonicAux hq).1.coeff i \u2208 (1 \u2294 Submodule.span R (Set.range q.coeff)) ^ p.natDegree * (1 \u2294 Submodule.span R (Set.range p.coeff)) \u2227 (p.divModByMonicAux hq).2.coeff i \u2208 (1 \u2294 Submodule.span R (Set.range q.coeff)) ^ p.natDegree * (1 \u2294 Submodule.span R (Set.range p.coeff))","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/CoeffMem.lean#L32-L80"},"Polynomial.coeff_modByMonic_mem_pow_natDegree_mul":{"signature":"theorem Polynomial.coeff_modByMonic_mem_pow_natDegree_mul {R : Type u_2} {S : Type u_3} [CommRing R] [Ring S] [Algebra R S] (p q : Polynomial S) (Mp : Submodule R S) (hp : \u2200 (i : \u2115), p.coeff i \u2208 Mp) (hp' : 1 \u2208 Mp) (Mq : Submodule R S) (hq : \u2200 (i : \u2115), q.coeff i \u2208 Mq) (hq' : 1 \u2208 Mq) (i : \u2115) :(p %\u2098 q).coeff i \u2208 Mq ^ p.natDegree * Mp","doc":"For polynomials p q : R[X], the coefficients of p %\u2098 q can be written as sums of products of coefficients of p and q. Precisely, each summand needs at most one coefficient of p and deg p coefficients of q.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/CoeffMem.lean#L82-L95"},"Polynomial.modByMonic_one":{"signature":"theorem Polynomial.modByMonic_one {R : Type u} [Ring R] (p : Polynomial R) :p %\u2098 1 = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L418-L420"},"Polynomial.natDegree_eq_natDegree":{"signature":"theorem Polynomial.natDegree_eq_natDegree {R : Type u} {S : Type v} [Semiring R] [Semiring S] {p : Polynomial R} {q : Polynomial S} (hpq : p.degree = q.degree) :p.natDegree = q.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L79-L80"},"Polynomial.degree_map_lt":{"signature":"theorem Polynomial.degree_map_lt {R : Type u} {S : Type v} [Semiring R] [Semiring S] {f : R \u2192+* S} {p : Polynomial R} (hp : f p.leadingCoeff = 0) (hp\u2080 : p \u2260 0) :(map f p).degree < p.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Degree.lean#L153-L157"},"Polynomial.isLocalization":{"signature":"theorem Polynomial.isLocalization {R : Type u_5} [CommSemiring R] (S : Submonoid R) (A : Type u_6) [CommSemiring A] [Algebra R A] [IsLocalization S A] :IsLocalization (Submonoid.map C S) (Polynomial A)","doc":"If A is the localization of R at a submonoid S, then A[X] is the localization of R[X] at S.map Polynomial.C. See also MvPolynomial.isLocalization for the multivariate case.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Localization/Algebra.lean#L187-L197"},"Polynomial.map_surjective":{"signature":"theorem Polynomial.map_surjective {R : Type u} {S : Type v} [Semiring R] [Semiring S] (f : R \u2192+* S) (hf : Function.Surjective \u21d1f) :Function.Surjective (map f)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Coeff.lean#L109-L114"},"Polynomial.ker_mapRingHom":{"signature":"theorem Polynomial.ker_mapRingHom {R : Type u} {S : Type u_1} [CommSemiring R] [Semiring S] (f : R \u2192+* S) :RingHom.ker (mapRingHom f) = Ideal.map C (RingHom.ker f)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Basic.lean#L527-L532"},"Polynomial.coeff_map_eq_comp":{"signature":"theorem Polynomial.coeff_map_eq_comp {R : Type u} {S : Type v} [Semiring R] [Semiring S] (p : Polynomial R) (f : R \u2192+* S) :(map f p).coeff = \u21d1f \u2218 p.coeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Coeff.lean#L83-L84"},"Polynomial.coeff_C_ne_zero":{"signature":"theorem Polynomial.coeff_C_ne_zero {R : Type u} {a : R} {n : \u2115} [Semiring R] (h : n \u2260 0) :(C a).coeff n = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L640-L640"},"Polynomial.evalEval":{"signature":"abbrev Polynomial.evalEval {R : Type u_1} [Semiring R] (x y : R) (p : Polynomial (Polynomial R)) :R","doc":"evalEval x y p is the evaluation p(x,y) of a two-variable polynomial p : R[X][Y].","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Bivariate.lean#L43-L44"},"Polynomial.continuous_eval\u2082":{"signature":"theorem Polynomial.continuous_eval\u2082 {R : Type u_1} {S : Type u_2} [Semiring R] [TopologicalSpace R] [IsTopologicalSemiring R] [Semiring S] (p : Polynomial S) (f : S \u2192+* R) :Continuous fun (x : R) => eval\u2082 f x p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Topology/Algebra/Polynomial.lean#L50-L54"},"Polynomial.continuous":{"signature":"theorem Polynomial.continuous {R : Type u_1} [Semiring R] [TopologicalSpace R] [IsTopologicalSemiring R] (p : Polynomial R) :Continuous fun (x : R) => eval x p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Topology/Algebra/Polynomial.lean#L56-L58"},"Polynomial.toContinuousMap":{"signature":"def Polynomial.toContinuousMap {R : Type u_1} [Semiring R] [TopologicalSpace R] [IsTopologicalSemiring R] (p : Polynomial R) :C(R, R)","doc":"Every polynomial with coefficients in a topological semiring gives a (bundled) continuous function.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Topology/ContinuousMap/Polynomial.lean#L40-L45"},"Polynomial.toContinuousMapOn":{"signature":"def Polynomial.toContinuousMapOn {R : Type u_1} [Semiring R] [TopologicalSpace R] [IsTopologicalSemiring R] (p : Polynomial R) (X : Set R) :C(\u2191X, R)","doc":"A polynomial as a continuous function, with domain restricted to some subset of the semiring of coefficients. (This is particularly useful when restricting to compact sets, e.g. [0,1].)","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Topology/ContinuousMap/Polynomial.lean#L51-L58"},"Polynomial.toContinuousMapOnAlgHom":{"signature":"def Polynomial.toContinuousMapOnAlgHom {R : Type u_1} [CommSemiring R] [TopologicalSpace R] [IsTopologicalSemiring R] (X : Set R) :Polynomial R \u2192\u2090[R] C(\u2191X, R)","doc":"The algebra map from R[X] to continuous functions C(X, R), for any subset X of R.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Topology/ContinuousMap/Polynomial.lean#L110-L129"},"Polynomial.natCast_mul":{"signature":"theorem Polynomial.natCast_mul {R : Type u} [Semiring R] (n : \u2115) (p : Polynomial R) :\u2191n * p = n \u2022 p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L821-L822"},"Polynomial.aeval_continuousMap_apply":{"signature":"theorem Polynomial.aeval_continuousMap_apply {R : Type u_1} {\u03b1 : Type u_2} [TopologicalSpace \u03b1] [CommSemiring R] [TopologicalSpace R] [IsTopologicalSemiring R] (g : Polynomial R) (f : C(\u03b1, R)) (x : \u03b1) :((aeval f) g) x = eval (f x) g","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Topology/ContinuousMap/Polynomial.lean#L74-L81"},"Polynomial.C_eq_algebraMap":{"signature":"theorem Polynomial.C_eq_algebraMap {R : Type u} [CommSemiring R] (r : R) :C r = (algebraMap R (Polynomial R)) r","doc":"When we have [CommSemiring R], the function C is the same as algebraMap R R[X]. (But note that C is defined when R is not necessarily commutative, in which case algebraMap is not available.)","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L81-L87"},"Polynomial.toContinuousMapOn_X_eq_restrict_id":{"signature":"theorem Polynomial.toContinuousMapOn_X_eq_restrict_id {R : Type u_1} [Semiring R] [TopologicalSpace R] [IsTopologicalSemiring R] (s : Set R) :X.toContinuousMapOn s = ContinuousMap.restrict s (ContinuousMap.id R)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Topology/ContinuousMap/Polynomial.lean#L61-L63"},"Polynomial.toContinuousMapOnAlgHom_apply":{"signature":"theorem Polynomial.toContinuousMapOnAlgHom_apply {R : Type u_1} [CommSemiring R] [TopologicalSpace R] [IsTopologicalSemiring R] (X : Set R) (p : Polynomial R) :(toContinuousMapOnAlgHom X) p = p.toContinuousMapOn X","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Topology/ContinuousMap/Polynomial.lean#L112-L112"},"Polynomial.expand_aeval":{"signature":"theorem Polynomial.expand_aeval {R : Type u} [CommSemiring R] {A : Type u_1} [Semiring A] [Algebra R A] (p : \u2115) (P : Polynomial R) (r : A) :(aeval r) ((expand R p) P) = (aeval (r ^ p)) P","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Expand.lean#L176-L180"},"Polynomial.natSepDegree_le_natDegree":{"signature":"theorem Polynomial.natSepDegree_le_natDegree {F : Type u} [Field F] (f : Polynomial F) :f.natSepDegree \u2264 f.natDegree","doc":"The separable degree of a polynomial is smaller than its degree.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/SeparableDegree.lean#L301-L306"},"Polynomial.natSepDegree_eq_zero":{"signature":"theorem Polynomial.natSepDegree_eq_zero {F : Type u} [Field F] (f : Polynomial F) (h : f.natDegree = 0) :f.natSepDegree = 0","doc":"A constant polynomial has zero separable degree.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/SeparableDegree.lean#L316-L318"},"Polynomial.natSepDegree_C":{"signature":"theorem Polynomial.natSepDegree_C {F : Type u} [Field F] (x : F) :(C x).natSepDegree = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/SeparableDegree.lean#L320-L321"},"Polynomial.natSepDegree_zero":{"signature":"theorem Polynomial.natSepDegree_zero {F : Type u} [Field F] :natSepDegree 0 = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/SeparableDegree.lean#L323-L325"},"Polynomial.aroots_X_sub_C":{"signature":"theorem Polynomial.aroots_X_sub_C {S : Type v} {T : Type w} [CommRing T] [CommRing S] [IsDomain S] [Algebra T S] (r : T) :(X - C r).aroots S = {(algebraMap T S) r}","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L463-L466"},"Polynomial.natSepDegree_X_sub_C":{"signature":"theorem Polynomial.natSepDegree_X_sub_C {F : Type u} [Field F] (x : F) :(X - C x).natSepDegree = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/SeparableDegree.lean#L308-L310"},"Polynomial.natSepDegree_le_of_dvd":{"signature":"theorem Polynomial.natSepDegree_le_of_dvd {F : Type u} [Field F] (f g : Polynomial F) (h1 : f \u2223 g) (h2 : g \u2260 0) :f.natSepDegree \u2264 g.natSepDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/SeparableDegree.lean#L467-L472"},"Polynomial.natSepDegree_X_pow_char_pow_sub_C":{"signature":"theorem Polynomial.natSepDegree_X_pow_char_pow_sub_C {F : Type u} [Field F] (q : \u2115) [ExpChar F q] (n : \u2115) (y : F) :(X ^ q ^ n - C y).natSepDegree = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/SeparableDegree.lean#L486-L488"},"Polynomial.natSepDegree_ne_zero":{"signature":"theorem Polynomial.natSepDegree_ne_zero {F : Type u} [Field F] (f : Polynomial F) (h : f.natDegree \u2260 0) :f.natSepDegree \u2260 0","doc":"A non-constant polynomial has non-zero separable degree.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/SeparableDegree.lean#L331-L337"},"Polynomial.natSepDegree_eq_zero_iff":{"signature":"theorem Polynomial.natSepDegree_eq_zero_iff {F : Type u} [Field F] (f : Polynomial F) :f.natSepDegree = 0 \u2194 f.natDegree = 0","doc":"A polynomial has zero separable degree if and only if it is constant.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/SeparableDegree.lean#L339-L341"},"Polynomial.natSepDegree_ne_zero_iff":{"signature":"theorem Polynomial.natSepDegree_ne_zero_iff {F : Type u} [Field F] (f : Polynomial F) :f.natSepDegree \u2260 0 \u2194 f.natDegree \u2260 0","doc":"A polynomial has non-zero separable degree if and only if it is non-constant.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/SeparableDegree.lean#L343-L345"},"Polynomial.hasStrictDerivAt":{"signature":"theorem Polynomial.hasStrictDerivAt {\ud835\udd5c : Type u} [NontriviallyNormedField \ud835\udd5c] (p : Polynomial \ud835\udd5c) (x : \ud835\udd5c) :HasStrictDerivAt (fun (x : \ud835\udd5c) => eval x p) (eval x (derivative p)) x","doc":"The derivative (in the analysis sense) of a polynomial p is given by p.derivative.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Calculus/Deriv/Polynomial.lean#L52-L58"},"Polynomial.hasDerivAt":{"signature":"theorem Polynomial.hasDerivAt {\ud835\udd5c : Type u} [NontriviallyNormedField \ud835\udd5c] (p : Polynomial \ud835\udd5c) (x : \ud835\udd5c) :HasDerivAt (fun (x : \ud835\udd5c) => eval x p) (eval x (derivative p)) x","doc":"The derivative (in the analysis sense) of a polynomial p is given by p.derivative.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Calculus/Deriv/Polynomial.lean#L65-L67"},"Polynomial.differentiableAt":{"signature":"theorem Polynomial.differentiableAt {\ud835\udd5c : Type u} [NontriviallyNormedField \ud835\udd5c] {x : \ud835\udd5c} (p : Polynomial \ud835\udd5c) :DifferentiableAt \ud835\udd5c (fun (x : \ud835\udd5c) => eval x p) x","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Calculus/Deriv/Polynomial.lean#L81-L82"},"Polynomial.differentiable":{"signature":"theorem Polynomial.differentiable {\ud835\udd5c : Type u} [NontriviallyNormedField \ud835\udd5c] (p : Polynomial \ud835\udd5c) :Differentiable \ud835\udd5c fun (x : \ud835\udd5c) => eval x p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Calculus/Deriv/Polynomial.lean#L94-L95"},"Polynomial.degree_pos_induction_on":{"signature":"theorem Polynomial.degree_pos_induction_on {R : Type u} [Semiring R] {P : Polynomial R \u2192 Prop} (p : Polynomial R) (h0 : 0 < p.degree) (hC : \u2200 {a : R}, a \u2260 0 \u2192 P (C a * X)) (hX : \u2200 {p : Polynomial R}, 0 < p.degree \u2192 P p \u2192 P (p * X)) (hadd : \u2200 {p : Polynomial R} {a : R}, 0 < p.degree \u2192 P p \u2192 P (p + C a)) :P p","doc":"A property holds for all polynomials of positive degree with coefficients in a semiring R if it holds for See natDegree_ne_zero_induction_on for a similar statement involving no explicit multiplication.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Inductions.lean#L155-L180"},"Polynomial.leadingCoeff_mul_X":{"signature":"theorem Polynomial.leadingCoeff_mul_X {R : Type u} [Semiring R] {p : Polynomial R} :(p * X).leadingCoeff = p.leadingCoeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L416-L418"},"Polynomial.tendsto_abv_eval\u2082_atTop":{"signature":"theorem Polynomial.tendsto_abv_eval\u2082_atTop {R : Type u_1} {S : Type u_2} {k : Type u_3} {\u03b1 : Type u_4} [Semiring R] [Ring S] [Field k] [LinearOrder k] [IsStrictOrderedRing k] (f : R \u2192+* S) (abv : S \u2192 k) [IsAbsoluteValue abv] (p : Polynomial R) (hd : 0 < p.degree) (hf : f p.leadingCoeff \u2260 0) {l : Filter \u03b1} {z : \u03b1 \u2192 S} (hz : Filter.Tendsto (abv \u2218 z) l Filter.atTop) :Filter.Tendsto (fun (x : \u03b1) => abv (eval\u2082 f (z x) p)) l Filter.atTop","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Topology/Algebra/Polynomial.lean#L98-L114"},"Polynomial.tendsto_abv_atTop":{"signature":"theorem Polynomial.tendsto_abv_atTop {R : Type u_1} {k : Type u_2} {\u03b1 : Type u_3} [Ring R] [Field k] [LinearOrder k] [IsStrictOrderedRing k] (abv : R \u2192 k) [IsAbsoluteValue abv] (p : Polynomial R) (h : 0 < p.degree) {l : Filter \u03b1} {z : \u03b1 \u2192 R} (hz : Filter.Tendsto (abv \u2218 z) l Filter.atTop) :Filter.Tendsto (fun (x : \u03b1) => abv (eval (z x) p)) l Filter.atTop","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Topology/Algebra/Polynomial.lean#L116-L121"},"Polynomial.tendsto_norm_atTop":{"signature":"theorem Polynomial.tendsto_norm_atTop {\u03b1 : Type u_1} {R : Type u_2} [NormedRing R] [IsAbsoluteValue norm] (p : Polynomial R) (h : 0 < p.degree) {l : Filter \u03b1} {z : \u03b1 \u2192 R} (hz : Filter.Tendsto (fun (x : \u03b1) => \u2016z x\u2016) l Filter.atTop) :Filter.Tendsto (fun (x : \u03b1) => \u2016eval (z x) p\u2016) l Filter.atTop","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Topology/Algebra/Polynomial.lean#L132-L134"},"Polynomial.zero_of_eval_zero":{"signature":"theorem Polynomial.zero_of_eval_zero {R : Type u} [CommRing R] [IsDomain R] [Infinite R] (p : Polynomial R) (h : \u2200 (x : R), eval x p = 0) :p = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L423-L427"},"Polynomial.funext":{"signature":"theorem Polynomial.funext {R : Type u} [CommRing R] [IsDomain R] [Infinite R] {p q : Polynomial R} (ext : \u2200 (r : R), eval r p = eval r q) :p = q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L429-L433"},"Polynomial.ext_iff_natDegree_le":{"signature":"theorem Polynomial.ext_iff_natDegree_le {R : Type u} [Semiring R] {p q : Polynomial R} {n : \u2115} (hp : p.natDegree \u2264 n) (hq : q.natDegree \u2264 n) :p = q \u2194 \u2200 i \u2264 n, p.coeff i = q.coeff i","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L93-L100"},"Polynomial.bUnion_roots_finite":{"signature":"theorem Polynomial.bUnion_roots_finite {R : Type u_1} {S : Type u_2} [Semiring R] [CommRing S] [IsDomain S] [DecidableEq S] (m : R \u2192+* S) (d : \u2115) {U : Set R} (h : U.Finite) :(\u22c3 (f : Polynomial R), \u22c3 (_ : f.natDegree \u2264 d \u2227 \u2200 (i : \u2115), f.coeff i \u2208 U), \u2191(map m f).roots.toFinset).Finite","doc":"The set of roots of all polynomials of bounded degree and having coefficients in a finite set is finite.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L571-L586"},"Polynomial.eq_one_of_roots_le":{"signature":"theorem Polynomial.eq_one_of_roots_le {F : Type u_3} {K : Type u_4} [CommRing F] [NormedField K] {p : Polynomial F} {f : F \u2192+* K} {B : \u211d} (hB : B < 0) (h1 : p.Monic) (h2 : (map f p).Splits) (h3 : \u2200 z \u2208 (map f p).roots, \u2016z\u2016 \u2264 B) :p = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Topology/Algebra/Polynomial.lean#L164-L170"},"Polynomial.X_pow_mul_C":{"signature":"theorem Polynomial.X_pow_mul_C {R : Type u} [Semiring R] (r : R) (n : \u2115) :X ^ n * C r = C r * X ^ n","doc":"Prefer putting constants to the left of X ^ n. This lemma is the loop-avoiding simp version of X_pow_mul.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L521-L526"},"Polynomial.coeff_eq_esymm_roots_of_card":{"signature":"theorem Polynomial.coeff_eq_esymm_roots_of_card {R : Type u_1} [CommRing R] [IsDomain R] {p : Polynomial R} (hroots : p.roots.card = p.natDegree) {k : \u2115} (h : k \u2264 p.natDegree) :p.coeff k = p.leadingCoeff * (-1) ^ (p.natDegree - k) * p.roots.esymm (p.natDegree - k)","doc":"Vieta's formula for the coefficients and the roots of a polynomial over an integral domain with as many roots as its degree.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Vieta.lean#L116-L124"},"Polynomial.coeff_eq_esymm_roots_of_splits":{"signature":"theorem Polynomial.coeff_eq_esymm_roots_of_splits {F : Type u_2} [Field F] {p : Polynomial F} (hsplit : p.Splits) {k : \u2115} (h : k \u2264 p.natDegree) :p.coeff k = p.leadingCoeff * (-1) ^ (p.natDegree - k) * p.roots.esymm (p.natDegree - k)","doc":"Vieta's formula for split polynomials over a field.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Vieta.lean#L126-L130"},"Polynomial.coeff_le_of_roots_le":{"signature":"theorem Polynomial.coeff_le_of_roots_le {F : Type u_3} {K : Type u_4} [CommRing F] [NormedField K] {p : Polynomial F} {f : F \u2192+* K} {B : \u211d} (i : \u2115) (h1 : p.Monic) (h2 : (map f p).Splits) (h3 : \u2200 z \u2208 (map f p).roots, \u2016z\u2016 \u2264 B) :\u2016(map f p).coeff i\u2016 \u2264 B ^ (p.natDegree - i) * \u2191(p.natDegree.choose i)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Topology/Algebra/Polynomial.lean#L172-L197"},"Polynomial.coeff_bdd_of_roots_le":{"signature":"theorem Polynomial.coeff_bdd_of_roots_le {F : Type u_3} {K : Type u_4} [CommRing F] [NormedField K] {B : \u211d} {d : \u2115} (f : F \u2192+* K) {p : Polynomial F} (h1 : p.Monic) (h2 : (map f p).Splits) (h3 : p.natDegree \u2264 d) (h4 : \u2200 z \u2208 (map f p).roots, \u2016z\u2016 \u2264 B) (i : \u2115) :\u2016(map f p).coeff i\u2016 \u2264 max B 1 ^ d * \u2191(d.choose (d / 2))","doc":"The coefficients of the monic polynomials of bounded degree with bounded roots are uniformly bounded.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Topology/Algebra/Polynomial.lean#L199-L217"},"Polynomial.sylvester":{"signature":"def Polynomial.sylvester {R : Type u_1} [Semiring R] (f g : Polynomial R) (m n : \u2115) :Matrix (Fin (m + n)) (Fin (m + n)) R","doc":"The Sylvester matrix of two polynomials f and g of degrees m and n respectively is a (m+n) \u00d7 (m+n) matrix with the coefficients of f and g arranged in a specific way. Here, m and n are free variables, not necessarily equal to the actual degrees of the polynomials f and g. Note that the natural definition would be a Matrix (Fin (m + n)) (Fin m \u2295 Fin n) R but we prefer having this as a square matrix to take determinants later on.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L51-L62"},"Polynomial.resultant":{"signature":"def Polynomial.resultant {R : Type u_1} [CommRing R] (f g : Polynomial R) (m : \u2115 := f.natDegree) (n : \u2115 := g.natDegree) :R","doc":"The resultant of two polynomials f and g is the determinant of the Sylvester matrix of f and g. The size arguments m and n are implemented as optParam, meaning that the default values are f.natDegree and g.natDegree respectively, but they can also be specified to be other values.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L135-L140"},"Polynomial.sylvester_zero_left_deg":{"signature":"theorem Polynomial.sylvester_zero_left_deg {R : Type u_1} [Semiring R] (f g : Polynomial R) (m : \u2115) :f.sylvester g 0 m = Matrix.diagonal fun (x : Fin (0 + m)) => f.coeff 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L66-L70"},"Polynomial.resultant_zero_left_deg":{"signature":"theorem Polynomial.resultant_zero_left_deg {R : Type u_1} [CommRing R] (f g : Polynomial R) (m : \u2115) :f.resultant g 0 m = f.coeff 0 ^ m","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L149-L150"},"Polynomial.sylvester_comm":{"signature":"theorem Polynomial.sylvester_comm {R : Type u_1} [Semiring R] (f g : Polynomial R) (m n : \u2115) :f.sylvester g m n = (Matrix.reindex (finCongr \u22ef) (finSumFinEquiv.symm.trans ((Equiv.sumComm (Fin n) (Fin m)).trans finSumFinEquiv))) (g.sylvester f n m)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L72-L76"},"Polynomial.resultant_comm":{"signature":"theorem Polynomial.resultant_comm {R : Type u_1} [CommRing R] (f g : Polynomial R) (m n : \u2115) :f.resultant g m n = (-1) ^ (m * n) * g.resultant f n m","doc":"Res(f, g) = (-1)\u1d50\u207f Res(g, f)","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L155-L169"},"Polynomial.resultant_zero_right_deg":{"signature":"theorem Polynomial.resultant_zero_right_deg {R : Type u_1} [CommRing R] (f g : Polynomial R) (m : \u2115) :f.resultant g m 0 = g.coeff 0 ^ m","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L171-L172"},"Polynomial.induction_of_Splits_of_injective_of_surjective":{"signature":"theorem Polynomial.induction_of_Splits_of_injective_of_surjective {R : Type u} [CommRing R] (p : Polynomial R) (P : {R : Type u} \u2192 [inst : CommRing R] \u2192 Polynomial R \u2192 Prop) (Splits : \u2200 (R : Type u) [inst : Field R] (p : Polynomial R), p.Splits \u2192 P p) (injective : \u2200 (R S : Type u) [inst : CommRing R] [inst_1 : CommRing S] (\u03c6 : R \u2192+* S), Function.Injective \u21d1\u03c6 \u2192 \u2200 (p : Polynomial R), P (map \u03c6 p) \u2192 P p) (surjective : \u2200 (R S : Type u) [inst : CommRing R] [inst_1 : CommRing S] (\u03c6 : R \u2192+* S), Function.Surjective \u21d1\u03c6 \u2192 \u2200 (p : Polynomial S), (\u2200 (q : Polynomial R), P q) \u2192 P p) :P p","doc":"An induction principle useful to prove statements about resultants. Let P be a predicate on a polynomial. If R \u2192 S injective implies (\u2200 p : S[X], P p) \u2192 (\u2200 p : R[X], P p), and if R \u2192 S surjective implies (\u2200 p : R[X], P p) \u2192 (\u2200 p : S[X], P p), then we may reduce to the case where R is a field and p splits.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L530-L555"},"Polynomial.resultant_zero_right":{"signature":"theorem Polynomial.resultant_zero_right {R : Type u_1} [CommRing R] (f : Polynomial R) (m n : \u2115) :f.resultant 0 m n = 0 ^ m * f.coeff 0 ^ n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L177-L182"},"Polynomial.sum_eq_of_subset":{"signature":"theorem Polynomial.sum_eq_of_subset {R : Type u} [Semiring R] {S : Type u_1} [AddCommMonoid S] {p : Polynomial R} (f : \u2115 \u2192 R \u2192 S) (hf : \u2200 (i : \u2115), f i 0 = 0) {s : Finset \u2115} (hs : p.support \u2286 s) :p.sum f = \u2211 n \u2208 s, f n (p.coeff n)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L832-L835"},"Polynomial.resultant_add_mul_right":{"signature":"theorem Polynomial.resultant_add_mul_right {R : Type u_1} [CommRing R] (f g p : Polynomial R) (m n : \u2115) (hp : p.natDegree + m \u2264 n) (hf : f.natDegree \u2264 m) :f.resultant (g + f * p) m n = f.resultant g m n","doc":"Res(f, g + fp) = Res(f, g) if deg f + deg p \u2264 deg g.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L240-L255"},"Polynomial.resultant_add_mul_left":{"signature":"theorem Polynomial.resultant_add_mul_left {R : Type u_1} [CommRing R] (f g p : Polynomial R) (m n : \u2115) (hk : p.natDegree + n \u2264 m) (hg : g.natDegree \u2264 n) :(f + g * p).resultant g m n = f.resultant g m n","doc":"Res(f + gp, g) = Res(f, g) if deg g + deg p \u2264 deg f.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L257-L262"},"Polynomial.resultant_zero_left":{"signature":"theorem Polynomial.resultant_zero_left {R : Type u_1} [CommRing R] (g : Polynomial R) (m n : \u2115) :resultant 0 g m n = 0 ^ n * g.coeff 0 ^ m","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L184-L187"},"Polynomial.resultant_succ_left_deg":{"signature":"theorem Polynomial.resultant_succ_left_deg {R : Type u_1} [CommRing R] (f g : Polynomial R) (m n : \u2115) (hf : f.natDegree \u2264 m) :f.resultant g (m + 1) n = (-1) ^ n * g.coeff n * f.resultant g m n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L299-L339"},"Polynomial.resultant_add_left_deg":{"signature":"theorem Polynomial.resultant_add_left_deg {R : Type u_1} [CommRing R] (f g : Polynomial R) (m n k : \u2115) (hf : f.natDegree \u2264 m) :f.resultant g (m + k) n = (-1) ^ (n * k) * g.coeff n ^ k * f.resultant g m n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L341-L345"},"Polynomial.resultant_C_mul_right":{"signature":"theorem Polynomial.resultant_C_mul_right {R : Type u_1} [CommRing R] (f g : Polynomial R) (m n : \u2115) (r : R) :f.resultant (C r * g) m n = r ^ m * f.resultant g m n","doc":"Res(f, a \u2022 g) = a ^ {deg f} * Res(f, g).","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L264-L291"},"Polynomial.sylvester_map_map":{"signature":"theorem Polynomial.sylvester_map_map {R : Type u_1} {S : Type u_2} [Semiring R] [Semiring S] (f g : Polynomial R) (m n : \u2115) (\u03c6 : R \u2192+* S) :(map \u03c6 f).sylvester (map \u03c6 g) m n = \u03c6.mapMatrix (f.sylvester g m n)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L78-L80"},"Polynomial.resultant_map_map":{"signature":"theorem Polynomial.resultant_map_map {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] (f g : Polynomial R) (m n : \u2115) (\u03c6 : R \u2192+* S) :(map \u03c6 f).resultant (map \u03c6 g) m n = \u03c6 (f.resultant g m n)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L144-L147"},"Polynomial.map_sub_sprod_roots_eq_prod_map_eval":{"signature":"theorem Polynomial.map_sub_sprod_roots_eq_prod_map_eval {R : Type u_1} [CommRing R] [IsDomain R] (s : Multiset R) (g : Polynomial R) (hg : g.Monic) (hg' : g.Splits) :(Multiset.map (fun (ij : R \u00d7 R) => ij.1 - ij.2) (s \u00d7\u02e2 g.roots)).prod = (Multiset.map (fun (x : R) => eval x g) s).prod","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L501-L509"},"Polynomial.map_sub_roots_sprod_eq_prod_map_eval":{"signature":"theorem Polynomial.map_sub_roots_sprod_eq_prod_map_eval {R : Type u_1} [CommRing R] [IsDomain R] (s : Multiset R) (g : Polynomial R) (hg : g.Monic) (hg' : g.Splits) :(Multiset.map (fun (ij : R \u00d7 R) => ij.1 - ij.2) (g.roots \u00d7\u02e2 s)).prod = (-1) ^ (s.card * g.roots.card) * (Multiset.map (fun (x : R) => eval x g) s).prod","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L511-L517"},"Polynomial.resultant_eq_prod_roots_sub":{"signature":"theorem Polynomial.resultant_eq_prod_roots_sub {K : Type u_3} [Field K] (f g : Polynomial K) (hf : f.Monic) (hg : g.Monic) (hf' : f.Splits) (hg' : g.Splits) :f.resultant g = (Multiset.map (fun (ij : K \u00d7 K) => ij.1 - ij.2) (f.roots \u00d7\u02e2 g.roots)).prod","doc":"If f and g are monic and splits, then Res(f, g) = \u220f (\u03b1 - \u03b2), where \u03b1 and \u03b2 runs through the roots of f and g respectively.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L407-L477"},"Polynomial.resultant_add_right_deg":{"signature":"theorem Polynomial.resultant_add_right_deg {R : Type u_1} [CommRing R] (f g : Polynomial R) (m n k : \u2115) (hg : g.natDegree \u2264 n) :f.resultant g m (n + k) = f.coeff m ^ k * f.resultant g m n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L347-L353"},"Polynomial.resultant_C_mul_left":{"signature":"theorem Polynomial.resultant_C_mul_left {R : Type u_1} [CommRing R] (f g : Polynomial R) (m n : \u2115) (r : R) :(C r * f).resultant g m n = r ^ n * f.resultant g m n","doc":"Res(a \u2022 f, g) = a ^ {deg g} * Res(f, g).","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L293-L297"},"Polynomial.resultant_eq_prod_eval":{"signature":"theorem Polynomial.resultant_eq_prod_eval {R : Type u_1} [CommRing R] [IsDomain R] (f g : Polynomial R) (n : \u2115) (hg : g.natDegree \u2264 n) (hf : f.Splits) :f.resultant g f.natDegree n = f.leadingCoeff ^ n * (Multiset.map (fun (x : R) => eval x g) f.roots).prod","doc":"If f splits with leading coeff a and degree n, then Res(f, g) = a\u207f * \u220f g(\u03b1) where \u03b1 runs through the roots of f.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L479-L526"},"Polynomial.mem_lifts_and_degree_eq":{"signature":"theorem Polynomial.mem_lifts_and_degree_eq {R : Type u} [Semiring R] {S : Type v} [Semiring S] {f : R \u2192+* S} {p : Polynomial S} (hlifts : p \u2208 lifts f) :\u2203 (q : Polynomial R), map f q = p \u2227 q.degree = p.degree","doc":"A polynomial lifts if and only if it can be lifted to a polynomial of the same degree.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Lifts.lean#L144-L156"},"Polynomial.resultant_mul_right":{"signature":"theorem Polynomial.resultant_mul_right {R : Type u_1} [CommRing R] (f g\u2081 g\u2082 : Polynomial R) (m : \u2115) (hm : f.natDegree \u2264 m) :f.resultant (g\u2081 * g\u2082) m (g\u2081.natDegree + g\u2082.natDegree) = f.resultant g\u2081 m * f.resultant g\u2082 m","doc":"Res(f, g\u2081 * g\u2082) = Res(f, g\u2081) * Res(f, g\u2082).","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L557-L581"},"Polynomial.resultant_C_right":{"signature":"theorem Polynomial.resultant_C_right {R : Type u_1} [CommRing R] (f : Polynomial R) (m n : \u2115) (r : R) :f.resultant (C r) m n = f.coeff m ^ n * r ^ m","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L368-L370"},"Polynomial.resultant_one_right":{"signature":"theorem Polynomial.resultant_one_right {R : Type u_1} [CommRing R] (f : Polynomial R) (m n : \u2115) :f.resultant 1 m n = f.coeff m ^ n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L375-L376"},"Polynomial.resultant_dvd_leadingCoeff_pow":{"signature":"theorem Polynomial.resultant_dvd_leadingCoeff_pow {R : Type u_1} [CommRing R] [IsDomain R] (f g : Polynomial R) (H : IsCoprime f g) :\u2203 (n : \u2115), f.resultant g \u2223 f.leadingCoeff ^ n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L610-L633"},"Polynomial.resultant_ne_zero":{"signature":"theorem Polynomial.resultant_ne_zero {R : Type u_1} [CommRing R] [IsDomain R] (f g : Polynomial R) (H : IsCoprime f g) :f.resultant g \u2260 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L635-L641"},"Polynomial.rootMultiplicity_sub_one_le_derivative_rootMultiplicity_of_ne_zero":{"signature":"theorem Polynomial.rootMultiplicity_sub_one_le_derivative_rootMultiplicity_of_ne_zero {R : Type u} [CommRing R] (p : Polynomial R) (t : R) (hnezero : derivative p \u2260 0) :rootMultiplicity t p - 1 \u2264 rootMultiplicity t (derivative p)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L41-L45"},"Polynomial.sylvesterDeriv":{"signature":"noncomputable def Polynomial.sylvesterDeriv {R : Type u_1} [Semiring R] (f : Polynomial R) :Matrix (Fin (f.natDegree - 1 + f.natDegree)) (Fin (f.natDegree - 1 + f.natDegree)) R","doc":"The Sylvester matrix for f and f.derivative, modified by dividing the bottom row by the leading coefficient of f. Important because its determinant is (up to a sign) the discriminant of f.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L82-L92"},"Polynomial.discr":{"signature":"noncomputable def Polynomial.discr {R : Type u_1} [CommRing R] (f : Polynomial R) :R","doc":"The discriminant of a polynomial, defined as the determinant of f.sylvesterDeriv modified by a sign. The sign is chosen so polynomials over \u211d with all roots real have non-negative discriminant.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L930-L934"},"Polynomial.evalEvalRingHom":{"signature":"abbrev Polynomial.evalEvalRingHom {R : Type u_1} [CommSemiring R] (x y : R) :Polynomial (Polynomial R) \u2192+* R","doc":"evalEval x y as a ring homomorphism.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Bivariate.lean#L144-L146"},"Polynomial.coe_evalEvalRingHom":{"signature":"theorem Polynomial.coe_evalEvalRingHom {R : Type u_1} [CommSemiring R] (x y : R) :\u21d1(evalEvalRingHom x y) = evalEval x y","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Bivariate.lean#L148-L148"},"Polynomial.evalEvalRingHom_eq":{"signature":"theorem Polynomial.evalEvalRingHom_eq {R : Type u_1} [CommSemiring R] (x : R) :evalEvalRingHom x = eval\u2082RingHom (evalRingHom x)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Bivariate.lean#L150-L151"},"Polynomial.eval\u2082_evalRingHom":{"signature":"theorem Polynomial.eval\u2082_evalRingHom {R : Type u_1} [CommSemiring R] (x : R) :eval\u2082 (evalRingHom x) = evalEval x","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Bivariate.lean#L153-L154"},"Polynomial.coe_aeval_eq_eval":{"signature":"theorem Polynomial.coe_aeval_eq_eval {R : Type u} [CommSemiring R] (r : R) :\u21d1(aeval r) = eval r","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L451-L453"},"Polynomial.rootSet_finite":{"signature":"theorem Polynomial.rootSet_finite {T : Type w} [CommRing T] (p : Polynomial T) (S : Type u_1) [CommRing S] [IsDomain S] [Algebra T S] :(p.rootSet S).Finite","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L567-L569"},"Polynomial.Gal":{"signature":"def Polynomial.Gal {F : Type u_1} [Field F] (p : Polynomial F) :Type u_1","doc":"The Galois group of a polynomial.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/PolynomialGaloisGroup.lean#L54-L57"},"Polynomial.instGroupGal":{"signature":"instance Polynomial.instGroupGal {F : Type u_1} [Field F] (p : Polynomial F) :Group p.Gal","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/PolynomialGaloisGroup.lean#L57-L57"},"Polynomial.trinomial":{"signature":"noncomputable def Polynomial.trinomial {R : Type u_1} [Semiring R] (k m n : \u2115) (u v w : R) :Polynomial R","doc":"Shorthand for a trinomial","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/UnitTrinomial.lean#L42-L44"},"Polynomial.trinomial_def":{"signature":"theorem Polynomial.trinomial_def {R : Type u_1} [Semiring R] (k m n : \u2115) (u v w : R) :trinomial k m n u v w = C u * X ^ k + C v * X ^ m + C w * X ^ n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/UnitTrinomial.lean#L46-L47"},"Polynomial.trinomial_middle_coeff":{"signature":"theorem Polynomial.trinomial_middle_coeff {R : Type u_1} [Semiring R] {k m n : \u2115} {u v w : R} (hkm : k < m) (hmn : m < n) :(trinomial k m n u v w).coeff m = v","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/UnitTrinomial.lean#L56-L59"},"Polynomial.comp_eq_sum_left":{"signature":"theorem Polynomial.comp_eq_sum_left {R : Type u} [Semiring R] {p q : Polynomial R} :p.comp q = p.sum fun (e : \u2115) (a : R) => C a * q ^ e","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L372-L372"},"Polynomial.degree_one_le":{"signature":"theorem Polynomial.degree_one_le {R : Type u} [Semiring R] :degree 1 \u2264 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L165-L165"},"Polynomial.degree_pow_le":{"signature":"theorem Polynomial.degree_pow_le {R : Type u} [Semiring R] (p : Polynomial R) (n : \u2115) :(p ^ n).degree \u2264 n \u2022 p.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L404-L406"},"Polynomial.natDegree_comp_le":{"signature":"theorem Polynomial.natDegree_comp_le {R : Type u} [Semiring R] {p q : Polynomial R} :(p.comp q).natDegree \u2264 p.natDegree * q.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L40-L63"},"Polynomial.natDegree_comp_eq_of_mul_ne_zero":{"signature":"theorem Polynomial.natDegree_comp_eq_of_mul_ne_zero {R : Type u} [Semiring R] {p q : Polynomial R} (h : p.leadingCoeff * q.leadingCoeff ^ p.natDegree \u2260 0) :(p.comp q).natDegree = p.natDegree * q.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L65-L70"},"Polynomial.comp_C":{"signature":"theorem Polynomial.comp_C {R : Type u} {a : R} [Semiring R] {p : Polynomial R} :p.comp (C a) = C (eval a p)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L383-L384"},"Polynomial.zero_comp":{"signature":"theorem Polynomial.zero_comp {R : Type u} [Semiring R] {p : Polynomial R} :comp 0 p = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L400-L401"},"Polynomial.natDegree_comp":{"signature":"theorem Polynomial.natDegree_comp {R : Type u} [Semiring R] {p q : Polynomial R} [NoZeroDivisors R] :(p.comp q).natDegree = p.natDegree * q.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L383-L390"},"Polynomial.leadingCoeff_comp":{"signature":"theorem Polynomial.leadingCoeff_comp {R : Type u} [Semiring R] {p q : Polynomial R} [NoZeroDivisors R] (hq : q.natDegree \u2260 0) :(p.comp q).leadingCoeff = p.leadingCoeff * q.leadingCoeff ^ p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L399-L401"},"Polynomial.one_comp":{"signature":"theorem Polynomial.one_comp {R : Type u} [Semiring R] {p : Polynomial R} :comp 1 p = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L406-L407"},"Polynomial.charP":{"signature":"instance Polynomial.charP {R : Type u} [Semiring R] {p : \u2115} [CharP R p] :CharP (Polynomial R) p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L372-L374"},"Polynomial.mul_comp":{"signature":"theorem Polynomial.mul_comp {R : Type u_1} [CommSemiring R] (p q r : Polynomial R) :(p * q).comp r = p.comp r * q.comp r","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L449-L452"},"Polynomial.X_comp":{"signature":"theorem Polynomial.X_comp {R : Type u} [Semiring R] {p : Polynomial R} :X.comp p = p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L379-L381"},"Polynomial.natCast_comp":{"signature":"theorem Polynomial.natCast_comp {R : Type u} [Semiring R] {p : Polynomial R} {n : \u2115} :(\u2191n).comp p = \u2191n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L390-L391"},"Polynomial.eval_geom_sum":{"signature":"theorem Polynomial.eval_geom_sum {R : Type u_1} [CommSemiring R] {n : \u2115} {x : R} :eval x (\u2211 i \u2208 Finset.range n, X ^ i) = \u2211 i \u2208 Finset.range n, x ^ i","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L688-L690"},"Polynomial.instIsLocalHomRingHomC":{"signature":"instance Polynomial.instIsLocalHomRingHomC {R : Type u_1} [CommRing R] :IsLocalHom C","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Nilpotent.lean#L191-L192"},"Polynomial.wfDvdMonoid":{"signature":"instance Polynomial.wfDvdMonoid {R : Type u_1} [CommSemiring R] [NoZeroDivisors R] [WfDvdMonoid R] :WfDvdMonoid (Polynomial R)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/UniqueFactorization.lean#L37-L66"},"Polynomial.uniqueFactorizationMonoid":{"signature":"instance Polynomial.uniqueFactorizationMonoid {D : Type u} [CommRing D] [UniqueFactorizationMonoid D] :UniqueFactorizationMonoid (Polynomial D)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/UniqueFactorization.lean#L93-L95"},"Polynomial.cyclotomic_three":{"signature":"theorem Polynomial.cyclotomic_three (R : Type u_1) [Ring R] :cyclotomic 3 R = X ^ 2 + X + 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean#L381-L383"},"Polynomial.divByMonic_eq_div":{"signature":"theorem Polynomial.divByMonic_eq_div {R : Type u} [Field R] {q : Polynomial R} (p : Polynomial R) (hq : q.Monic) :p /\u2098 q = p / q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L343-L345"},"Polynomial.mul_div_eq_iff_isRoot":{"signature":"theorem Polynomial.mul_div_eq_iff_isRoot {R : Type u} {a : R} [Field R] {p : Polynomial R} :(X - C a) * (p / (X - C a)) = p \u2194 p.IsRoot a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L350-L351"},"Polynomial.derivative_sq":{"signature":"theorem Polynomial.derivative_sq {R : Type u} [CommSemiring R] (p : Polynomial R) :derivative (p ^ 2) = C 2 * p * derivative p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L460-L461"},"Polynomial.derivative_X_sub_C":{"signature":"theorem Polynomial.derivative_X_sub_C {R : Type u} [Ring R] (c : R) :derivative (X - C c) = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L619-L620"},"Polynomial.derivative_X_sub_C_sq":{"signature":"theorem Polynomial.derivative_X_sub_C_sq {R : Type u} [CommRing R] (c : R) :derivative ((X - C c) ^ 2) = C 2 * (X - C c)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L668-L669"},"Polynomial.idealX":{"signature":"def Polynomial.idealX (K : Type u_1) [Field K] :IsDedekindDomain.HeightOneSpectrum (Polynomial K)","doc":"This is the principal ideal generated by X in the ring of polynomials over a field K, regarded as an element of the height-one-spectrum.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/RatFunc/AsPolynomial.lean#L240-L245"},"Polynomial.toPowerSeries":{"signature":"def Polynomial.toPowerSeries {R : Type u_1} [Semiring R] :Polynomial R \u2192 PowerSeries R","doc":"The natural inclusion from polynomials into formal power series.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/PowerSeries/Basic.lean#L797-L800"},"Polynomial.coeff_coe":{"signature":"theorem Polynomial.coeff_coe {R : Type u_1} [Semiring R] (\u03c6 : Polynomial R) (n : \u2115) :(PowerSeries.coeff n) \u2191\u03c6 = \u03c6.coeff n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/PowerSeries/Basic.lean#L809-L811"},"Polynomial.coe_monomial":{"signature":"theorem Polynomial.coe_monomial {R : Type u_1} [Semiring R] (n : \u2115) (a : R) :\u2191((monomial n) a) = (PowerSeries.monomial n) a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/PowerSeries/Basic.lean#L813-L817"},"Polynomial.coe_one":{"signature":"theorem Polynomial.coe_one {R : Type u_1} [Semiring R] :\u21911 = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/PowerSeries/Basic.lean#L823-L826"},"Polynomial.coe_mul":{"signature":"theorem Polynomial.coe_mul {R : Type u_1} [Semiring R] (\u03c6 \u03c8 : Polynomial R) :\u2191(\u03c6 * \u03c8) = \u2191\u03c6 * \u2191\u03c8","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/PowerSeries/Basic.lean#L833-L835"},"Polynomial.coe_zero":{"signature":"theorem Polynomial.coe_zero {R : Type u_1} [Semiring R] :\u21910 = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/PowerSeries/Basic.lean#L819-L821"},"Polynomial.coe_add":{"signature":"theorem Polynomial.coe_add {R : Type u_1} [Semiring R] (\u03c6 \u03c8 : Polynomial R) :\u2191(\u03c6 + \u03c8) = \u2191\u03c6 + \u2191\u03c8","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/PowerSeries/Basic.lean#L828-L831"},"Polynomial.coe_C":{"signature":"theorem Polynomial.coe_C {R : Type u_1} [Semiring R] (a : R) :\u2191(C a) = PowerSeries.C a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/PowerSeries/Basic.lean#L841-L844"},"Polynomial.coe_injective":{"signature":"theorem Polynomial.coe_injective (R : Type u_1) [Semiring R] :Function.Injective toPowerSeries","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/PowerSeries/Basic.lean#L862-L864"},"Polynomial.algebraMap_hahnSeries_injective":{"signature":"theorem Polynomial.algebraMap_hahnSeries_injective (\u0393 : Type u_1) {R : Type u_2} [CommSemiring R] [Semiring \u0393] [PartialOrder \u0393] [IsStrictOrderedRing \u0393] :Function.Injective \u21d1(algebraMap (Polynomial R) (HahnSeries \u0393 R))","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/HahnSeries/PowerSeries.lean#L232-L234"},"Polynomial.coe_X":{"signature":"theorem Polynomial.coe_X {R : Type u_1} [Semiring R] :\u2191X = PowerSeries.X","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/PowerSeries/Basic.lean#L846-L848"},"Polynomial.coe_inj":{"signature":"theorem Polynomial.coe_inj {R : Type u_1} [Semiring R] {\u03c6 \u03c8 : Polynomial R} :\u2191\u03c6 = \u2191\u03c8 \u2194 \u03c6 = \u03c8","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/PowerSeries/Basic.lean#L868-L870"},"Polynomial.coe_eq_zero_iff":{"signature":"theorem Polynomial.coe_eq_zero_iff {R : Type u_1} [Semiring R] {\u03c6 : Polynomial R} :\u2191\u03c6 = 0 \u2194 \u03c6 = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/PowerSeries/Basic.lean#L872-L873"},"Polynomial.normUnit_X":{"signature":"theorem Polynomial.normUnit_X {R : Type u} [CommRing R] [NoZeroDivisors R] [NormalizationMonoid R] :normUnit X = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L241-L243"},"Polynomial.X_eq_normalize":{"signature":"theorem Polynomial.X_eq_normalize {R : Type u} [CommRing R] [NoZeroDivisors R] [NormalizationMonoid R] :X = normalize X","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L245-L246"},"Polynomial.coeff_X_pow_mul'":{"signature":"theorem Polynomial.coeff_X_pow_mul' {R : Type u} [Semiring R] (p : Polynomial R) (n d : \u2115) :(X ^ n * p).coeff d = if n \u2264 d then p.coeff (d - n) else 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L246-L248"},"Polynomial.X_pow_dvd_iff":{"signature":"theorem Polynomial.X_pow_dvd_iff {R : Type u} [Semiring R] {f : Polynomial R} {n : \u2115} :X ^ n \u2223 f \u2194 \u2200 d < n, f.coeff d = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L44-L56"},"Polynomial.idealX_span":{"signature":"theorem Polynomial.idealX_span (K : Type u_1) [Field K] :(idealX K).asIdeal = Ideal.span {X}","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/RatFunc/AsPolynomial.lean#L247-L248"},"Polynomial.valuation_of_mk":{"signature":"theorem Polynomial.valuation_of_mk (K : Type u_1) [Field K] (f : Polynomial K) {g : Polynomial K} (hg : g \u2260 0) :(IsDedekindDomain.HeightOneSpectrum.valuation (RatFunc K) (idealX K)) (RatFunc.mk f g) = (idealX K).intValuation f / (idealX K).intValuation g","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/RatFunc/AsPolynomial.lean#L256-L259"},"Polynomial.sub_comp":{"signature":"theorem Polynomial.sub_comp {R : Type u} [Ring R] {p q r : Polynomial R} :(p - q).comp r = p.comp r - q.comp r","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L762-L764"},"Polynomial.card_le_degree_of_subset_roots":{"signature":"theorem Polynomial.card_le_degree_of_subset_roots {R : Type u} [CommRing R] [IsDomain R] {p : Polynomial R} {Z : Finset R} (h : Z.val \u2286 p.roots) :Z.card \u2264 p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L136-L138"},"Polynomial.dickson":{"signature":"noncomputable def Polynomial.dickson {R : Type u_1} [CommRing R] (k : \u2115) (a : R) :\u2115 \u2192 Polynomial R","doc":"dickson is the n-th (generalised) Dickson polynomial of the k-th kind associated to the element a \u2208 R.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Dickson.lean#L61-L66"},"Polynomial.dickson_add_two":{"signature":"theorem Polynomial.dickson_add_two {R : Type u_1} [CommRing R] (k : \u2115) (a : R) (n : \u2115) :dickson k a (n + 2) = X * dickson k a (n + 1) - C a * dickson k a n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Dickson.lean#L79-L81"},"Polynomial.irreducible_of_degree_eq_one_of_isRelPrime_coeff":{"signature":"theorem Polynomial.irreducible_of_degree_eq_one_of_isRelPrime_coeff {R : Type u} [CommRing R] [IsDomain R] {p : Polynomial R} (hp : p.degree = 1) (hc : IsRelPrime (p.coeff 0) (p.coeff 1)) :Irreducible p","doc":"A degree 1 polynomial C a * X + C b is irreducible if a, b are relatively prime.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/RingDivision.lean#L225-L247"},"Polynomial.degree_add_le_of_le":{"signature":"theorem Polynomial.degree_add_le_of_le {R : Type u} [Semiring R] {p q : Polynomial R} {a b : WithBot \u2115} (hp : p.degree \u2264 a) (hq : q.degree \u2264 b) :(p + q).degree \u2264 max a b","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L337-L339"},"Polynomial.degree_mul_le_of_le":{"signature":"theorem Polynomial.degree_mul_le_of_le {R : Type u} [Semiring R] {p q : Polynomial R} {a b : WithBot \u2115} (hp : p.degree \u2264 a) (hq : q.degree \u2264 b) :(p * q).degree \u2264 a + b","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L401-L402"},"Polynomial.coeff_mul_add_eq_of_natDegree_le":{"signature":"theorem Polynomial.coeff_mul_add_eq_of_natDegree_le {R : Type u} [Semiring R] {df dg : \u2115} {f g : Polynomial R} (hdf : f.natDegree \u2264 df) (hdg : g.natDegree \u2264 dg) :(f * g).coeff (df + dg) = f.coeff df * g.coeff dg","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L442-L454"},"Polynomial.irreducible_C_mul_X_add_C":{"signature":"theorem Polynomial.irreducible_C_mul_X_add_C {R : Type u} [CommRing R] [IsDomain R] {a b : R} (ha : a \u2260 0) (hab : IsRelPrime a b) :Irreducible (C a * X + C b)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/RingDivision.lean#L249-L253"},"Polynomial.smul_pow":{"signature":"def Polynomial.smul_pow {R : Type u_1} [Semiring R] {S : Type u_2} [AddCommMonoid S] [Pow S \u2115] [MulActionWithZero R S] (x : S) :\u2115 \u2192 R \u2192 S","doc":"Scalar multiplication together with taking a natural number power.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Smeval.lean#L51-L52"},"Polynomial.smeval":{"signature":"def Polynomial.smeval {R : Type u_3} [Semiring R] (p : Polynomial R) {S : Type u_4} [AddCommMonoid S] [Pow S \u2115] [MulActionWithZero R S] (x : S) :S","doc":"Evaluate a polynomial p in the scalar semiring R at an element x in the target S using scalar multiple R-action.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Smeval.lean#L54-L56"},"Polynomial.continuousAt":{"signature":"theorem Polynomial.continuousAt {R : Type u_1} [Semiring R] [TopologicalSpace R] [IsTopologicalSemiring R] (p : Polynomial R) {a : R} :ContinuousAt (fun (x : R) => eval x p) a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Topology/Algebra/Polynomial.lean#L60-L62"},"Polynomial.eval\u2082_at_natCast":{"signature":"theorem Polynomial.eval\u2082_at_natCast {R : Type u} [Semiring R] {p : Polynomial R} {S : Type u_1} [Semiring S] (f : R \u2192+* S) (n : \u2115) :eval\u2082 f (\u2191n) p = f (eval (\u2191n) p)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L264-L268"},"Polynomial.smeval_def":{"signature":"theorem Polynomial.smeval_def {R : Type u_3} [Semiring R] (p : Polynomial R) {S : Type u_4} [AddCommMonoid S] [Pow S \u2115] [MulActionWithZero R S] (x : S) :p.smeval x = p.sum (smul_pow x)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Smeval.lean#L54-L56"},"Polynomial.smeval_eq_sum":{"signature":"theorem Polynomial.smeval_eq_sum {R : Type u_1} [Semiring R] (p : Polynomial R) {S : Type u_2} [AddCommMonoid S] [Pow S \u2115] [MulActionWithZero R S] (x : S) :p.smeval x = p.sum (smul_pow x)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Smeval.lean#L58-L58"},"Polynomial.eval_eq_smeval":{"signature":"theorem Polynomial.eval_eq_smeval {R : Type u_1} [Semiring R] (r : R) (p : Polynomial R) :eval r p = p.smeval r","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Smeval.lean#L69-L71"},"Polynomial.smeval_C":{"signature":"theorem Polynomial.smeval_C {R : Type u_1} [Semiring R] (r : R) {S : Type u_2} [AddCommMonoid S] [Pow S \u2115] [MulActionWithZero R S] (x : S) :(C r).smeval x = r \u2022 x ^ 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Smeval.lean#L60-L62"},"Polynomial.smeval_one":{"signature":"theorem Polynomial.smeval_one (R : Type u_1) [Semiring R] {S : Type u_2} [AddCommMonoid S] [Pow S \u2115] [MulActionWithZero R S] (x : S) :smeval 1 x = 1 \u2022 x ^ 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Smeval.lean#L86-L89"},"Polynomial.smeval_add":{"signature":"theorem Polynomial.smeval_add (R : Type u_1) [Semiring R] (p q : Polynomial R) {S : Type u_2} [AddCommMonoid S] [Pow S \u2115] [Module R S] (x : S) :(p + q).smeval x = p.smeval x + q.smeval x","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Smeval.lean#L108-L113"},"Polynomial.smeval_monomial":{"signature":"theorem Polynomial.smeval_monomial {R : Type u_1} [Semiring R] (r : R) {S : Type u_2} [AddCommMonoid S] [Pow S \u2115] [MulActionWithZero R S] (x : S) (n : \u2115) :((monomial n) r).smeval x = r \u2022 x ^ n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Smeval.lean#L64-L67"},"Polynomial.smeval_mul_X":{"signature":"theorem Polynomial.smeval_mul_X (R : Type u_1) [Semiring R] (p : Polynomial R) {S : Type u_2} [NonAssocSemiring S] [Module R S] [Pow S \u2115] (x : S) [NatPowAssoc S] [IsScalarTower R S S] :(p * X).smeval x = p.smeval x * x","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Smeval.lean#L243-L248"},"Polynomial.C_mul_monomial":{"signature":"theorem Polynomial.C_mul_monomial {R : Type u} {a b : R} {n : \u2115} [Semiring R] :C a * (monomial n) b = (monomial n) (a * b)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L472-L474"},"Polynomial.smeval_C_mul":{"signature":"theorem Polynomial.smeval_C_mul (R : Type u_1) [Semiring R] (r : R) (p : Polynomial R) {S : Type u_2} [NonAssocSemiring S] [Module R S] [Pow S \u2115] (x : S) :(C r * p).smeval x = r \u2022 p.smeval x","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Smeval.lean#L182-L185"},"Polynomial.ascPochhammer_smeval_cast":{"signature":"theorem Polynomial.ascPochhammer_smeval_cast (R : Type u_1) [Semiring R] {S : Type u_2} [NonAssocSemiring S] [Pow S \u2115] [Module R S] [IsScalarTower R S S] [NatPowAssoc S] (x : S) (n : \u2115) :(ascPochhammer R n).smeval x = (ascPochhammer \u2115 n).smeval x","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Binomial.lean#L187-L196"},"Polynomial.integralNormalization_eval\u2082_leadingCoeff_mul":{"signature":"theorem Polynomial.integralNormalization_eval\u2082_leadingCoeff_mul {R : Type u} {S : Type v} [Semiring R] {p : Polynomial R} [CommSemiring S] (h : 1 \u2264 p.natDegree) (f : R \u2192+* S) (x : S) :eval\u2082 f (f p.leadingCoeff * x) p.integralNormalization = f p.leadingCoeff ^ (p.natDegree - 1) * eval\u2082 f x p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/IntegralNormalization.lean#L148-L151"},"Polynomial.degreeLTEquiv":{"signature":"def Polynomial.degreeLTEquiv (R : Type u_2) [Semiring R] (n : \u2115) :\u21a5(degreeLT R n) \u2243\u2097[R] Fin n \u2192 R","doc":"The first n coefficients on degreeLT n form a linear equivalence with Fin n \u2192 R.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Basic.lean#L111-L145"},"Polynomial.degreeLTEquiv_eq_zero_iff_eq_zero":{"signature":"theorem Polynomial.degreeLTEquiv_eq_zero_iff_eq_zero {R : Type u} [Semiring R] {n : \u2115} {p : Polynomial R} (hp : p \u2208 degreeLT R n) :(degreeLTEquiv R n) \u27e8p, hp\u27e9 = 0 \u2194 p = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Basic.lean#L147-L148"},"Polynomial.eval_eq_sum_degreeLTEquiv":{"signature":"theorem Polynomial.eval_eq_sum_degreeLTEquiv {R : Type u} [Semiring R] {n : \u2115} {p : Polynomial R} (hp : p \u2208 degreeLT R n) (x : R) :eval x p = \u2211 i : Fin n, (degreeLTEquiv R n) \u27e8p, hp\u27e9 i * x ^ \u2191i","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Basic.lean#L150-L153"},"Polynomial.eq_zero_of_degree_lt_of_eval_finset_eq_zero":{"signature":"theorem Polynomial.eq_zero_of_degree_lt_of_eval_finset_eq_zero {R : Type u_1} [CommRing R] [IsDomain R] {f : Polynomial R} (s : Finset R) (degree_f_lt : f.degree < \u2191s.card) (eval_f : \u2200 x \u2208 s, eval x f = 0) :f = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/LinearAlgebra/Lagrange.lean#L48-L56"},"Polynomial.eq_of_degree_sub_lt_of_eval_finset_eq":{"signature":"theorem Polynomial.eq_of_degree_sub_lt_of_eval_finset_eq {R : Type u_1} [CommRing R] [IsDomain R] {f g : Polynomial R} (s : Finset R) (degree_fg_lt : (f - g).degree < \u2191s.card) (eval_fg : \u2200 x \u2208 s, eval x f = eval x g) :f = g","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/LinearAlgebra/Lagrange.lean#L58-L63"},"Polynomial.eq_of_degrees_lt_of_eval_finset_eq":{"signature":"theorem Polynomial.eq_of_degrees_lt_of_eval_finset_eq {R : Type u_1} [CommRing R] [IsDomain R] {f g : Polynomial R} (s : Finset R) (degree_f_lt : f.degree < \u2191s.card) (degree_g_lt : g.degree < \u2191s.card) (eval_fg : \u2200 x \u2208 s, eval x f = eval x g) :f = g","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/LinearAlgebra/Lagrange.lean#L65-L69"},"Polynomial.card_rootSet_eq_natDegree":{"signature":"theorem Polynomial.card_rootSet_eq_natDegree {F : Type u} [Field F] {K : Type v} [Field K] [Algebra F K] {p : Polynomial F} (hsep : p.Separable) (hsplit : (map (algebraMap F K) p).Splits) :Fintype.card \u2191(p.rootSet K) = p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Separable.lean#L457-L462"},"Polynomial.eval\u2082_at_zero":{"signature":"theorem Polynomial.eval\u2082_at_zero {R : Type u} {S : Type v} [Semiring R] {p : Polynomial R} [Semiring S] (f : R \u2192+* S) :eval\u2082 f 0 p = f (p.coeff 0)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Coeff.lean#L40-L44"},"Polynomial.coeff_zero_eq_aeval_zero'":{"signature":"theorem Polynomial.coeff_zero_eq_aeval_zero' {R : Type u} {A : Type z} [CommSemiring R] [Semiring A] [Algebra R A] (p : Polynomial R) :(algebraMap R A) (p.coeff 0) = (aeval 0) p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L473-L474"},"Polynomial.eval\u2082_smulOneHom_eq_smeval":{"signature":"theorem Polynomial.eval\u2082_smulOneHom_eq_smeval (R : Type u_3) [Semiring R] {S : Type u_4} [Semiring S] [Module R S] [IsScalarTower R S S] (p : Polynomial R) (x : S) :eval\u2082 RingHom.smulOneHom x p = p.smeval x","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Smeval.lean#L73-L78"},"Polynomial.finite_setOf_isRoot":{"signature":"theorem Polynomial.finite_setOf_isRoot {R : Type u} [CommRing R] [IsDomain R] {p : Polynomial R} (hp : p \u2260 0) :{x : R | p.IsRoot x}.Finite","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L140-L143"},"Polynomial.eq_zero_of_infinite_isRoot":{"signature":"theorem Polynomial.eq_zero_of_infinite_isRoot {R : Type u} [CommRing R] [IsDomain R] (p : Polynomial R) (h : {x : R | p.IsRoot x}.Infinite) :p = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L145-L146"},"Polynomial.algHom_ext_iff":{"signature":"theorem Polynomial.algHom_ext_iff {R : Type u} {B : Type u_2} [CommSemiring R] [Semiring B] [Algebra R B] {f g : Polynomial R \u2192\u2090[R] B} :f = g \u2194 f X = g X","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L254-L254"},"Polynomial.coeff_expand_mul'":{"signature":"theorem Polynomial.coeff_expand_mul' {R : Type u} [CommSemiring R] {p : \u2115} (hp : 0 < p) (f : Polynomial R) (n : \u2115) :((expand R p) f).coeff (p * n) = f.coeff n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Expand.lean#L114-L116"},"Polynomial.derivative'":{"signature":"def Polynomial.derivative' {R : Type u_1} [CommSemiring R] :Derivation R (Polynomial R) (Polynomial R)","doc":"Polynomial.derivative as a derivation.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivation.lean#L32-L39"},"Polynomial.derivation_ext":{"signature":"theorem Polynomial.derivation_ext {R : Type u_1} {A : Type u_2} [CommSemiring R] [AddCommMonoid A] [Module R A] [Module (Polynomial R) A] {D\u2081 D\u2082 : Derivation R (Polynomial R) A} (h : D\u2081 X = D\u2082 X) :D\u2081 = D\u2082","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivation.lean#L53-L56"},"Polynomial.mkDerivation":{"signature":"def Polynomial.mkDerivation (R : Type u_1) {A : Type u_2} [CommSemiring R] [AddCommMonoid A] [Module R A] [Module (Polynomial R) A] [IsScalarTower R (Polynomial R) A] :A \u2192\u2097[R] Derivation R (Polynomial R) A","doc":"The derivation on R[X] that takes the value a on X.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivation.lean#L61-L65"},"Polynomial.mkDerivation_apply":{"signature":"theorem Polynomial.mkDerivation_apply (R : Type u_1) {A : Type u_2} [CommSemiring R] [AddCommMonoid A] [Module R A] [Module (Polynomial R) A] [IsScalarTower R (Polynomial R) A] (a : A) (f : Polynomial R) :((mkDerivation R) a) f = derivative f \u2022 a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivation.lean#L67-L69"},"Polynomial.discr_of_degree_eq_two":{"signature":"theorem Polynomial.discr_of_degree_eq_two {R : Type u_1} [CommRing R] {f : Polynomial R} (hf : f.degree = 2) :f.discr = f.coeff 1 ^ 2 - 4 * f.coeff 0 * f.coeff 2","doc":"Standard formula for the discriminant of a quadratic polynomial.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L956-L972"},"Polynomial.coeff_zero_multiset_prod":{"signature":"theorem Polynomial.coeff_zero_multiset_prod {R : Type u} [CommSemiring R] (t : Multiset (Polynomial R)) :t.prod.coeff 0 = (Multiset.map (fun (f : Polynomial R) => f.coeff 0) t).prod","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/BigOperators.lean#L235-L237"},"Polynomial.coeff_zero_prod":{"signature":"theorem Polynomial.coeff_zero_prod {R : Type u} {\u03b9 : Type w} (s : Finset \u03b9) [CommSemiring R] (f : \u03b9 \u2192 Polynomial R) :(\u220f i \u2208 s, f i).coeff 0 = \u220f i \u2208 s, (f i).coeff 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/BigOperators.lean#L239-L240"},"Polynomial.cyclotomic_coeff_zero":{"signature":"theorem Polynomial.cyclotomic_coeff_zero (R : Type u_1) [CommRing R] {n : \u2115} (hn : 1 < n) :(cyclotomic n R).coeff 0 = 1","doc":"The constant term of cyclotomic n R is 1 if 2 \u2264 n.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean#L533-L561"},"Polynomial.cyclotomic_two":{"signature":"theorem Polynomial.cyclotomic_two (R : Type u_1) [Ring R] :cyclotomic 2 R = X + 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean#L378-L379"},"Polynomial.cyclotomic_pos":{"signature":"theorem Polynomial.cyclotomic_pos {n : \u2115} (hn : 2 < n) {R : Type u_1} [CommRing R] [LinearOrder R] [IsStrictOrderedRing R] (x : R) :0 < eval x (cyclotomic n R)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Eval.lean#L67-L109"},"Polynomial.cyclotomic_pos_and_nonneg":{"signature":"theorem Polynomial.cyclotomic_pos_and_nonneg (n : \u2115) {R : Type u_1} [CommRing R] [LinearOrder R] [IsStrictOrderedRing R] (x : R) :(1 < x \u2192 0 < eval x (cyclotomic n R)) \u2227 (1 \u2264 x \u2192 0 \u2264 eval x (cyclotomic n R))","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Eval.lean#L111-L120"},"Polynomial.cyclotomic_pos'":{"signature":"theorem Polynomial.cyclotomic_pos' (n : \u2115) {R : Type u_1} [CommRing R] [LinearOrder R] [IsStrictOrderedRing R] {x : R} (hx : 1 < x) :0 < eval x (cyclotomic n R)","doc":"Cyclotomic polynomials are always positive on inputs larger than one. Similar to cyclotomic_pos but with the condition on the input rather than index of the cyclotomic polynomial.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Eval.lean#L122-L128"},"Polynomial.cyclotomic_nonneg":{"signature":"theorem Polynomial.cyclotomic_nonneg (n : \u2115) {R : Type u_1} [CommRing R] [LinearOrder R] [IsStrictOrderedRing R] {x : R} (hx : 1 \u2264 x) :0 \u2264 eval x (cyclotomic n R)","doc":"Cyclotomic polynomials are always nonnegative on inputs one or more.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Eval.lean#L130-L134"},"Polynomial.sub_one_pow_totient_lt_cyclotomic_eval":{"signature":"theorem Polynomial.sub_one_pow_totient_lt_cyclotomic_eval {n : \u2115} {q : \u211d} (hn' : 2 \u2264 n) (hq' : 1 < q) :(q - 1) ^ n.totient < eval q (cyclotomic n \u211d)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Eval.lean#L172-L217"},"Polynomial.sub_one_pow_totient_lt_natAbs_cyclotomic_eval":{"signature":"theorem Polynomial.sub_one_pow_totient_lt_natAbs_cyclotomic_eval {n q : \u2115} (hn' : 1 < n) (hq : q \u2260 1) :(q - 1) ^ n.totient < (eval (\u2191q) (cyclotomic n \u2124)).natAbs","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Eval.lean#L297-L306"},"Polynomial.sub_one_lt_natAbs_cyclotomic_eval":{"signature":"theorem Polynomial.sub_one_lt_natAbs_cyclotomic_eval {n q : \u2115} (hn' : 1 < n) (hq : q \u2260 1) :q - 1 < (eval (\u2191q) (cyclotomic n \u2124)).natAbs","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Eval.lean#L308-L312"},"Polynomial.aeval_algebraMap_apply_eq_algebraMap_eval":{"signature":"theorem Polynomial.aeval_algebraMap_apply_eq_algebraMap_eval {R : Type u} {A : Type z} [CommSemiring R] [Semiring A] [Algebra R A] (x : R) (p : Polynomial R) :(aeval ((algebraMap R A) x)) p = (algebraMap R A) (eval x p)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L415-L417"},"Polynomial.coeff_det_X_add_C_zero":{"signature":"theorem Polynomial.coeff_det_X_add_C_zero {n : Type u_1} {\u03b1 : Type u_2} [DecidableEq n] [Fintype n] [CommRing \u03b1] (A B : Matrix n n \u03b1) :(X \u2022 A.map \u21d1C + B.map \u21d1C).det.coeff 0 = B.det","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/LinearAlgebra/Matrix/Polynomial.lean#L62-L70"},"Polynomial.isProperMap_eval":{"signature":"theorem Polynomial.isProperMap_eval {R : Type u_2} [NormedRing R] [IsAbsoluteValue norm] [ProperSpace R] (p : Polynomial R) (h : 0 < p.degree) :IsProperMap fun (x : R) => eval x p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Topology/Algebra/Polynomial.lean#L142-L145"},"Polynomial.isClosedMap_eval":{"signature":"theorem Polynomial.isClosedMap_eval {R : Type u_2} [NormedRing R] [IsAbsoluteValue norm] [ProperSpace R] (p : Polynomial R) :IsClosedMap fun (x : R) => eval x p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Topology/Algebra/Polynomial.lean#L147-L150"},"Polynomial.aeval_apply_smul_mem_of_le_comap'":{"signature":"theorem Polynomial.aeval_apply_smul_mem_of_le_comap' {R : Type u} {A : Type z} {M : Type u_3} [CommSemiring R] [AddCommMonoid M] [Module R M] {q : Submodule R M} {m : M} [Semiring A] [Algebra R A] [Module A M] [IsScalarTower R A M] (hm : m \u2208 q) (p : Polynomial R) (a : A) (hq : q \u2264 Submodule.comap ((Algebra.lsmul R R M) a) q) :(aeval a) p \u2022 m \u2208 q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L693-L705"},"Polynomial.newtonMap":{"signature":"def Polynomial.newtonMap {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [Algebra R S] (P : Polynomial R) (x : S) :S","doc":"Given a single-variable polynomial P with derivative P', this is the map: x \u21a6 x - P(x) / P'(x). When P'(x) is not a unit we use a junk-value pattern and send x \u21a6 x.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Dynamics/Newton.lean#L45-L48"},"Polynomial.newtonMap_apply":{"signature":"theorem Polynomial.newtonMap_apply {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [Algebra R S] (P : Polynomial R) {x : S} :P.newtonMap x = x - Ring.inverse ((aeval x) (derivative P)) * (aeval x) P","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Dynamics/Newton.lean#L50-L52"},"Polynomial.powSubPowFactor":{"signature":"def Polynomial.powSubPowFactor {R : Type u} [CommRing R] (x y : R) (i : \u2115) :{ z : R // x ^ i - y ^ i = z * (x - y) }","doc":"x^n - y^n can be expressed as z * (x - y) for some z in the ring.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Identities.lean#L92-L100"},"Polynomial.evalSubFactor":{"signature":"def Polynomial.evalSubFactor {R : Type u} [CommRing R] (f : Polynomial R) (x y : R) :{ z : R // eval x f - eval y f = z * (x - y) }","doc":"For any polynomial f, f.eval x - f.eval y can be expressed as z * (x - y) for some z in the ring.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Identities.lean#L102-L111"},"Polynomial.isNilpotent_aeval_sub_of_isNilpotent_sub":{"signature":"theorem Polynomial.isNilpotent_aeval_sub_of_isNilpotent_sub {R : Type u_2} {S : Type u_3} [CommRing R] [CommRing S] [Algebra R S] (P : Polynomial R) {a b : S} (h : IsNilpotent (a - b)) :IsNilpotent ((aeval a) P - (aeval b) P)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Nilpotent.lean#L203-L207"},"Polynomial.isNilpotent_iterate_newtonMap_sub_of_isNilpotent":{"signature":"theorem Polynomial.isNilpotent_iterate_newtonMap_sub_of_isNilpotent {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [Algebra R S] {P : Polynomial R} {x : S} (h : IsNilpotent ((aeval x) P)) (n : \u2115) :IsNilpotent (P.newtonMap^[n] x - x)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Dynamics/Newton.lean#L64-L72"},"Polynomial.powAddExpansion":{"signature":"def Polynomial.powAddExpansion {R : Type u_1} [CommSemiring R] (x y : R) (n : \u2115) :{ k : R // (x + y) ^ n = x ^ n + \u2191n * x ^ (n - 1) * y + k * y ^ 2 }","doc":"(x + y)^n can be expressed as x^n + n*x^(n-1)*y + k * y^2 for some k in the ring.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Identities.lean#L37-L51"},"Polynomial.sum_add'":{"signature":"theorem Polynomial.sum_add' {R : Type u} [Semiring R] {S : Type u_1} [AddCommMonoid S] (p : Polynomial R) (f g : \u2115 \u2192 R \u2192 S) :p.sum (f + g) = p.sum f + p.sum g","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L870-L871"},"Polynomial.sum_add":{"signature":"theorem Polynomial.sum_add {R : Type u} [Semiring R] {S : Type u_1} [AddCommMonoid S] (p : Polynomial R) (f g : \u2115 \u2192 R \u2192 S) :(p.sum fun (n : \u2115) (x : R) => f n x + g n x) = p.sum f + p.sum g","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L873-L875"},"Polynomial.eval_sum":{"signature":"theorem Polynomial.eval_sum {R : Type u} [Semiring R] (p : Polynomial R) (f : \u2115 \u2192 R \u2192 Polynomial R) (x : R) :eval x (p.sum f) = p.sum fun (n : \u2115) (a : R) => eval x (f n a)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L334-L336"},"Polynomial.eval_mul_X":{"signature":"theorem Polynomial.eval_mul_X {R : Type u} [Semiring R] {p : Polynomial R} {x : R} :eval x (p * X) = eval x p * x","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L317-L318"},"Polynomial.eval_mul_X_pow":{"signature":"theorem Polynomial.eval_mul_X_pow {R : Type u} [Semiring R] {p : Polynomial R} {x : R} {k : \u2115} :eval x (p * X ^ k) = eval x p * x ^ k","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L320-L324"},"Polynomial.derivative_eval":{"signature":"theorem Polynomial.derivative_eval {R : Type u} [Semiring R] (p : Polynomial R) (x : R) :eval x (derivative p) = p.sum fun (n : \u2115) (a : R) => a * \u2191n * x ^ (n - 1)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L263-L265"},"Polynomial.binomExpansion":{"signature":"def Polynomial.binomExpansion {R : Type u} [CommRing R] (f : Polynomial R) (x y : R) :{ k : R // eval (x + y) f = eval x f + eval x (derivative f) * y + k * y ^ 2 }","doc":"A polynomial f evaluated at x + y can be expressed as the evaluation of f at x, plus y times the (polynomial) derivative of f at x, plus some element k : R times y^2.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Identities.lean#L78-L90"},"Polynomial.isUnit_aeval_of_isUnit_aeval_of_isNilpotent_sub":{"signature":"theorem Polynomial.isUnit_aeval_of_isUnit_aeval_of_isNilpotent_sub {R : Type u_2} {S : Type u_3} [CommRing R] [CommRing S] [Algebra R S] {P : Polynomial R} {a b : S} (hb : IsUnit ((aeval b) P)) (hab : IsNilpotent (a - b)) :IsUnit ((aeval a) P)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Nilpotent.lean#L211-L216"},"Polynomial.aeval_pow_two_pow_dvd_aeval_iterate_newtonMap":{"signature":"theorem Polynomial.aeval_pow_two_pow_dvd_aeval_iterate_newtonMap {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [Algebra R S] {P : Polynomial R} {x : S} (h : IsNilpotent ((aeval x) P)) (h' : IsUnit ((aeval x) (derivative P))) (n : \u2115) :(aeval x) P ^ 2 ^ n \u2223 (aeval (P.newtonMap^[n] x)) P","doc":"This is really an auxiliary result, en route to Polynomial.existsUnique_nilpotent_sub_and_aeval_eq_zero.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Dynamics/Newton.lean#L82-L102"},"Polynomial.existsUnique_nilpotent_sub_and_aeval_eq_zero":{"signature":"theorem Polynomial.existsUnique_nilpotent_sub_and_aeval_eq_zero {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [Algebra R S] {P : Polynomial R} {x : S} (h : IsNilpotent ((aeval x) P)) (h' : IsUnit ((aeval x) (derivative P))) :\u2203! r : S, IsNilpotent (x - r) \u2227 (aeval r) P = 0","doc":"If x is almost a root of P in the sense that P(x) is nilpotent (and P'(x) is a unit) then we may write x as a sum x = n + r where n is nilpotent and r is a root of P. Moreover, n and r are unique. This can be used to prove the Jordan-Chevalley decomposition of linear endomorphims.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Dynamics/Newton.lean#L104-L129"},"Polynomial.piEquiv":{"signature":"def Polynomial.piEquiv {\u03b9 : Type u_2} [Finite \u03b9] (R : \u03b9 \u2192 Type u_1) [(i : \u03b9) \u2192 Semiring (R i)] :Polynomial ((i : \u03b9) \u2192 R i) \u2243+* ((i : \u03b9) \u2192 Polynomial (R i))","doc":"The polynomial ring over a finite product of rings is isomorphic to the product of polynomial rings over individual rings.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Coeff.lean#L92-L101"},"Polynomial.derivative_natCast":{"signature":"theorem Polynomial.derivative_natCast {R : Type u} [Semiring R] {n : \u2115} :derivative \u2191n = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L193-L196"},"Polynomial.natDegree_add_eq_right_of_degree_lt":{"signature":"theorem Polynomial.natDegree_add_eq_right_of_degree_lt {R : Type u} [Semiring R] {p q : Polynomial R} (h : p.degree < q.degree) :(p + q).natDegree = q.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L192-L194"},"Polynomial.natDegree_add_eq_right_of_natDegree_lt":{"signature":"theorem Polynomial.natDegree_add_eq_right_of_natDegree_lt {R : Type u} [Semiring R] {p q : Polynomial R} (h : p.natDegree < q.natDegree) :(p + q).natDegree = q.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L196-L198"},"Polynomial.tendsto_div_exp_atTop":{"signature":"theorem Polynomial.tendsto_div_exp_atTop (p : Polynomial \u211d) :Filter.Tendsto (fun (x : \u211d) => eval x p / Real.exp x) Filter.atTop (nhds 0)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/SpecialFunctions/PolynomialExp.lean#L30-L34"},"Polynomial.eval_eq_zero_of_dvd_of_eval_eq_zero":{"signature":"theorem Polynomial.eval_eq_zero_of_dvd_of_eval_eq_zero {R : Type u} [CommSemiring R] {p q : Polynomial R} {x : R} :p \u2223 q \u2192 eval x p = 0 \u2192 eval x q = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L685-L686"},"Polynomial.aeval_eq_zero_of_dvd_aeval_eq_zero":{"signature":"theorem Polynomial.aeval_eq_zero_of_dvd_aeval_eq_zero {S : Type v} {T : Type w} [CommSemiring S] [CommSemiring T] [Algebra S T] {p q : Polynomial S} (h\u2081 : p \u2223 q) {a : T} (h\u2082 : (aeval a) p = 0) :(aeval a) q = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L483-L486"},"Polynomial.lcoeff_apply":{"signature":"theorem Polynomial.lcoeff_apply {R : Type u} [Semiring R] (n : \u2115) (f : Polynomial R) :(lcoeff R n) f = f.coeff n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L84-L86"},"Polynomial.coeff_X_add_C_pow":{"signature":"theorem Polynomial.coeff_X_add_C_pow {R : Type u} [Semiring R] (r : R) (n k : \u2115) :((X + C r) ^ n).coeff k = r ^ (n - k) * \u2191(n.choose k)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L288-L297"},"Polynomial.coeff_X_add_one_pow":{"signature":"theorem Polynomial.coeff_X_add_one_pow (R : Type u_1) [Semiring R] (n k : \u2115) :((X + 1) ^ n).coeff k = \u2191(n.choose k)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L299-L300"},"Polynomial.coeff_mul_natCast":{"signature":"theorem Polynomial.coeff_mul_natCast {R : Type u} [Semiring R] {p : Polynomial R} {a k : \u2115} :(p * \u2191a).coeff k = p.coeff k * \u2191a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L167-L168"},"Polynomial.natDegree_quadratic_le":{"signature":"theorem Polynomial.natDegree_quadratic_le {R : Type u} {a b c : R} [Semiring R] :(C a * X ^ 2 + C b * X + C c).natDegree \u2264 2","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/SmallDegree.lean#L108-L109"},"Polynomial.eq_neg_mul_add_of_roots_quadratic_eq_pair":{"signature":"theorem Polynomial.eq_neg_mul_add_of_roots_quadratic_eq_pair {R : Type u_1} [CommRing R] [IsDomain R] {a b c x1 x2 : R} (hroots : (C a * X ^ 2 + C b * X + C c).roots = {x1, x2}) :b = -a * (x1 + x2)","doc":"Vieta's formula for quadratics.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/SmallDegreeVieta.lean#L29-L39"},"Polynomial.natDegree_map_eq_iff":{"signature":"theorem Polynomial.natDegree_map_eq_iff {R : Type u} {S : Type v} [Semiring R] [Semiring S] {f : R \u2192+* S} {p : Polynomial R} :(map f p).natDegree = p.natDegree \u2194 f p.leadingCoeff \u2260 0 \u2228 p.natDegree = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L270-L275"},"Polynomial.integralNormalization_map":{"signature":"theorem Polynomial.integralNormalization_map {R : Type u} [Semiring R] {A : Type u_1} [Semiring A] (f : R \u2192+* A) (p : Polynomial R) (H : f p.leadingCoeff \u2260 0) :(map f p).integralNormalization = map f p.integralNormalization","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/IntegralNormalization.lean#L176-L180"},"Polynomial.bernoulli":{"signature":"def Polynomial.bernoulli (n : \u2115) :Polynomial \u211a","doc":"The Bernoulli polynomials are defined in terms of the negative Bernoulli numbers.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/NumberTheory/BernoulliPolynomials.lean#L50-L52"},"Polynomial.bernoulli_def":{"signature":"theorem Polynomial.bernoulli_def (n : \u2115) :bernoulli n = \u2211 i \u2208 Finset.range (n + 1), (monomial i) (_root_.bernoulli (n - i) * \u2191(n.choose i))","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/NumberTheory/BernoulliPolynomials.lean#L54-L60"},"Polynomial.IsMonicOfDegree":{"signature":"structure Polynomial.IsMonicOfDegree {R : Type u_1} [Semiring R] (p : Polynomial R) (n : \u2115) :Prop","doc":"This says that p has natDegree n and is monic.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/IsMonicOfDegree.lean#L30-L34"},"Polynomial.isMonicOfDegree_iff'":{"signature":"theorem Polynomial.isMonicOfDegree_iff' {R : Type u_1} [Semiring R] (p : Polynomial R) (n : \u2115) :p.IsMonicOfDegree n \u2194 p.natDegree = n \u2227 p.Monic","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/IsMonicOfDegree.lean#L31-L31"},"Polynomial.isMonicOfDegree_zero_iff":{"signature":"theorem Polynomial.isMonicOfDegree_zero_iff {R : Type u_1} [Semiring R] {p : Polynomial R} :p.IsMonicOfDegree 0 \u2194 p = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/IsMonicOfDegree.lean#L36-L41"},"Polynomial.isMonicOfDegree_iff_of_subsingleton":{"signature":"theorem Polynomial.isMonicOfDegree_iff_of_subsingleton {R : Type u_1} [Semiring R] [Subsingleton R] {p : Polynomial R} {n : \u2115} :p.IsMonicOfDegree n \u2194 n = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/IsMonicOfDegree.lean#L47-L53"},"Polynomial.update":{"signature":"def Polynomial.update {R : Type u} [Semiring R] (p : Polynomial R) (n : \u2115) (a : R) :Polynomial R","doc":"Replace the coefficient of a p : R[X] at a given degree n : \u2115 by a given value a : R. If a = 0, this is equal to p.erase n If p.natDegree < n and a \u2260 0, this increases the degree to n.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L972-L976"},"Polynomial.support_update":{"signature":"theorem Polynomial.support_update {R : Type u} [Semiring R] (p : Polynomial R) (n : \u2115) (a : R) [Decidable (a = 0)] :(p.update n a).support = if a = 0 then p.support.erase n else insert n p.support","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L999-L1003"},"Polynomial.degree_update_le":{"signature":"theorem Polynomial.degree_update_le {R : Type u} [Semiring R] (p : Polynomial R) (n : \u2115) (a : R) :(p.update n a).degree \u2264 max p.degree \u2191n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L381-L387"},"Polynomial.degree_linear_lt":{"signature":"theorem Polynomial.degree_linear_lt {R : Type u} {a b : R} [Semiring R] :(C a * X + C b).degree < 2","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/SmallDegree.lean#L72-L73"},"Polynomial.degree_linear_lt_degree_C_mul_X_sq":{"signature":"theorem Polynomial.degree_linear_lt_degree_C_mul_X_sq {R : Type u} {a b c : R} [Semiring R] (ha : a \u2260 0) :(C b * X + C c).degree < (C a * X ^ 2).degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/SmallDegree.lean#L98-L100"},"Polynomial.leadingCoeff_quadratic":{"signature":"theorem Polynomial.leadingCoeff_quadratic {R : Type u} {a b c : R} [Semiring R] (ha : a \u2260 0) :(C a * X ^ 2 + C b * X + C c).leadingCoeff = a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/SmallDegree.lean#L114-L117"},"Polynomial.degree_quadratic":{"signature":"theorem Polynomial.degree_quadratic {R : Type u} {a b c : R} [Semiring R] (ha : a \u2260 0) :(C a * X ^ 2 + C b * X + C c).degree = 2","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/SmallDegree.lean#L102-L106"},"Polynomial.natDegree_quadratic":{"signature":"theorem Polynomial.natDegree_quadratic {R : Type u} {a b c : R} [Semiring R] (ha : a \u2260 0) :(C a * X ^ 2 + C b * X + C c).natDegree = 2","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/SmallDegree.lean#L111-L112"},"Polynomial.MonicDegreeEq":{"signature":"abbrev Polynomial.MonicDegreeEq (R : Type u) (n : \u2115) [Semiring R] :Type u","doc":"The type of monic polynomials of degree n. The implementation is slightly different because it is useful to still contain X ^ n when R is trivial. See MonicDegreeEq.mk for the usual constructor.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Monic.lean#L233-L239"},"Polynomial.freeMonic":{"signature":"def Polynomial.freeMonic (R : Type u_1) [CommRing R] (n : \u2115) :Polynomial (MvPolynomial (Fin n) R)","doc":"The free monic polynomial of degree n, as a polynomial in R[X\u2081,...,X\u2099][X].","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/UniversalFactorizationRing.lean#L47-L49"},"Polynomial.coeff_freeMonic":{"signature":"theorem Polynomial.coeff_freeMonic (R : Type u_1) [CommRing R] (n k : \u2115) :(freeMonic R n).coeff k = if h : k < n then MvPolynomial.X \u27e8k, h\u27e9 else if k = n then 1 else 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/UniversalFactorizationRing.lean#L51-L58"},"Polynomial.degree_freeMonic":{"signature":"theorem Polynomial.degree_freeMonic (R : Type u_1) [CommRing R] (n : \u2115) [Nontrivial R] :(freeMonic R n).degree = \u2191n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/UniversalFactorizationRing.lean#L60-L63"},"Polynomial.natDegree_freeMonic":{"signature":"theorem Polynomial.natDegree_freeMonic (R : Type u_1) [CommRing R] (n : \u2115) [Nontrivial R] :(freeMonic R n).natDegree = n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/UniversalFactorizationRing.lean#L65-L66"},"Polynomial.monic_freeMonic":{"signature":"theorem Polynomial.monic_freeMonic (R : Type u_1) [CommRing R] (n : \u2115) :(freeMonic R n).Monic","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/UniversalFactorizationRing.lean#L68-L70"},"Polynomial.UniversalFactorizationRing":{"signature":"def Polynomial.UniversalFactorizationRing {R : Type u_1} [CommRing R] {n : \u2115} (m k : \u2115) (hn : n = m + k) (p : MonicDegreeEq R n) :Type u_1","doc":"The universal factorization ring of a monic polynomial p of degree n. This is the representing object of the functor S \u21a6 \"factorizations of p into (monic deg m) * (monic deg k) in S\". See UniversalFactorizationRing.homEquiv.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/UniversalFactorizationRing.lean#L388-L396"},"Polynomial.instCommRingUniversalFactorizationRing":{"signature":"instance Polynomial.instCommRingUniversalFactorizationRing {R : Type u_1} [CommRing R] {n : \u2115} (m k : \u2115) (hn : n = m + k) (p : MonicDegreeEq R n) :CommRing (UniversalFactorizationRing m k hn p)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/UniversalFactorizationRing.lean#L396-L396"},"Polynomial.instAlgebraUniversalFactorizationRing":{"signature":"instance Polynomial.instAlgebraUniversalFactorizationRing {R : Type u_1} [CommRing R] {n : \u2115} (m k : \u2115) (hn : n = m + k) (p : MonicDegreeEq R n) :Algebra R (UniversalFactorizationRing m k hn p)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/UniversalFactorizationRing.lean#L396-L396"},"Polynomial.UniversalCoprimeFactorizationRing":{"signature":"abbrev Polynomial.UniversalCoprimeFactorizationRing {R : Type u_1} [CommRing R] {n : \u2115} (m k : \u2115) (hn : n = m + k) (p : MonicDegreeEq R n) :Type u_1","doc":"The universal coprime factorization ring of a monic polynomial p of degree n. This is the representing object of the functor S \u21a6 \"factorizations of p into coprime (monic deg m) * (monic deg k) in S\". See UniversalCoprimeFactorizationRing.homEquiv.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/UniversalFactorizationRing.lean#L528-L533"},"Polynomial.adjSylvester":{"signature":"noncomputable def Polynomial.adjSylvester {m n : \u2115} {R : Type u_1} [CommRing R] (f g : Polynomial R) :\u21a5(degreeLT R (m + n)) \u2192\u2097[R] \u21a5(degreeLT R m) \u00d7 \u21a5(degreeLT R n)","doc":"The adjugate map of the sylvester map. It takes P to (p, q) such that f * q + g * p = Res(f, g) * P.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L852-L858"},"Polynomial.sylvesterMap":{"signature":"noncomputable def Polynomial.sylvesterMap {m n : \u2115} {R : Type u_1} [CommRing R] (f g : Polynomial R) (hf : f.natDegree \u2264 m) (hg : g.natDegree \u2264 n) :\u21a5(degreeLT R m) \u00d7 \u21a5(degreeLT R n) \u2192\u2097[R] \u21a5(degreeLT R (m + n))","doc":"The map (p, q) \u21a6 f * q + g * p whose associated matrix is Syl(f, g).","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L808-L824"},"Polynomial.toMatrix_sylvesterMap":{"signature":"theorem Polynomial.toMatrix_sylvesterMap {m n : \u2115} {R : Type u_1} [CommRing R] (f g : Polynomial R) (hf : f.natDegree \u2264 m) (hg : g.natDegree \u2264 n) :(LinearMap.toMatrix ((degreeLT.basis R m).prod (degreeLT.basis R n)) (degreeLT.basis R (m + n))) (f.sylvesterMap g hf hg) = (Matrix.reindex (Equiv.refl (Fin (m + n))) finSumFinEquiv.symm) (f.sylvester g m n)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L826-L842"},"Polynomial.toMatrix_sylvesterMap'":{"signature":"theorem Polynomial.toMatrix_sylvesterMap' {m n : \u2115} {R : Type u_1} [CommRing R] (f g : Polynomial R) (hf : f.natDegree \u2264 m) (hg : g.natDegree \u2264 n) :(LinearMap.toMatrix (((degreeLT.basis R m).prod (degreeLT.basis R n)).reindex finSumFinEquiv) (degreeLT.basis R (m + n))) (f.sylvesterMap g hf hg) = f.sylvester g m n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L844-L850"},"Polynomial.sylveserMap_comp_adjSylvester":{"signature":"theorem Polynomial.sylveserMap_comp_adjSylvester {m n : \u2115} {R : Type u_1} [CommRing R] (f g : Polynomial R) (hf : f.natDegree \u2264 m) (hg : g.natDegree \u2264 n) :f.sylvesterMap g hf hg \u2218\u2097 f.adjSylvester g = f.resultant g m n \u2022 LinearMap.id","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L860-L866"},"Polynomial.exists_mul_add_mul_eq_C_resultant":{"signature":"theorem Polynomial.exists_mul_add_mul_eq_C_resultant {m n : \u2115} {R : Type u_1} [CommRing R] (f g : Polynomial R) (hf : f.natDegree \u2264 m) (hg : g.natDegree \u2264 n) (H : m \u2260 0 \u2228 n \u2260 0) :\u2203 (p : Polynomial R) (q : Polynomial R), p.degree < \u2191n \u2227 q.degree < \u2191m \u2227 f * p + g * q = C (f.resultant g m n)","doc":"Note that if n = m = 0 then resultant = 1 but f and g aren't necessarily coprime.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L876-L886"},"Polynomial.isUnit_resultant_iff_isCoprime":{"signature":"theorem Polynomial.isUnit_resultant_iff_isCoprime {R : Type u_1} [CommRing R] {f g : Polynomial R} (hf : f.Monic) :IsUnit (f.resultant g) \u2194 IsCoprime f g","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L888-L909"},"Polynomial.natDegree_le_iff_coeff_eq_zero":{"signature":"theorem Polynomial.natDegree_le_iff_coeff_eq_zero {R : Type u} {n : \u2115} [Semiring R] {p : Polynomial R} :p.natDegree \u2264 n \u2194 \u2200 (N : \u2115), n < N \u2192 p.coeff N = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L79-L80"},"Polynomial.instEtaleUniversalCoprimeFactorizationRing":{"signature":"instance Polynomial.instEtaleUniversalCoprimeFactorizationRing {R : Type u_1} [CommRing R] {n : \u2115} (m k : \u2115) (hn : n = m + k) (p : MonicDegreeEq R n) :Algebra.Etale R (UniversalCoprimeFactorizationRing m k hn p)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/UniversalFactorizationRing.lean#L563-L575"},"Polynomial.natDegree_pos_of_nextCoeff_ne_zero":{"signature":"theorem Polynomial.natDegree_pos_of_nextCoeff_ne_zero {R : Type u} [Semiring R] {p : Polynomial R} (h : p.nextCoeff \u2260 0) :0 < p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L310-L311"},"Polynomial.natDegree_eraseLead":{"signature":"theorem Polynomial.natDegree_eraseLead {R : Type u_1} [Semiring R] {f : Polynomial R} (h : f.nextCoeff \u2260 0) :f.eraseLead.natDegree = f.natDegree - 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/EraseLead.lean#L218-L222"},"Polynomial.degree_gcd_le_left":{"signature":"theorem Polynomial.degree_gcd_le_left {R : Type u_1} [CommRing R] [NormalizedGCDMonoid R] {p : Polynomial R} (hp : p \u2260 0) (q : Polynomial R) :(gcd p q).degree \u2264 p.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Content.lean#L464-L466"},"Polynomial.degree_gcd_le_right":{"signature":"theorem Polynomial.degree_gcd_le_right {R : Type u_1} [CommRing R] [NormalizedGCDMonoid R] (p : Polynomial R) {q : Polynomial R} (hq : q \u2260 0) :(gcd p q).degree \u2264 q.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Content.lean#L468-L470"},"Polynomial.C_mul_dvd":{"signature":"theorem Polynomial.C_mul_dvd {R : Type u} {a : R} [Field R] {p q : Polynomial R} (ha : a \u2260 0) :C a * p \u2223 q \u2194 p \u2223 q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L578-L582"},"Polynomial.div_def":{"signature":"theorem Polynomial.div_def {R : Type u} [Field R] {p q : Polynomial R} :p / q = C q.leadingCoeff\u207b\u00b9 * (p /\u2098 (q * C q.leadingCoeff\u207b\u00b9))","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L334-L335"},"Polynomial.leadingCoeff_divByMonic_of_monic":{"signature":"theorem Polynomial.leadingCoeff_divByMonic_of_monic {R : Type u} [CommRing R] {p q : Polynomial R} (hmonic : q.Monic) (hdegree : q.degree \u2264 p.degree) :(p /\u2098 q).leadingCoeff = p.leadingCoeff","doc":"Division by a monic polynomial doesn't change the leading coefficient.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L760-L770"},"Polynomial.leadingCoeff_div":{"signature":"theorem Polynomial.leadingCoeff_div {R : Type u} [Field R] {p q : Polynomial R} (hpq : q.degree \u2264 p.degree) :(p / q).leadingCoeff = p.leadingCoeff / q.leadingCoeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L556-L563"},"Polynomial.div_C_mul":{"signature":"theorem Polynomial.div_C_mul {R : Type u} {a : R} [Field R] {p q : Polynomial R} :p / (C a * q) = C a\u207b\u00b9 * (p / q)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L565-L570"},"Polynomial.natDegree_sub_eq_of_prod_eq":{"signature":"theorem Polynomial.natDegree_sub_eq_of_prod_eq {R : Type u} [Semiring R] [NoZeroDivisors R] {p\u2081 p\u2082 q\u2081 q\u2082 : Polynomial R} (hp\u2081 : p\u2081 \u2260 0) (hq\u2081 : q\u2081 \u2260 0) (hp\u2082 : p\u2082 \u2260 0) (hq\u2082 : q\u2082 \u2260 0) (h_eq : p\u2081 * q\u2082 = p\u2082 * q\u2081) :\u2191p\u2081.natDegree - \u2191q\u2081.natDegree = \u2191p\u2082.natDegree - \u2191q\u2082.natDegree","doc":"This lemma is useful for working with the intDegree of a rational function.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Domain.lean#L88-L94"},"Polynomial.trailingDegree":{"signature":"def Polynomial.trailingDegree {R : Type u} [Semiring R] (p : Polynomial R) :\u2115\u221e","doc":"trailingDegree p is the multiplicity of x in the polynomial p, i.e. the smallest X-exponent in p. trailingDegree p = some n when p \u2260 0 and n is the smallest power of X that appears in p, otherwise trailingDegree 0 = \u22a4.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L43-L49"},"Polynomial.natTrailingDegree":{"signature":"def Polynomial.natTrailingDegree {R : Type u} [Semiring R] (p : Polynomial R) :\u2115","doc":"natTrailingDegree p forces trailingDegree p to \u2115, by defining natTrailingDegree \u22a4 = 0.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L54-L57"},"Polynomial.contract_mul_expand":{"signature":"theorem Polynomial.contract_mul_expand {R : Type u} [CommSemiring R] {p : \u2115} (hp : p \u2260 0) (f g : Polynomial R) :contract p (f * (expand R p) g) = contract p f * g","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Expand.lean#L214-L229"},"Polynomial.contract_add":{"signature":"theorem Polynomial.contract_add {R : Type u} [CommSemiring R] {p : \u2115} (hp : p \u2260 0) (f g : Polynomial R) :contract p (f + g) = contract p f + contract p g","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Expand.lean#L210-L212"},"Polynomial.contract_C":{"signature":"theorem Polynomial.contract_C {R : Type u} [CommSemiring R] (p : \u2115) (r : R) :contract p (C r) = C r","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Expand.lean#L208-L208"},"Polynomial.isCoprime_expand":{"signature":"theorem Polynomial.isCoprime_expand {R : Type u} [CommSemiring R] {f g : Polynomial R} {p : \u2115} (hp : p \u2260 0) :IsCoprime ((expand R p) f) ((expand R p) g) \u2194 IsCoprime f g","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Expand.lean#L231-L234"},"Polynomial.wronskian":{"signature":"def Polynomial.wronskian {R : Type u_1} [CommRing R] (a b : Polynomial R) :Polynomial R","doc":"Wronskian of a pair of polynomials, W(a, b) = ab' - a'b.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Wronskian.lean#L42-L44"},"Polynomial.wronskianBilin":{"signature":"def Polynomial.wronskianBilin (R : Type u_1) [CommRing R] :Polynomial R \u2192\u2097[R] Polynomial R \u2192\u2097[R] Polynomial R","doc":"Polynomial.wronskian as a bilinear map.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Wronskian.lean#L47-L49"},"Polynomial.wronskian_add_right":{"signature":"theorem Polynomial.wronskian_add_right {R : Type u_1} [CommRing R] (a b c : Polynomial R) :a.wronskian (b + c) = a.wronskian b + a.wronskian c","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Wronskian.lean#L67-L68"},"Polynomial.eval_C_mul":{"signature":"theorem Polynomial.eval_C_mul {R : Type u} {a : R} [Semiring R] {p : Polynomial R} {x : R} :eval x (C a * p) = a * eval x p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L307-L311"},"Polynomial.natDegree_pos_of_eval\u2082_root":{"signature":"theorem Polynomial.natDegree_pos_of_eval\u2082_root {R : Type u} {S : Type v} [Semiring R] [Semiring S] {p : Polynomial R} (hp : p \u2260 0) (f : R \u2192+* S) {z : S} (hz : eval\u2082 f z p = 0) (inj : \u2200 (x : R), f x = 0 \u2192 x = 0) :0 < p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L240-L246"},"Polynomial.natDegree_pos_of_aeval_root":{"signature":"theorem Polynomial.natDegree_pos_of_aeval_root {R : Type u} {S : Type v} [CommRing R] [Semiring S] [Algebra R S] {p : Polynomial R} (hp : p \u2260 0) {z : S} (hz : (aeval z) p = 0) (inj : \u2200 (x : R), (algebraMap R S) x = 0 \u2192 x = 0) :0 < p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/RingDivision.lean#L44-L46"},"Polynomial.natDegree_pos_of_monic_of_aeval_eq_zero":{"signature":"theorem Polynomial.natDegree_pos_of_monic_of_aeval_eq_zero {R : Type u} {S : Type v} [CommRing R] [Nontrivial R] [Semiring S] [Algebra R S] [FaithfulSMul R S] {p : Polynomial R} (hp : p.Monic) {x : S} (hx : (aeval x) p = 0) :0 < p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/RingDivision.lean#L146-L150"},"Polynomial.map_ofNat":{"signature":"theorem Polynomial.map_ofNat {R : Type u} {S : Type v} [Semiring R] [Semiring S] (f : R \u2192+* S) (n : \u2115) [n.AtLeastTwo] :map f (OfNat.ofNat n) = OfNat.ofNat n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L538-L541"},"Polynomial.eval_ofNat":{"signature":"theorem Polynomial.eval_ofNat {R : Type u} [Semiring R] (n : \u2115) [n.AtLeastTwo] (a : R) :eval a (OfNat.ofNat n) = OfNat.ofNat n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L282-L285"},"Polynomial.logMahlerMeasure":{"signature":"noncomputable def Polynomial.logMahlerMeasure (p : Polynomial \u2102) :\u211d","doc":"The logarithmic Mahler measure of a polynomial p defined as (2 * \u03c0)\u207b\u00b9 * \u222b x \u2208 (0, 2 * \u03c0), log \u2016p (e ^ (i * x))\u2016","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/MahlerMeasure.lean#L43-L45"},"Polynomial.mahlerMeasure":{"signature":"noncomputable def Polynomial.mahlerMeasure (p : Polynomial \u2102) :\u211d","doc":"The Mahler measure of a polynomial p defined as e ^ (logMahlerMeasure p) if p is nonzero and 0 otherwise","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/MahlerMeasure.lean#L70-L72"},"Polynomial.cardinalMk_eq_max":{"signature":"theorem Polynomial.cardinalMk_eq_max {R : Type u} [Semiring R] [Nontrivial R] :Cardinal.mk (Polynomial R) = max (Cardinal.mk R) Cardinal.aleph0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Cardinal.lean#L29-L33"},"Polynomial.cardinalMk_le_max":{"signature":"theorem Polynomial.cardinalMk_le_max {R : Type u} [Semiring R] :Cardinal.mk (Polynomial R) \u2264 max (Cardinal.mk R) Cardinal.aleph0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Cardinal.lean#L35-L38"},"Polynomial.natTrailingDegree_le_trailingDegree":{"signature":"theorem Polynomial.natTrailingDegree_le_trailingDegree {R : Type u} [Semiring R] {p : Polynomial R} :\u2191p.natTrailingDegree \u2264 p.trailingDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L109-L111"},"Polynomial.eq_zero_of_natDegree_lt_card_of_eval_eq_zero":{"signature":"theorem Polynomial.eq_zero_of_natDegree_lt_card_of_eval_eq_zero {R : Type u_1} [CommRing R] [IsDomain R] (p : Polynomial R) {\u03b9 : Type u_2} [Fintype \u03b9] {f : \u03b9 \u2192 R} (hf : Function.Injective f) (heval : \u2200 (i : \u03b9), eval (f i) p = 0) (hcard : p.natDegree < Fintype.card \u03b9) :p = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L690-L707"},"Polynomial.eq_of_natDegree_lt_card_of_eval_eq":{"signature":"theorem Polynomial.eq_of_natDegree_lt_card_of_eval_eq {R : Type u_1} [CommRing R] [IsDomain R] (p q : Polynomial R) {\u03b9 : Type u_2} [Fintype \u03b9] {f : \u03b9 \u2192 R} (hf : Function.Injective f) (heval : \u2200 (i : \u03b9), eval (f i) p = eval (f i) q) (hcard : max p.natDegree q.natDegree < Fintype.card \u03b9) :p = q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L709-L716"},"Polynomial.preHilbertPoly":{"signature":"noncomputable def Polynomial.preHilbertPoly (F : Type u_1) [Field F] (d k : \u2115) :Polynomial F","doc":"For any field F and natural numbers d and k, Polynomial.preHilbertPoly F d k is defined as (d.factorial : F)\u207b\u00b9 \u2022 ((ascPochhammer F d).comp (X - (C (k : F)) + 1)). This is the most basic form of Hilbert polynomials. Polynomial.preHilbertPoly \u211a d 0 is exactly the Hilbert polynomial of the polynomial ring \u211a[X_0,...,X_d] viewed as a graded module over itself. In fact, Polynomial.preHilbertPoly F d k is the same as Polynomial.hilbertPoly ((X : F[X]) ^ k) (d + 1) for any field F and d k : \u2115 (see the lemma Polynomial.hilbertPoly_X_pow_succ). See also the lemma Polynomial.preHilbertPoly_eq_choose_sub_add, which states that if CharZero F, then for any d k n : \u2115 with k \u2264 n, (Polynomial.preHilbertPoly F d k).eval (n : F) equals (n - k + d).choose d.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/HilbertPoly.lean#L51-L64"},"Polynomial.natDegree_smul":{"signature":"theorem Polynomial.natDegree_smul {R : Type u} [Semiring R] (p : Polynomial R) {S : Type u_1} [Semiring S] [IsDomain S] [Module S R] [Module.IsTorsionFree S R] {a : S} (ha : a \u2260 0) :(a \u2022 p).natDegree = p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Domain.lean#L43-L51"},"Polynomial.natDegree_preHilbertPoly":{"signature":"theorem Polynomial.natDegree_preHilbertPoly (F : Type u_1) [Field F] [CharZero F] (d k : \u2115) :(preHilbertPoly F d k).natDegree = d","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/HilbertPoly.lean#L66-L70"},"Polynomial.coeff_list_prod_of_natDegree_le":{"signature":"theorem Polynomial.coeff_list_prod_of_natDegree_le {S : Type u_1} [Semiring S] (l : List (Polynomial S)) (n : \u2115) (hl : \u2200 p \u2208 l, p.natDegree \u2264 n) :l.prod.coeff (l.length * n) = (List.map (fun (p : Polynomial S) => p.coeff n) l).prod","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/BigOperators.lean#L110-L123"},"Polynomial.coeff_multiset_prod_of_natDegree_le":{"signature":"theorem Polynomial.coeff_multiset_prod_of_natDegree_le {R : Type u} [CommSemiring R] (t : Multiset (Polynomial R)) (n : \u2115) (hl : \u2200 p \u2208 t, p.natDegree \u2264 n) :t.prod.coeff (t.card * n) = (Multiset.map (fun (p : Polynomial R) => p.coeff n) t).prod","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/BigOperators.lean#L222-L225"},"Polynomial.coeff_prod_of_natDegree_le":{"signature":"theorem Polynomial.coeff_prod_of_natDegree_le {R : Type u} {\u03b9 : Type w} (s : Finset \u03b9) [CommSemiring R] (f : \u03b9 \u2192 Polynomial R) (n : \u2115) (h : \u2200 p \u2208 s, (f p).natDegree \u2264 n) :(\u220f i \u2208 s, f i).coeff (s.card * n) = \u220f i \u2208 s, (f i).coeff n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/BigOperators.lean#L227-L233"},"Polynomial.coeff_det_X_add_C_card":{"signature":"theorem Polynomial.coeff_det_X_add_C_card {n : Type u_1} {\u03b1 : Type u_2} [DecidableEq n] [Fintype n] [CommRing \u03b1] (A B : Matrix n n \u03b1) :(X \u2022 A.map \u21d1C + B.map \u21d1C).det.coeff (Fintype.card n) = A.det","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/LinearAlgebra/Matrix/Polynomial.lean#L72-L84"},"Polynomial.natDegree_list_sum_le":{"signature":"theorem Polynomial.natDegree_list_sum_le {S : Type u_1} [Semiring S] (l : List (Polynomial S)) :l.sum.natDegree \u2264 List.foldr max 0 (List.map natDegree l)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/BigOperators.lean#L49-L53"},"Polynomial.natDegree_multiset_sum_le":{"signature":"theorem Polynomial.natDegree_multiset_sum_le {S : Type u_1} [Semiring S] (l : Multiset (Polynomial S)) :l.sum.natDegree \u2264 Multiset.foldr max 0 (Multiset.map natDegree l)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/BigOperators.lean#L55-L57"},"Polynomial.natDegree_sum_le":{"signature":"theorem Polynomial.natDegree_sum_le {\u03b9 : Type w} (s : Finset \u03b9) {S : Type u_1} [Semiring S] (f : \u03b9 \u2192 Polynomial S) :(\u2211 i \u2208 s, f i).natDegree \u2264 Finset.fold max 0 (natDegree \u2218 f) s","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/BigOperators.lean#L59-L61"},"Polynomial.natDegree_det_X_add_C_le":{"signature":"theorem Polynomial.natDegree_det_X_add_C_le {n : Type u_1} {\u03b1 : Type u_2} [DecidableEq n] [Fintype n] [CommRing \u03b1] (A B : Matrix n n \u03b1) :(X \u2022 A.map \u21d1C + B.map \u21d1C).det.natDegree \u2264 Fintype.card n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/LinearAlgebra/Matrix/Polynomial.lean#L40-L60"},"Polynomial.leadingCoeff_det_X_one_add_C":{"signature":"theorem Polynomial.leadingCoeff_det_X_one_add_C {n : Type u_1} {\u03b1 : Type u_2} [DecidableEq n] [Fintype n] [CommRing \u03b1] (A : Matrix n n \u03b1) :(X \u2022 1 + A.map \u21d1C).det.leadingCoeff = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/LinearAlgebra/Matrix/Polynomial.lean#L86-L99"},"Polynomial.modByMonic_eq_of_dvd_sub":{"signature":"theorem Polynomial.modByMonic_eq_of_dvd_sub {R : Type u} [CommRing R] {p\u2081 p\u2082 q : Polynomial R} (hq : q.Monic) (h : q \u2223 p\u2081 - p\u2082) :p\u2081 %\u2098 q = p\u2082 %\u2098 q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L664-L668"},"Polynomial.toFinsupp_apply":{"signature":"theorem Polynomial.toFinsupp_apply {R : Type u} [Semiring R] (f : Polynomial R) (i : \u2115) :f.toFinsupp i = f.coeff i","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L578-L578"},"Polynomial.eta":{"signature":"theorem Polynomial.eta {R : Type u} [Semiring R] (f : Polynomial R) :{ toFinsupp := f.toFinsupp } = f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L98-L99"},"Polynomial.toFinsupp_X_pow":{"signature":"theorem Polynomial.toFinsupp_X_pow {R : Type u} [Semiring R] (n : \u2115) :(X ^ n).toFinsupp = AddMonoidAlgebra.single n 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L794-L796"},"Polynomial.splits_mul":{"signature":"theorem Polynomial.splits_mul {R : Type u_1} [Semiring R] {f g : Polynomial R} (hf : f.Splits) (hg : g.Splits) :(f * g).Splits","doc":"Alias of Polynomial.Splits.mul.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L678-L679"},"Polynomial.eval\u2082_C_X":{"signature":"theorem Polynomial.eval\u2082_C_X {R : Type u} [Semiring R] {p : Polynomial R} :eval\u2082 C X p = p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Coeff.lean#L46-L49"},"Polynomial.degree_C_mul_X":{"signature":"theorem Polynomial.degree_C_mul_X {R : Type u} {a : R} [Semiring R] (ha : a \u2260 0) :(C a * X).degree = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L197-L198"},"Polynomial.degree_C_lt":{"signature":"theorem Polynomial.degree_C_lt {R : Type u} {a : R} [Semiring R] :(C a).degree < 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L162-L163"},"Polynomial.degree_C_lt_degree_C_mul_X":{"signature":"theorem Polynomial.degree_C_lt_degree_C_mul_X {R : Type u} {a b : R} [Semiring R] (ha : a \u2260 0) :(C b).degree < (C a * X).degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L505-L506"},"Polynomial.leadingCoeff_linear":{"signature":"theorem Polynomial.leadingCoeff_linear {R : Type u} {a b : R} [Semiring R] (ha : a \u2260 0) :(C a * X + C b).leadingCoeff = a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/SmallDegree.lean#L85-L88"},"Polynomial.eval_multiset_prod_X_sub_C_derivative":{"signature":"theorem Polynomial.eval_multiset_prod_X_sub_C_derivative {R : Type u} [CommRing R] [DecidableEq R] {S : Multiset R} {r : R} (hr : r \u2208 S) :eval r (derivative (Multiset.map (fun (a : R) => X - C a) S).prod) = (Multiset.map (fun (a : R) => r - a) (S.erase r)).prod","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L656-L662"},"Polynomial.cyclotomic_prime_pow_mul_X_pow_sub_one":{"signature":"theorem Polynomial.cyclotomic_prime_pow_mul_X_pow_sub_one (R : Type u_1) [CommRing R] (p k : \u2115) [hn : Fact (Nat.Prime p)] :cyclotomic (p ^ (k + 1)) R * (X ^ p ^ k - 1) = X ^ p ^ (k + 1) - 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean#L528-L531"},"Polynomial.aeval_natCast":{"signature":"theorem Polynomial.aeval_natCast {R : Type u} {A : Type z} [CommSemiring R] [Semiring A] [Algebra R A] (x : A) (n : \u2115) :(aeval x) \u2191n = \u2191n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L296-L297"},"Polynomial.IsEisensteinAt":{"signature":"structure Polynomial.IsEisensteinAt {R : Type u} [CommSemiring R] (f : Polynomial R) (\ud835\udcdf : Ideal R) :Prop","doc":"Given an ideal \ud835\udcdf of a commutative semiring R, we say that a polynomial f : R[X] is Eisenstein at \ud835\udcdf if f.leadingCoeff \u2209 \ud835\udcdf, \u2200 n, n < f.natDegree \u2192 f.coeff n \u2208 \ud835\udcdf and f.coeff 0 \u2209 \ud835\udcdf ^ 2.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Eisenstein/Basic.lean#L51-L58"},"Polynomial.monomial_mul_X":{"signature":"theorem Polynomial.monomial_mul_X {R : Type u} [Semiring R] (n : \u2115) (r : R) :(monomial n) r * X = (monomial (n + 1)) r","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L544-L546"},"Polynomial.mul_X_comp":{"signature":"theorem Polynomial.mul_X_comp {R : Type u} [Semiring R] {p r : Polynomial R} :(p * X).comp r = p.comp r * r","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L417-L421"},"Polynomial.X_pow_comp":{"signature":"theorem Polynomial.X_pow_comp {R : Type u} [Semiring R] {p : Polynomial R} {k : \u2115} :(X ^ k).comp p = p ^ k","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L423-L427"},"Polynomial.geom_sum_X_comp_X_add_one_eq_sum":{"signature":"theorem Polynomial.geom_sum_X_comp_X_add_one_eq_sum {R : Type u} [Semiring R] (n : \u2115) :(\u2211 i \u2208 Finset.range n, X ^ i).comp (X + 1) = \u2211 i \u2208 Finset.range n, \u2191(n.choose (i + 1)) * X ^ i","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Basic.lean#L246-L261"},"Polynomial.cyclotomic_mul_prime_dvd_eq_pow":{"signature":"theorem Polynomial.cyclotomic_mul_prime_dvd_eq_pow (R : Type u_1) {p n : \u2115} [hp : Fact (Nat.Prime p)] [Ring R] [CharP R p] (hn : p \u2223 n) :cyclotomic (n * p) R = cyclotomic n R ^ p","doc":"If R is of characteristic p and p \u2223 n, then cyclotomic (n * p) R = (cyclotomic n R) ^ p.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Expand.lean#L144-L153"},"Polynomial.pow_comp":{"signature":"theorem Polynomial.pow_comp {R : Type u_1} [CommSemiring R] (p q : Polynomial R) (n : \u2115) :(p ^ n).comp q = p.comp q ^ n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L459-L463"},"Polynomial.ofMultiset":{"signature":"def Polynomial.ofMultiset {R : Type u} [CommRing R] :AddChar (Multiset R) (Polynomial R)","doc":"The map sending a collection of roots into a polynomial, as a morphism.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L1194-L1198"},"Polynomial.roots_ofMultiset":{"signature":"theorem Polynomial.roots_ofMultiset {R : Type u} [CommRing R] [IsDomain R] (s : Multiset R) :(ofMultiset s).roots = s","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L282-L283"},"Polynomial.rightInverse_ofMultiset_roots":{"signature":"theorem Polynomial.rightInverse_ofMultiset_roots (R : Type u) [CommRing R] [IsDomain R] :Function.RightInverse (\u21d1ofMultiset) roots","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L286-L287"},"Polynomial.aeval_prod":{"signature":"theorem Polynomial.aeval_prod {R : Type u} {A : Type z} {B : Type u_2} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (x : A \u00d7 B) :aeval x = (aeval x.1).prod (aeval x.2)","doc":"Polynomial evaluation on a pair is a product of the evaluations on the components.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L419-L422"},"Polynomial.aeval_prod_apply":{"signature":"theorem Polynomial.aeval_prod_apply {R : Type u} {A : Type z} {B : Type u_2} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (x : A \u00d7 B) (p : Polynomial R) :(aeval x) p = ((aeval x.1) p, (aeval x.2) p)","doc":"Polynomial evaluation on a pair is a pair of evaluations.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L424-L426"},"Polynomial.comp_assoc":{"signature":"theorem Polynomial.comp_assoc {R : Type u_1} [CommSemiring R] (\u03c6 \u03c8 \u03c7 : Polynomial R) :(\u03c6.comp \u03c8).comp \u03c7 = \u03c6.comp (\u03c8.comp \u03c7)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L465-L469"},"Polynomial.smeval_X_mul":{"signature":"theorem Polynomial.smeval_X_mul (R : Type u_1) [Semiring R] (p : Polynomial R) {S : Type u_2} [NonAssocSemiring S] [Module R S] [Pow S \u2115] (x : S) [NatPowAssoc S] [SMulCommClass R S S] :(X * p).smeval x = x * p.smeval x","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Smeval.lean#L210-L215"},"Polynomial.smeval_X_pow_assoc":{"signature":"theorem Polynomial.smeval_X_pow_assoc (R : Type u_1) [Semiring R] (p : Polynomial R) {S : Type u_2} [NonAssocSemiring S] [Module R S] [Pow S \u2115] (x : S) [NatPowAssoc S] [SMulCommClass R S S] (m n : \u2115) :x ^ m * x ^ n * p.smeval x = x ^ m * (x ^ n * p.smeval x)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Smeval.lean#L217-L221"},"Polynomial.smeval_X_pow_mul":{"signature":"theorem Polynomial.smeval_X_pow_mul (R : Type u_1) [Semiring R] (p : Polynomial R) {S : Type u_2} [NonAssocSemiring S] [Module R S] [Pow S \u2115] (x : S) [NatPowAssoc S] [SMulCommClass R S S] (n : \u2115) :(X ^ n * p).smeval x = x ^ n * p.smeval x","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Smeval.lean#L223-L228"},"Polynomial.smeval_monomial_mul":{"signature":"theorem Polynomial.smeval_monomial_mul (R : Type u_1) [Semiring R] (r : R) (p : Polynomial R) {S : Type u_2} [NonAssocSemiring S] [Module R S] [Pow S \u2115] (x : S) [NatPowAssoc S] [SMulCommClass R S S] (n : \u2115) :((monomial n) r * p).smeval x = r \u2022 (x ^ n * p.smeval x)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Smeval.lean#L230-L237"},"Polynomial.smeval_mul":{"signature":"theorem Polynomial.smeval_mul (R : Type u_1) [Semiring R] (p q : Polynomial R) {S : Type u_2} [NonAssocSemiring S] [Module R S] [Pow S \u2115] (x : S) [NatPowAssoc S] [IsScalarTower R S S] [SMulCommClass R S S] :(p * q).smeval x = p.smeval x * q.smeval x","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Smeval.lean#L266-L270"},"Polynomial.smeval_pow":{"signature":"theorem Polynomial.smeval_pow (R : Type u_1) [Semiring R] (p : Polynomial R) {S : Type u_2} [NonAssocSemiring S] [Module R S] [Pow S \u2115] (x : S) [NatPowAssoc S] [IsScalarTower R S S] [SMulCommClass R S S] (n : \u2115) :(p ^ n).smeval x = p.smeval x ^ n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Smeval.lean#L272-L276"},"Polynomial.smeval_comp":{"signature":"theorem Polynomial.smeval_comp (R : Type u_1) [Semiring R] (p q : Polynomial R) {S : Type u_2} [NonAssocSemiring S] [Module R S] [Pow S \u2115] (x : S) [NatPowAssoc S] [IsScalarTower R S S] [SMulCommClass R S S] :(p.comp q).smeval x = p.smeval (q.smeval x)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Smeval.lean#L278-L281"},"Polynomial.smeval_X":{"signature":"theorem Polynomial.smeval_X (R : Type u_1) [Semiring R] {S : Type u_2} [AddCommMonoid S] [Pow S \u2115] [MulActionWithZero R S] (x : S) :X.smeval x = x ^ 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Smeval.lean#L91-L94"},"Polynomial.smeval_zero":{"signature":"theorem Polynomial.smeval_zero (R : Type u_1) [Semiring R] {S : Type u_2} [AddCommMonoid S] [Pow S \u2115] [MulActionWithZero R S] (x : S) :smeval 0 x = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Smeval.lean#L82-L84"},"Polynomial.smeval_neg":{"signature":"theorem Polynomial.smeval_neg (R : Type u_1) [Ring R] {S : Type u_2} [AddCommGroup S] [Pow S \u2115] [Module R S] (p : Polynomial R) (x : S) :(-p).smeval x = -p.smeval x","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Smeval.lean#L153-L155"},"Polynomial.smeval_sub":{"signature":"theorem Polynomial.smeval_sub (R : Type u_1) [Ring R] {S : Type u_2} [AddCommGroup S] [Pow S \u2115] [Module R S] (p q : Polynomial R) (x : S) :(p - q).smeval x = p.smeval x - q.smeval x","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Smeval.lean#L157-L159"},"Polynomial.smeval_natCast":{"signature":"theorem Polynomial.smeval_natCast (R : Type u_1) [Semiring R] {S : Type u_2} [AddCommMonoid S] [Pow S \u2115] [Module R S] (x : S) (n : \u2115) :(\u2191n).smeval x = n \u2022 x ^ 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Smeval.lean#L115-L118"},"Polynomial.descPochhammer_smeval_eq_descFactorial":{"signature":"theorem Polynomial.descPochhammer_smeval_eq_descFactorial {R : Type u_1} [NonAssocRing R] [Pow R \u2115] [NatPowAssoc R] (n k : \u2115) :(descPochhammer \u2124 k).smeval \u2191n = \u2191(n.descFactorial k)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Binomial.lean#L216-L226"},"Polynomial.descPochhammer_smeval_eq_ascPochhammer":{"signature":"theorem Polynomial.descPochhammer_smeval_eq_ascPochhammer {R : Type u_1} [NonAssocRing R] [Pow R \u2115] [NatPowAssoc R] (r : R) (n : \u2115) :(descPochhammer \u2124 n).smeval r = (ascPochhammer \u2115 n).smeval (r - \u2191n + 1)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Binomial.lean#L206-L214"},"Polynomial.ascPochhammer_smeval_neg_eq_descPochhammer":{"signature":"theorem Polynomial.ascPochhammer_smeval_neg_eq_descPochhammer {R : Type u_1} [NonAssocRing R] [Pow R \u2115] [NatPowAssoc R] (r : R) (k : \u2115) :(ascPochhammer \u2115 k).smeval (-r) = (\u2191k).negOnePow \u2022 (descPochhammer \u2124 k).smeval r","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Binomial.lean#L228-L237"},"Polynomial.ascPochhammer_smeval_eq_eval":{"signature":"theorem Polynomial.ascPochhammer_smeval_eq_eval {R : Type u_1} [Semiring R] (r : R) (n : \u2115) :(ascPochhammer \u2115 n).smeval r = eval r (ascPochhammer R n)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Binomial.lean#L200-L202"},"Polynomial.eval_natCast_mul":{"signature":"theorem Polynomial.eval_natCast_mul {R : Type u} [Semiring R] {p : Polynomial R} {x : R} {n : \u2115} :eval x (\u2191n * p) = \u2191n * eval x p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L313-L315"},"Polynomial.mahlerMeasure_zero":{"signature":"theorem Polynomial.mahlerMeasure_zero :mahlerMeasure 0 = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/MahlerMeasure.lean#L84-L85"},"Polynomial.logMahlerMeasure_zero":{"signature":"theorem Polynomial.logMahlerMeasure_zero :logMahlerMeasure 0 = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/MahlerMeasure.lean#L50-L52"},"Polynomial.logMahlerMeasure_eq_log_MahlerMeasure":{"signature":"theorem Polynomial.logMahlerMeasure_eq_log_MahlerMeasure {p : Polynomial \u2102} :p.logMahlerMeasure = Real.log p.mahlerMeasure","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/MahlerMeasure.lean#L80-L82"},"Polynomial.mahlerMeasure_def_of_ne_zero":{"signature":"theorem Polynomial.mahlerMeasure_def_of_ne_zero {p : Polynomial \u2102} (hp : p \u2260 0) :p.mahlerMeasure = Real.exp ((2 * Real.pi)\u207b\u00b9 * \u222b (x : \u211d) in 0..2 * Real.pi, Real.log \u2016eval (circleMap 0 1 x) p\u2016)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/MahlerMeasure.lean#L75-L77"},"Polynomial.mahlerMeasure_eq_zero_iff":{"signature":"theorem Polynomial.mahlerMeasure_eq_zero_iff (p : Polynomial \u2102) :p.mahlerMeasure = 0 \u2194 p = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/MahlerMeasure.lean#L104-L108"},"Polynomial.intervalIntegrable_mahlerMeasure":{"signature":"theorem Polynomial.intervalIntegrable_mahlerMeasure (p : Polynomial \u2102) :IntervalIntegrable (fun (x : \u211d) => Real.log \u2016eval (circleMap 0 1 x) p\u2016) MeasureTheory.volume 0 (2 * Real.pi)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/MahlerMeasure.lean#L110-L114"},"Polynomial.mahlerMeasure_mul":{"signature":"theorem Polynomial.mahlerMeasure_mul (p q : Polynomial \u2102) :(p * q).mahlerMeasure = p.mahlerMeasure * q.mahlerMeasure","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/MahlerMeasure.lean#L118-L137"},"Polynomial.logMahlerMeasure_mul_eq_add_logMahlerMeasure":{"signature":"theorem Polynomial.logMahlerMeasure_mul_eq_add_logMahlerMeasure {p q : Polynomial \u2102} (hpq : p * q \u2260 0) :(p * q).logMahlerMeasure = p.logMahlerMeasure + q.logMahlerMeasure","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/MahlerMeasure.lean#L146-L148"},"Polynomial.logMahlerMeasure_const":{"signature":"theorem Polynomial.logMahlerMeasure_const (z : \u2102) :(C z).logMahlerMeasure = Real.log \u2016z\u2016","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/MahlerMeasure.lean#L58-L60"},"Polynomial.mahlerMeasure_const":{"signature":"theorem Polynomial.mahlerMeasure_const (z : \u2102) :(C z).mahlerMeasure = \u2016z\u2016","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/MahlerMeasure.lean#L90-L95"},"Polynomial.logMahlerMeasure_one":{"signature":"theorem Polynomial.logMahlerMeasure_one :logMahlerMeasure 1 = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/MahlerMeasure.lean#L54-L56"},"Polynomial.mahlerMeasure_one":{"signature":"theorem Polynomial.mahlerMeasure_one :mahlerMeasure 1 = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/MahlerMeasure.lean#L87-L88"},"Polynomial.prod_mahlerMeasure_eq_mahlerMeasure_prod":{"signature":"theorem Polynomial.prod_mahlerMeasure_eq_mahlerMeasure_prod (s : Multiset (Polynomial \u2102)) :s.prod.mahlerMeasure = (Multiset.map (fun (p : Polynomial \u2102) => p.mahlerMeasure) s).prod","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/MahlerMeasure.lean#L139-L144"},"Polynomial.logMahlerMeasure_X_sub_C":{"signature":"theorem Polynomial.logMahlerMeasure_X_sub_C (z : \u2102) :(X - C z).logMahlerMeasure = \u2016z\u2016.posLog","doc":"The logarithmic Mahler measure of X - C z is the log\u207a of the absolute value of z.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/MahlerMeasure.lean#L158-L161"},"Polynomial.mahlerMeasure_pos_of_ne_zero":{"signature":"theorem Polynomial.mahlerMeasure_pos_of_ne_zero {p : Polynomial \u2102} (hp : p \u2260 0) :0 < p.mahlerMeasure","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/MahlerMeasure.lean#L101-L102"},"Polynomial.mahlerMeasure_X_sub_C":{"signature":"theorem Polynomial.mahlerMeasure_X_sub_C (z : \u2102) :(X - C z).mahlerMeasure = max 1 \u2016z\u2016","doc":"The Mahler measure of X - C z equals max 1 \u2016z\u2016.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/MahlerMeasure.lean#L177-L185"},"Polynomial.logMahlerMeasure_eq_log_leadingCoeff_add_sum_log_roots":{"signature":"theorem Polynomial.logMahlerMeasure_eq_log_leadingCoeff_add_sum_log_roots (p : Polynomial \u2102) :p.logMahlerMeasure = Real.log \u2016p.leadingCoeff\u2016 + (Multiset.map (fun (a : \u2102) => \u2016a\u2016.posLog) p.roots).sum","doc":"The logarithmic Mahler measure of a polynomial is the log of the absolute value of its leading coefficient plus the sum of the logs of the absolute values of its roots lying outside the unit disk.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/MahlerMeasure.lean#L204-L215"},"Polynomial.mahlerMeasure_eq_leadingCoeff_mul_prod_roots":{"signature":"theorem Polynomial.mahlerMeasure_eq_leadingCoeff_mul_prod_roots (p : Polynomial \u2102) :p.mahlerMeasure = \u2016p.leadingCoeff\u2016 * (Multiset.map (fun (a : \u2102) => max 1 \u2016a\u2016) p.roots).prod","doc":"The Mahler measure of a polynomial is the absolute value of its leading coefficient times the product of the absolute values of its roots lying outside the unit disk.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/MahlerMeasure.lean#L217-L228"},"Polynomial.norm_coeff_le_choose_mul_mahlerMeasure":{"signature":"theorem Polynomial.norm_coeff_le_choose_mul_mahlerMeasure (n : \u2115) (p : Polynomial \u2102) :\u2016p.coeff n\u2016 \u2264 \u2191(p.natDegree.choose n) * p.mahlerMeasure","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/MahlerMeasure.lean#L279-L325"},"Polynomial.mirror":{"signature":"noncomputable def Polynomial.mirror {R : Type u_1} [Semiring R] (p : Polynomial R) :Polynomial R","doc":"mirror of a polynomial: reverses the coefficients while preserving Polynomial.natDegree","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Mirror.lean#L38-L40"},"Polynomial.natTrailingDegree_eq_of_trailingDegree_eq_some":{"signature":"theorem Polynomial.natTrailingDegree_eq_of_trailingDegree_eq_some {R : Type u} [Semiring R] {p : Polynomial R} {n : \u2115} (h : p.trailingDegree = \u2191n) :p.natTrailingDegree = n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L105-L107"},"Polynomial.support_add":{"signature":"theorem Polynomial.support_add {R : Type u} [Semiring R] {p q : Polynomial R} :(p + q).support \u2286 p.support \u222a q.support","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L428-L429"},"Polynomial.support_trinomial'":{"signature":"theorem Polynomial.support_trinomial' {R : Type u} [Semiring R] (k m n : \u2115) (x y z : R) :(C x * X ^ k + C y * X ^ m + C z * X ^ n).support \u2286 {k, m, n}","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L777-L787"},"Polynomial.trailingDegree_le_of_ne_zero":{"signature":"theorem Polynomial.trailingDegree_le_of_ne_zero {R : Type u} {n : \u2115} [Semiring R] {p : Polynomial R} (h : p.coeff n \u2260 0) :p.trailingDegree \u2264 \u2191n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L118-L119"},"Polynomial.trinomial_trailing_coeff'":{"signature":"theorem Polynomial.trinomial_trailing_coeff' {R : Type u_1} [Semiring R] {k m n : \u2115} {u v w : R} (hkm : k < m) (hmn : m < n) :(trinomial k m n u v w).coeff k = u","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/UnitTrinomial.lean#L61-L64"},"Polynomial.trinomial_natTrailingDegree":{"signature":"theorem Polynomial.trinomial_natTrailingDegree {R : Type u_1} [Semiring R] {k m n : \u2115} {u v w : R} (hkm : k < m) (hmn : m < n) (hu : u \u2260 0) :(trinomial k m n u v w).natTrailingDegree = k","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/UnitTrinomial.lean#L79-L90"},"Polynomial.trinomial_leading_coeff'":{"signature":"theorem Polynomial.trinomial_leading_coeff' {R : Type u_1} [Semiring R] {k m n : \u2115} {u v w : R} (hkm : k < m) (hmn : m < n) :(trinomial k m n u v w).coeff n = w","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/UnitTrinomial.lean#L51-L54"},"Polynomial.trinomial_natDegree":{"signature":"theorem Polynomial.trinomial_natDegree {R : Type u_1} [Semiring R] {k m n : \u2115} {u v w : R} (hkm : k < m) (hmn : m < n) (hw : w \u2260 0) :(trinomial k m n u v w).natDegree = n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/UnitTrinomial.lean#L66-L77"},"Polynomial.trinomial_mirror":{"signature":"theorem Polynomial.trinomial_mirror {R : Type u_1} [Semiring R] {k m n : \u2115} {u v w : R} (hkm : k < m) (hmn : m < n) (hu : u \u2260 0) (hw : w \u2260 0) :(trinomial k m n u v w).mirror = trinomial k (n - m + k) n w v u","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/UnitTrinomial.lean#L104-L111"},"Polynomial.binomial_eq_binomial":{"signature":"theorem Polynomial.binomial_eq_binomial {R : Type u} [Semiring R] {k l m n : \u2115} {u v : R} (hu : u \u2260 0) (hv : v \u2260 0) :C u * X ^ k + C v * X ^ l = C u * X ^ m + C v * X ^ n \u2194 k = m \u2227 l = n \u2228 u = v \u2227 k = n \u2227 l = m \u2228 u + v = 0 \u2227 k = l \u2227 m = n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L815-L819"},"Polynomial.mirror_zero":{"signature":"theorem Polynomial.mirror_zero {R : Type u_1} [Semiring R] :mirror 0 = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Mirror.lean#L42-L43"},"Polynomial.trailingCoeff":{"signature":"def Polynomial.trailingCoeff {R : Type u} [Semiring R] (p : Polynomial R) :R","doc":"trailingCoeff p gives the coefficient of the smallest power of X in p.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L59-L61"},"Polynomial.natTrailingDegree_zero":{"signature":"theorem Polynomial.natTrailingDegree_zero {R : Type u} [Semiring R] :natTrailingDegree 0 = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L85-L87"},"Polynomial.trailingDegree_eq_top":{"signature":"theorem Polynomial.trailingDegree_eq_top {R : Type u} [Semiring R] {p : Polynomial R} :p.trailingDegree = \u22a4 \u2194 p = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L89-L91"},"Polynomial.trailingDegree_eq_natTrailingDegree":{"signature":"theorem Polynomial.trailingDegree_eq_natTrailingDegree {R : Type u} [Semiring R] {p : Polynomial R} (hp : p \u2260 0) :p.trailingDegree = \u2191p.natTrailingDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L93-L95"},"Polynomial.trailingCoeff_eq_zero":{"signature":"theorem Polynomial.trailingCoeff_eq_zero {R : Type u} [Semiring R] {p : Polynomial R} :p.trailingCoeff = 0 \u2194 p = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L245-L251"},"Polynomial.natTrailingDegree_le_natDegree":{"signature":"theorem Polynomial.natTrailingDegree_le_natDegree {R : Type u} [Semiring R] (p : Polynomial R) :p.natTrailingDegree \u2264 p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L271-L274"},"Polynomial.trailingCoeff_nonzero_iff_nonzero":{"signature":"theorem Polynomial.trailingCoeff_nonzero_iff_nonzero {R : Type u} [Semiring R] {p : Polynomial R} :p.trailingCoeff \u2260 0 \u2194 p \u2260 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L253-L254"},"Polynomial.reverse_natDegree_le":{"signature":"theorem Polynomial.reverse_natDegree_le {R : Type u_1} [Semiring R] (f : Polynomial R) :f.reverse.natDegree \u2264 f.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Reverse.lean#L236-L241"},"Polynomial.natTrailingDegree_le_of_ne_zero":{"signature":"theorem Polynomial.natTrailingDegree_le_of_ne_zero {R : Type u} {n : \u2115} [Semiring R] {p : Polynomial R} (h : p.coeff n \u2260 0) :p.natTrailingDegree \u2264 n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L121-L122"},"Polynomial.natDegree_eq_reverse_natDegree_add_natTrailingDegree":{"signature":"theorem Polynomial.natDegree_eq_reverse_natDegree_add_natTrailingDegree {R : Type u_1} [Semiring R] (f : Polynomial R) :f.natDegree = f.reverse.natDegree + f.natTrailingDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Reverse.lean#L243-L255"},"Polynomial.reverse_natDegree":{"signature":"theorem Polynomial.reverse_natDegree {R : Type u_1} [Semiring R] (f : Polynomial R) :f.reverse.natDegree = f.natDegree - f.natTrailingDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Reverse.lean#L257-L258"},"Polynomial.reverse_leadingCoeff":{"signature":"theorem Polynomial.reverse_leadingCoeff {R : Type u_1} [Semiring R] (f : Polynomial R) :f.reverse.leadingCoeff = f.trailingCoeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Reverse.lean#L260-L262"},"Polynomial.mirror_natDegree":{"signature":"theorem Polynomial.mirror_natDegree {R : Type u_1} [Semiring R] (p : Polynomial R) :p.mirror.natDegree = p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Mirror.lean#L59-L65"},"Polynomial.coeff_eq_zero_of_lt_trailingDegree":{"signature":"theorem Polynomial.coeff_eq_zero_of_lt_trailingDegree {R : Type u} {n : \u2115} [Semiring R] {p : Polynomial R} (h : \u2191n < p.trailingDegree) :p.coeff n = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L218-L219"},"Polynomial.trailingDegree_zero":{"signature":"theorem Polynomial.trailingDegree_zero {R : Type u} [Semiring R] :trailingDegree 0 = \u22a4","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L77-L79"},"Polynomial.coeff_eq_zero_of_lt_natTrailingDegree":{"signature":"theorem Polynomial.coeff_eq_zero_of_lt_natTrailingDegree {R : Type u} [Semiring R] {p : Polynomial R} {n : \u2115} (h : n < p.natTrailingDegree) :p.coeff n = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L221-L228"},"Polynomial.revAtFun_eq":{"signature":"theorem Polynomial.revAtFun_eq (N i : \u2115) :revAtFun N i = (revAt N) i","doc":"We prefer to use the bundled revAt over unbundled revAtFun.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Reverse.lean#L56-L59"},"Polynomial.nonempty_support_iff":{"signature":"theorem Polynomial.nonempty_support_iff {R : Type u} [Semiring R] {p : Polynomial R} :p.support.Nonempty \u2194 p \u2260 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Support.lean#L115-L116"},"Polynomial.mul_X_pow_eq_zero":{"signature":"theorem Polynomial.mul_X_pow_eq_zero {R : Type u} [Semiring R] {p : Polynomial R} {n : \u2115} (H : p * X ^ n = 0) :p = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L276-L277"},"Polynomial.natTrailingDegree_eq_support_min'":{"signature":"theorem Polynomial.natTrailingDegree_eq_support_min' {R : Type u} [Semiring R] {p : Polynomial R} (h : p \u2260 0) :p.natTrailingDegree = p.support.min' \u22ef","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L262-L264"},"Polynomial.natTrailingDegree_mul_X_pow":{"signature":"theorem Polynomial.natTrailingDegree_mul_X_pow {R : Type u} [Semiring R] {p : Polynomial R} (hp : p \u2260 0) (n : \u2115) :(p * X ^ n).natTrailingDegree = p.natTrailingDegree + n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L276-L287"},"Polynomial.trailingDegree_eq_zero":{"signature":"theorem Polynomial.trailingDegree_eq_zero {R : Type u} [Semiring R] {p : Polynomial R} :p.trailingDegree = 0 \u2194 p.coeff 0 \u2260 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L137-L139"},"Polynomial.natTrailingDegree_eq_zero":{"signature":"theorem Polynomial.natTrailingDegree_eq_zero {R : Type u} [Semiring R] {p : Polynomial R} :p.natTrailingDegree = 0 \u2194 p = 0 \u2228 p.coeff 0 \u2260 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L141-L142"},"Polynomial.natTrailingDegree_reverse":{"signature":"theorem Polynomial.natTrailingDegree_reverse {R : Type u_1} [Semiring R] (f : Polynomial R) :f.reverse.natTrailingDegree = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Reverse.lean#L264-L266"},"Polynomial.mirror_natTrailingDegree":{"signature":"theorem Polynomial.mirror_natTrailingDegree {R : Type u_1} [Semiring R] (p : Polynomial R) :p.mirror.natTrailingDegree = p.natTrailingDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Mirror.lean#L67-L71"},"Polynomial.coeff_mirror":{"signature":"theorem Polynomial.coeff_mirror {R : Type u_1} [Semiring R] (p : Polynomial R) (n : \u2115) :p.mirror.coeff n = p.coeff ((revAt (p.natDegree + p.natTrailingDegree)) n)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Mirror.lean#L73-L88"},"Polynomial.mirror_mirror":{"signature":"theorem Polynomial.mirror_mirror {R : Type u_1} [Semiring R] (p : Polynomial R) :p.mirror.mirror = p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Mirror.lean#L112-L114"},"Polynomial.mirror_involutive":{"signature":"theorem Polynomial.mirror_involutive {R : Type u_1} [Semiring R] :Function.Involutive mirror","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Mirror.lean#L118-L119"},"Polynomial.mirror_eq_iff":{"signature":"theorem Polynomial.mirror_eq_iff {R : Type u_1} [Semiring R] {p q : Polynomial R} :p.mirror = q \u2194 p = q.mirror","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Mirror.lean#L121-L122"},"Polynomial.mapAlgHom":{"signature":"def Polynomial.mapAlgHom {R : Type u} {A : Type z} {B : Type u_2} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A \u2192\u2090[R] B) :Polynomial A \u2192\u2090[R] Polynomial B","doc":"Polynomial.map as an AlgHom for noncommutative algebras. This is the algebra version of Polynomial.mapRingHom.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L164-L169"},"Polynomial.mapAlgHom_comp":{"signature":"theorem Polynomial.mapAlgHom_comp {R : Type u} {A : Type z} {B : Type u_2} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (C : Type u_3) [Semiring C] [Algebra R C] (f : B \u2192\u2090[R] C) (g : A \u2192\u2090[R] B) :(mapAlgHom f).comp (mapAlgHom g) = mapAlgHom (f.comp g)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L184-L187"},"Polynomial.mapAlgHom_id":{"signature":"theorem Polynomial.mapAlgHom_id {R : Type u} {A : Type z} [CommSemiring R] [Semiring A] [Algebra R A] :mapAlgHom (AlgHom.id R A) = AlgHom.id R (Polynomial A)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L175-L177"},"Polynomial.mapAlgEquiv":{"signature":"def Polynomial.mapAlgEquiv {R : Type u} {A : Type z} {B : Type u_2} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A \u2243\u2090[R] B) :Polynomial A \u2243\u2090[R] Polynomial B","doc":"If A and B are isomorphic as R-algebras, then so are their polynomial rings","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L205-L207"},"Polynomial.rootSet_mapsTo":{"signature":"theorem Polynomial.rootSet_mapsTo {T : Type w} [CommRing T] {p : Polynomial T} [IsDomain T] {S : Type u_1} {S' : Type u_2} [CommRing S] [IsDomain S] [Algebra T S] [CommRing S'] [IsDomain S'] [Algebra T S'] [Module.IsTorsionFree T S'] (f : S \u2192\u2090[T] S') :Set.MapsTo (\u21d1f) (p.rootSet S) (p.rootSet S')","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L626-L632"},"Polynomial.card_support_eq'":{"signature":"theorem Polynomial.card_support_eq' {R : Type u_1} [Semiring R] {n : \u2115} (k : Fin n \u2192 \u2115) (x : Fin n \u2192 R) (hk : Function.Injective k) (hx : \u2200 (i : Fin n), x i \u2260 0) :(\u2211 i : Fin n, C (x i) * X ^ k i).support.card = n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/EraseLead.lean#L388-L400"},"Polynomial.card_support_eq":{"signature":"theorem Polynomial.card_support_eq {R : Type u_1} [Semiring R] {f : Polynomial R} {n : \u2115} :f.support.card = n \u2194 \u2203 (k : Fin n \u2192 \u2115) (x : Fin n \u2192 R) (_ : StrictMono k) (_ : \u2200 (i : Fin n), x i \u2260 0), f = \u2211 i : Fin n, C (x i) * X ^ k i","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/EraseLead.lean#L402-L446"},"Polynomial.X_pow_sub_one_splits":{"signature":"theorem Polynomial.X_pow_sub_one_splits {K : Type u_1} [Field K] {\u03b6 : K} {n : \u2115} (h : IsPrimitiveRoot \u03b6 n) :(X ^ n - C 1).Splits","doc":"If there is a primitive n-th root of unity in K, then X ^ n - 1 splits.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean#L143-L146"},"Polynomial.aeval_eq_prod_aroots_sub_of_splits":{"signature":"theorem Polynomial.aeval_eq_prod_aroots_sub_of_splits {R : Type u_1} [CommRing R] {f : Polynomial R} {A : Type u_2} [CommRing A] [IsDomain A] [Algebra R A] [IsSimpleRing R] (hf : (map (algebraMap R A) f).Splits) (x : A) :(aeval x) f = (algebraMap R A) f.leadingCoeff * (Multiset.map (fun (x_1 : A) => x - x_1) (f.aroots A)).prod","doc":"Alias of Polynomial.Splits.aeval_eq_prod_aroots.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L805-L806"},"Polynomial.exists_monic_irreducible_factor":{"signature":"theorem Polynomial.exists_monic_irreducible_factor {F : Type u_1} [Field F] (f : Polynomial F) (hu : \u00acIsUnit f) :\u2203 (g : Polynomial F), g.Monic \u2227 Irreducible g \u2227 g \u2223 f","doc":"A polynomial over a field which is not a unit must have a monic irreducible factor. See also WfDvdMonoid.exists_irreducible_factor.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/UniqueFactorization.lean#L149-L158"},"Polynomial.aroots_pow":{"signature":"theorem Polynomial.aroots_pow {S : Type v} {T : Type w} [CommRing T] [CommRing S] [IsDomain S] [Algebra T S] (p : Polynomial T) (n : \u2115) :(p ^ n).aroots S = n \u2022 p.aroots S","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L505-L508"},"Polynomial.natSepDegree_pow":{"signature":"theorem Polynomial.natSepDegree_pow {F : Type u} [Field F] (f : Polynomial F) {n : \u2115} :(f ^ n).natSepDegree = if n = 0 then 0 else f.natSepDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/SeparableDegree.lean#L400-L406"},"Polynomial.natSepDegree_pow_of_ne_zero":{"signature":"theorem Polynomial.natSepDegree_pow_of_ne_zero {F : Type u} [Field F] (f : Polynomial F) {n : \u2115} (hn : n \u2260 0) :(f ^ n).natSepDegree = f.natSepDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/SeparableDegree.lean#L408-L409"},"Polynomial.aroots_mul":{"signature":"theorem Polynomial.aroots_mul {S : Type v} {T : Type w} [CommRing T] [IsDomain T] [CommRing S] [IsDomain S] [Algebra T S] [Module.IsTorsionFree T S] {p q : Polynomial T} (hpq : p * q \u2260 0) :(p * q).aroots S = p.aroots S + q.aroots S","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L455-L461"},"Polynomial.natSepDegree_mul_eq_iff":{"signature":"theorem Polynomial.natSepDegree_mul_eq_iff {F : Type u} [Field F] (f g : Polynomial F) :(f * g).natSepDegree = f.natSepDegree + g.natSepDegree \u2194 f = 0 \u2227 g = 0 \u2228 IsCoprime f g","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/SeparableDegree.lean#L430-L461"},"Polynomial.natSepDegree_mul_of_isCoprime":{"signature":"theorem Polynomial.natSepDegree_mul_of_isCoprime {F : Type u} [Field F] (f g : Polynomial F) (hc : IsCoprime f g) :(f * g).natSepDegree = f.natSepDegree + g.natSepDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/SeparableDegree.lean#L463-L465"},"Polynomial.derivative_neg":{"signature":"theorem Polynomial.derivative_neg {R : Type u} [Ring R] (f : Polynomial R) :derivative (-f) = -derivative f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L608-L610"},"Polynomial.natDegree_C_mul_le":{"signature":"theorem Polynomial.natDegree_C_mul_le {R : Type u} [Semiring R] (a : R) (f : Polynomial R) :(C a * f).natDegree \u2264 f.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L96-L97"},"Polynomial.lifts_iff_set_range":{"signature":"theorem Polynomial.lifts_iff_set_range {R : Type u} [Semiring R] {S : Type v} [Semiring S] {f : R \u2192+* S} (p : Polynomial S) :p \u2208 lifts f \u2194 p \u2208 Set.range (map f)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Lifts.lean#L66-L67"},"Polynomial.support_binomial'":{"signature":"theorem Polynomial.support_binomial' {R : Type u} [Semiring R] (k m : \u2115) (x y : R) :(C x * X ^ k + C y * X ^ m).support \u2286 {k, m}","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L769-L775"},"Polynomial.support_binomial":{"signature":"theorem Polynomial.support_binomial {R : Type u} [Semiring R] {k m : \u2115} (hkm : k \u2260 m) {x y : R} (hx : x \u2260 0) (hy : y \u2260 0) :(C x * X ^ k + C y * X ^ m).support = {k, m}","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L195-L200"},"Polynomial.card_support_binomial":{"signature":"theorem Polynomial.card_support_binomial {R : Type u} [Semiring R] {k m : \u2115} (h : k \u2260 m) {x y : R} (hx : x \u2260 0) (hy : y \u2260 0) :(C x * X ^ k + C y * X ^ m).support.card = 2","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L211-L213"},"Polynomial.card_support_eq_two":{"signature":"theorem Polynomial.card_support_eq_two {R : Type u_1} [Semiring R] {f : Polynomial R} :f.support.card = 2 \u2194 \u2203 (k : \u2115) (m : \u2115) (_ : k < m) (x : R) (y : R) (_ : x \u2260 0) (_ : y \u2260 0), f = C x * X ^ k + C y * X ^ m","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/EraseLead.lean#L456-L466"},"Polynomial.hermite":{"signature":"noncomputable def Polynomial.hermite :\u2115 \u2192 Polynomial \u2124","doc":"the probabilists' Hermite polynomials.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Hermite/Basic.lean#L47-L50"},"Polynomial.eval_eq_sum_range'":{"signature":"theorem Polynomial.eval_eq_sum_range' {R : Type u} [Semiring R] {p : Polynomial R} {n : \u2115} (hn : p.natDegree < n) (x : R) :eval x p = \u2211 i \u2208 Finset.range n, p.coeff i * x ^ i","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Degree.lean#L67-L69"},"Polynomial.eq_zero_of_forall_eval_zero_of_natDegree_lt_card":{"signature":"theorem Polynomial.eq_zero_of_forall_eval_zero_of_natDegree_lt_card {R : Type u} [CommRing R] [IsDomain R] (f : Polynomial R) (hf : \u2200 (r : R), eval r f = 0) (hfR : \u2191f.natDegree < Cardinal.mk R) :f = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L731-L737"},"Polynomial.exists_eval_ne_zero_of_natDegree_lt_card":{"signature":"theorem Polynomial.exists_eval_ne_zero_of_natDegree_lt_card {R : Type u} [CommRing R] [IsDomain R] (f : Polynomial R) (hf : f \u2260 0) (hfR : \u2191f.natDegree < Cardinal.mk R) :\u2203 (r : R), eval r f \u2260 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L740-L743"},"Polynomial.mahlerMeasure_le_sum_norm_coeff":{"signature":"theorem Polynomial.mahlerMeasure_le_sum_norm_coeff (p : Polynomial \u2102) :p.mahlerMeasure \u2264 p.sum fun (x : \u2115) (a : \u2102) => \u2016a\u2016","doc":"The Mahler measure of a polynomial is bounded above by the sum of the norms of its coefficients.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/MahlerMeasure.lean#L249-L276"},"Polynomial.comp_X":{"signature":"theorem Polynomial.comp_X {R : Type u} [Semiring R] {p : Polynomial R} :p.comp X = p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L374-L377"},"Polynomial.algEquivOfCompEqX":{"signature":"def Polynomial.algEquivOfCompEqX {R : Type u} [CommSemiring R] (p q : Polynomial R) (hpq : p.comp q = X) (hqp : q.comp p = X) :Polynomial R \u2243\u2090[R] Polynomial R","doc":"Two polynomials p and q such that p(q(X))=X and q(p(X))=X induces an automorphism of the polynomial algebra.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L322-L327"},"Polynomial.neg_comp":{"signature":"theorem Polynomial.neg_comp {R : Type u} [Ring R] {p q : Polynomial R} :(-p).comp q = -p.comp q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L758-L760"},"Polynomial.algEquivAevalNegX":{"signature":"def Polynomial.algEquivAevalNegX {R : Type u_3} [CommRing R] :Polynomial R \u2243\u2090[R] Polynomial R","doc":"The involutive automorphism of the polynomial algebra given by p(X) \u21a6 p(-X).","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L369-L372"},"Polynomial.algEquivAevalNegX_symm_apply":{"signature":"theorem Polynomial.algEquivAevalNegX_symm_apply {R : Type u_3} [CommRing R] (a : Polynomial R) :algEquivAevalNegX.symm a = (aeval (-X)) a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L370-L370"},"Polynomial.X_pow_mul_assoc":{"signature":"theorem Polynomial.X_pow_mul_assoc {R : Type u} [Semiring R] {p q : Polynomial R} {n : \u2115} :p * X ^ n * q = p * q * X ^ n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L528-L529"},"Polynomial.X_pow_mul_assoc_C":{"signature":"theorem Polynomial.X_pow_mul_assoc_C {R : Type u} [Semiring R] {p : Polynomial R} {n : \u2115} (r : R) :p * X ^ n * C r = p * C r * X ^ n","doc":"Prefer putting constants to the left of X ^ n. This lemma is the loop-avoiding simp version of X_pow_mul_assoc.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L531-L536"},"Polynomial.scaleRoots_ne_zero":{"signature":"theorem Polynomial.scaleRoots_ne_zero {R : Type u_1} [Semiring R] {p : Polynomial R} (hp : p \u2260 0) (s : R) :p.scaleRoots s \u2260 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ScaleRoots.lean#L46-L52"},"Polynomial.scaleRoots_aeval_eq_zero":{"signature":"theorem Polynomial.scaleRoots_aeval_eq_zero {R : Type u_1} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] {p : Polynomial R} {a : A} {r : R} (ha : (aeval a) p = 0) :(aeval ((algebraMap R A) r * a)) (p.scaleRoots r) = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ScaleRoots.lean#L161-L165"},"Polynomial.eq_zero_of_degree_lt_of_eval_index_eq_zero":{"signature":"theorem Polynomial.eq_zero_of_degree_lt_of_eval_index_eq_zero {R : Type u_1} [CommRing R] [IsDomain R] {f : Polynomial R} {\u03b9 : Type u_2} {v : \u03b9 \u2192 R} (s : Finset \u03b9) (hvs : Set.InjOn v \u2191s) (degree_f_lt : f.degree < \u2191s.card) (eval_f : \u2200 i \u2208 s, eval (v i) f = 0) :f = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/LinearAlgebra/Lagrange.lean#L94-L101"},"Polynomial.eq_of_degree_sub_lt_of_eval_index_eq":{"signature":"theorem Polynomial.eq_of_degree_sub_lt_of_eval_index_eq {R : Type u_1} [CommRing R] [IsDomain R] {f g : Polynomial R} {\u03b9 : Type u_2} {v : \u03b9 \u2192 R} (s : Finset \u03b9) (hvs : Set.InjOn v \u2191s) (degree_fg_lt : (f - g).degree < \u2191s.card) (eval_fg : \u2200 i \u2208 s, eval (v i) f = eval (v i) g) :f = g","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/LinearAlgebra/Lagrange.lean#L103-L109"},"Polynomial.coeff_pow_of_natDegree_le":{"signature":"theorem Polynomial.coeff_pow_of_natDegree_le {R : Type u} {m n : \u2115} [Semiring R] {p : Polynomial R} (pn : p.natDegree \u2264 n) :(p ^ m).coeff (m * n) = p.coeff n ^ m","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L154-L161"},"Polynomial.separable_C_mul_X_pow_add_C_mul_X_add_C":{"signature":"theorem Polynomial.separable_C_mul_X_pow_add_C_mul_X_add_C {R : Type u} [CommRing R] {n : \u2115} (a b c : R) (hn : \u2191n = 0) (hb : IsUnit b) :(C a * X ^ n + C b * X + C c).Separable","doc":"If n = 0 in R and b is a unit, then a * X ^ n + b * X + c is separable.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Separable.lean#L265-L275"},"Polynomial.C_mul_comp":{"signature":"theorem Polynomial.C_mul_comp {R : Type u} {a : R} [Semiring R] {p r : Polynomial R} :(C a * p).comp r = C a * p.comp r","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L435-L439"},"Polynomial.natCast_mul_comp":{"signature":"theorem Polynomial.natCast_mul_comp {R : Type u} [Semiring R] {p r : Polynomial R} {n : \u2115} :(\u2191n * p).comp r = \u2191n * p.comp r","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L441-L443"},"Polynomial.aeval_X_left_apply":{"signature":"theorem Polynomial.aeval_X_left_apply {R : Type u} [CommSemiring R] (p : Polynomial R) :(aeval X) p = p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L386-L387"},"Polynomial.derivative_X_pow_succ":{"signature":"theorem Polynomial.derivative_X_pow_succ {R : Type u} [Semiring R] (n : \u2115) :derivative (X ^ (n + 1)) = C (\u2191n + 1) * X ^ n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L102-L105"},"Polynomial.iterate_derivative_zero":{"signature":"theorem Polynomial.iterate_derivative_zero {R : Type u} [Semiring R] {k : \u2115} :(\u21d1derivative)^[k] 0 = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L76-L77"},"Polynomial.iterate_derivative_eq_zero":{"signature":"theorem Polynomial.iterate_derivative_eq_zero {R : Type u} [Semiring R] {p : Polynomial R} {x : \u2115} (hx : p.natDegree < x) :(\u21d1derivative)^[x] p = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L203-L212"},"Polynomial.iterate_derivative_smul":{"signature":"theorem Polynomial.iterate_derivative_smul {R : Type u} [Semiring R] {S : Type u_1} [SMulZeroClass S R] [IsScalarTower S R R] (s : S) (p : Polynomial R) (k : \u2115) :(\u21d1derivative)^[k] (s \u2022 p) = s \u2022 (\u21d1derivative)^[k] p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L143-L148"},"Polynomial.derivativeFinsupp":{"signature":"noncomputable def Polynomial.derivativeFinsupp {R : Type u} [Semiring R] :Polynomial R \u2192\u2097[R] \u2115 \u2192\u2080 Polynomial R","doc":"Iterated derivatives as a finite support function.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L394-L402"},"Polynomial.sumIDeriv":{"signature":"noncomputable def Polynomial.sumIDeriv {R : Type u_1} [Semiring R] :Polynomial R \u2192\u2097[R] Polynomial R","doc":"Sum of iterated derivatives of a polynomial, as a linear map This definition does not allow different weights for the derivatives. It is likely that it could be extended to allow them, but this was not needed for the initial use case (the integration by parts of the integral $I_i$ in the Lindemann-Weierstrass theorem).","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/SumIteratedDerivative.lean#L53-L63"},"Polynomial.support_derivativeFinsupp_subset_range":{"signature":"theorem Polynomial.support_derivativeFinsupp_subset_range {R : Type u} [Semiring R] {p : Polynomial R} {n : \u2115} (h : p.natDegree < n) :(derivativeFinsupp p).support \u2286 Finset.range n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L407-L411"},"Polynomial.sumIDeriv_apply":{"signature":"theorem Polynomial.sumIDeriv_apply {R : Type u_1} [Semiring R] (p : Polynomial R) :sumIDeriv p = \u2211 i \u2208 Finset.range (p.natDegree + 1), (\u21d1derivative)^[i] p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/SumIteratedDerivative.lean#L65-L68"},"Polynomial.sumIDeriv_eq_self_add":{"signature":"theorem Polynomial.sumIDeriv_eq_self_add {R : Type u_1} [Semiring R] (p : Polynomial R) :sumIDeriv p = p + derivative (sumIDeriv p)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/SumIteratedDerivative.lean#L102-L106"},"Polynomial.annIdeal":{"signature":"noncomputable def Polynomial.annIdeal (R : Type u_1) {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] (a : A) :Ideal (Polynomial R)","doc":"annIdeal R a is the annihilating ideal of all p : R[X] such that p(a) = 0. The informal notation p(a) stands for Polynomial.aeval a p. Again informally, the annihilating ideal of a is { p \u2208 R[X] | p(a) = 0 }. This is an ideal in R[X]. The formal definition uses the kernel of the aeval map.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/LinearAlgebra/AnnihilatingPolynomial.lean#L49-L56"},"Polynomial.annIdealGenerator":{"signature":"noncomputable def Polynomial.annIdealGenerator (\ud835\udd5c : Type u_1) {A : Type u_2} [Field \ud835\udd5c] [Ring A] [Algebra \ud835\udd5c A] (a : A) :Polynomial \ud835\udd5c","doc":"annIdealGenerator \ud835\udd5c a is the monic generator of annIdeal \ud835\udd5c a if one exists, otherwise 0. Since \ud835\udd5c[X] is a principal ideal domain there is a polynomial g such that span \ud835\udd5c {g} = annIdeal a. This picks some generator. We prefer the monic generator of the ideal.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/LinearAlgebra/AnnihilatingPolynomial.lean#L72-L80"},"Polynomial.annIdealGenerator_eq_zero_iff":{"signature":"theorem Polynomial.annIdealGenerator_eq_zero_iff {\ud835\udd5c : Type u_1} {A : Type u_2} [Field \ud835\udd5c] [Ring A] [Algebra \ud835\udd5c A] {a : A} :annIdealGenerator \ud835\udd5c a = 0 \u2194 annIdeal \ud835\udd5c a = \u22a5","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/LinearAlgebra/AnnihilatingPolynomial.lean#L86-L89"},"Polynomial.mem_annIdeal_iff_aeval_eq_zero":{"signature":"theorem Polynomial.mem_annIdeal_iff_aeval_eq_zero {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] {a : A} {p : Polynomial R} :p \u2208 annIdeal R a \u2194 (aeval a) p = 0","doc":"It is useful to refer to ideal membership sometimes and the annihilation condition other times.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/LinearAlgebra/AnnihilatingPolynomial.lean#L58-L61"},"Polynomial.monic_annIdealGenerator":{"signature":"theorem Polynomial.monic_annIdealGenerator (\ud835\udd5c : Type u_1) {A : Type u_2} [Field \ud835\udd5c] [Ring A] [Algebra \ud835\udd5c A] (a : A) (hg : annIdealGenerator \ud835\udd5c a \u2260 0) :(annIdealGenerator \ud835\udd5c a).Monic","doc":"The generator we chose for the annihilating ideal is monic when the ideal is non-zero.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/LinearAlgebra/AnnihilatingPolynomial.lean#L114-L117"},"Polynomial.annIdealGenerator_mem":{"signature":"theorem Polynomial.annIdealGenerator_mem (\ud835\udd5c : Type u_1) {A : Type u_2} [Field \ud835\udd5c] [Ring A] [Algebra \ud835\udd5c A] (a : A) :annIdealGenerator \ud835\udd5c a \u2208 annIdeal \ud835\udd5c a","doc":"The annihilating ideal generator is a member of the annihilating ideal.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/LinearAlgebra/AnnihilatingPolynomial.lean#L106-L108"},"Polynomial.annIdealGenerator_aeval_eq_zero":{"signature":"theorem Polynomial.annIdealGenerator_aeval_eq_zero (\ud835\udd5c : Type u_1) {A : Type u_2} [Field \ud835\udd5c] [Ring A] [Algebra \ud835\udd5c A] (a : A) :(aeval a) (annIdealGenerator \ud835\udd5c a) = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/LinearAlgebra/AnnihilatingPolynomial.lean#L126-L127"},"Polynomial.span_singleton_annIdealGenerator":{"signature":"theorem Polynomial.span_singleton_annIdealGenerator (\ud835\udd5c : Type u_1) {A : Type u_2} [Field \ud835\udd5c] [Ring A] [Algebra \ud835\udd5c A] (a : A) :Ideal.span {annIdealGenerator \ud835\udd5c a} = annIdeal \ud835\udd5c a","doc":"annIdealGenerator \ud835\udd5c a is indeed a generator.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/LinearAlgebra/AnnihilatingPolynomial.lean#L93-L104"},"Polynomial.mem_iff_annIdealGenerator_dvd":{"signature":"theorem Polynomial.mem_iff_annIdealGenerator_dvd {\ud835\udd5c : Type u_1} {A : Type u_2} [Field \ud835\udd5c] [Ring A] [Algebra \ud835\udd5c A] {p : Polynomial \ud835\udd5c} {a : A} :p \u2208 annIdeal \ud835\udd5c a \u2194 annIdealGenerator \ud835\udd5c a \u2223 p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/LinearAlgebra/AnnihilatingPolynomial.lean#L131-L133"},"Polynomial.degree_annIdealGenerator_le_of_mem":{"signature":"theorem Polynomial.degree_annIdealGenerator_le_of_mem {\ud835\udd5c : Type u_1} {A : Type u_2} [Field \ud835\udd5c] [Ring A] [Algebra \ud835\udd5c A] (a : A) (p : Polynomial \ud835\udd5c) (hp : p \u2208 annIdeal \ud835\udd5c a) (hpn0 : p \u2260 0) :(annIdealGenerator \ud835\udd5c a).degree \u2264 p.degree","doc":"The generator of the annihilating ideal has minimal degree among the non-zero members of the annihilating ideal","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/LinearAlgebra/AnnihilatingPolynomial.lean#L135-L139"},"Polynomial.annIdealGenerator_eq_minpoly":{"signature":"theorem Polynomial.annIdealGenerator_eq_minpoly (\ud835\udd5c : Type u_1) {A : Type u_2} [Field \ud835\udd5c] [Ring A] [Algebra \ud835\udd5c A] (a : A) :annIdealGenerator \ud835\udd5c a = minpoly \ud835\udd5c a","doc":"The generator of the annihilating ideal is the minimal polynomial.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/LinearAlgebra/AnnihilatingPolynomial.lean#L143-L153"},"Polynomial.rootMultiplicity_pos":{"signature":"theorem Polynomial.rootMultiplicity_pos {R : Type u} [CommRing R] {p : Polynomial R} (hp : p \u2260 0) {x : R} :0 < rootMultiplicity x p \u2194 p.IsRoot x","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L639-L641"},"Polynomial.derivative_X_sub_C_pow":{"signature":"theorem Polynomial.derivative_X_sub_C_pow {R : Type u} [CommRing R] (c : R) (m : \u2115) :derivative ((X - C c) ^ m) = C \u2191m * (X - C c) ^ (m - 1)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L664-L666"},"Polynomial.derivative_rootMultiplicity_of_root_of_mem_nonZeroDivisors":{"signature":"theorem Polynomial.derivative_rootMultiplicity_of_root_of_mem_nonZeroDivisors {R : Type u} [CommRing R] {p : Polynomial R} {t : R} (hpt : p.IsRoot t) (hnzd : \u2191(rootMultiplicity t p) \u2208 nonZeroDivisors R) :rootMultiplicity t (derivative p) = rootMultiplicity t p - 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L47-L64"},"Polynomial.derivative_rootMultiplicity_of_root":{"signature":"theorem Polynomial.derivative_rootMultiplicity_of_root {R : Type u} [CommRing R] [NoZeroDivisors R] [CharZero R] {p : Polynomial R} {t : R} (hpt : p.IsRoot t) :rootMultiplicity t (derivative p) = rootMultiplicity t p - 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L154-L159"},"Polynomial.monic_mul_C_of_leadingCoeff_mul_eq_one":{"signature":"theorem Polynomial.monic_mul_C_of_leadingCoeff_mul_eq_one {R : Type u} [Semiring R] {p : Polynomial R} {b : R} (hp : p.leadingCoeff * b = 1) :(p * C b).Monic","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Monic.lean#L69-L73"},"Polynomial.eval\u2082_mul_eq_zero_of_left":{"signature":"theorem Polynomial.eval\u2082_mul_eq_zero_of_left {R : Type u} {S : Type v} [Semiring R] {p : Polynomial R} [CommSemiring S] (f : R \u2192+* S) (x : S) (q : Polynomial R) (hp : eval\u2082 f x p = 0) :eval\u2082 f x (p * q) = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L202-L204"},"Polynomial.mapRingHom_comp_C":{"signature":"theorem Polynomial.mapRingHom_comp_C {R : Type u_1} {S : Type u_2} [Semiring R] [Semiring S] (f : R \u2192+* S) :(mapRingHom f).comp C = C.comp f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L550-L551"},"Polynomial.residueFieldMapCAlgEquiv":{"signature":"noncomputable def Polynomial.residueFieldMapCAlgEquiv {R : Type u_1} [CommRing R] (I : Ideal R) [I.IsPrime] (J : Ideal (Polynomial R)) [J.IsPrime] [J.LiesOver I] (hJ : J = Ideal.map C I) :J.ResidueField \u2243\u2090[I.ResidueField] RatFunc I.ResidueField","doc":"\u03ba(I[X]) \u2243\u2090[\u03ba(I)] \u03ba(I)(X).","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/LocalRing/ResidueField/Polynomial.lean#L29-L75"},"Polynomial.residueFieldMapCAlgEquiv_symm_C":{"signature":"theorem Polynomial.residueFieldMapCAlgEquiv_symm_C {R : Type u_1} [CommRing R] (I : Ideal R) [I.IsPrime] (J : Ideal (Polynomial R)) [J.IsPrime] [J.LiesOver I] (hJ : J = Ideal.map C I) (r : I.ResidueField) :(residueFieldMapCAlgEquiv I J hJ).symm (RatFunc.C r) = (algebraMap I.ResidueField J.ResidueField) r","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/LocalRing/ResidueField/Polynomial.lean#L83-L86"},"Polynomial.natDegree_natCast":{"signature":"theorem Polynomial.natDegree_natCast {R : Type u} [Semiring R] (n : \u2115) :(\u2191n).natDegree = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L178-L180"},"Polynomial.natDegree_sub_le_of_le":{"signature":"theorem Polynomial.natDegree_sub_le_of_le {R : Type u} {n m : \u2115} [Ring R] {p q : Polynomial R} (hp : p.natDegree \u2264 m) (hq : q.natDegree \u2264 n) :(p - q).natDegree \u2264 max m n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L539-L541"},"Polynomial.X_pow_sub_X_sub_one_irreducible_aux":{"signature":"theorem Polynomial.X_pow_sub_X_sub_one_irreducible_aux {n : \u2115} (z : \u2102) :\u00ac(z ^ n = z + 1 \u2227 z ^ n + z ^ 2 = 0)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Selmer.lean#L33-L47"},"Polynomial.eq_of_degrees_lt_of_eval_index_eq":{"signature":"theorem Polynomial.eq_of_degrees_lt_of_eval_index_eq {R : Type u_1} [CommRing R] [IsDomain R] {f g : Polynomial R} {\u03b9 : Type u_2} {v : \u03b9 \u2192 R} (s : Finset \u03b9) (hvs : Set.InjOn v \u2191s) (degree_f_lt : f.degree < \u2191s.card) (degree_g_lt : g.degree < \u2191s.card) (eval_fg : \u2200 i \u2208 s, eval (v i) f = eval (v i) g) :f = g","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/LinearAlgebra/Lagrange.lean#L111-L115"},"Polynomial.rootMultiplicity_sub_one_le_derivative_rootMultiplicity":{"signature":"theorem Polynomial.rootMultiplicity_sub_one_le_derivative_rootMultiplicity {R : Type u} [CommRing R] [NoZeroDivisors R] [CharZero R] (p : Polynomial R) (t : R) :rootMultiplicity t p - 1 \u2264 rootMultiplicity t (derivative p)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L161-L166"},"Polynomial.toFinsupp_algebraMap":{"signature":"theorem Polynomial.toFinsupp_algebraMap {R : Type u} {A : Type z} [CommSemiring R] [Semiring A] [Algebra R A] (r : R) :((algebraMap R (Polynomial A)) r).toFinsupp = (algebraMap R (AddMonoidAlgebra A \u2115)) r","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L72-L76"},"Polynomial.ofFinsupp_algebraMap":{"signature":"theorem Polynomial.ofFinsupp_algebraMap {R : Type u} {A : Type z} [CommSemiring R] [Semiring A] [Algebra R A] (r : R) :{ toFinsupp := (algebraMap R (AddMonoidAlgebra A \u2115)) r } = (algebraMap R (Polynomial A)) r","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L78-L79"},"Polynomial.toMvPolynomial":{"signature":"noncomputable def Polynomial.toMvPolynomial {R : Type u_1} {\u03c3 : Type u_3} [CommSemiring R] (i : \u03c3) :Polynomial R \u2192\u2090[R] MvPolynomial \u03c3 R","doc":"The embedding of R[X] into R[X\u1d62] as an R-algebra homomorphism.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/MvPolynomial/Equiv.lean#L775-L777"},"Polynomial.toMvPolynomial_X":{"signature":"theorem Polynomial.toMvPolynomial_X {R : Type u_1} {\u03c3 : Type u_3} [CommSemiring R] (i : \u03c3) :(toMvPolynomial i) X = MvPolynomial.X i","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/MvPolynomial/Equiv.lean#L783-L785"},"Polynomial.degree_list_prod_le":{"signature":"theorem Polynomial.degree_list_prod_le {S : Type u_1} [Semiring S] (l : List (Polynomial S)) :l.prod.degree \u2264 (List.map degree l).sum","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/BigOperators.lean#L107-L108"},"Polynomial.degree_multiset_prod_le":{"signature":"theorem Polynomial.degree_multiset_prod_le {R : Type u} [CommSemiring R] (t : Multiset (Polynomial R)) :t.prod.degree \u2264 (Multiset.map degree t).sum","doc":"The degree of a product of polynomials is at most the sum of the degrees, where the degree of the zero polynomial is \u22a5.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/BigOperators.lean#L137-L141"},"Polynomial.degree_prod_le":{"signature":"theorem Polynomial.degree_prod_le {R : Type u} {\u03b9 : Type w} (s : Finset \u03b9) [CommSemiring R] (f : \u03b9 \u2192 Polynomial R) :(\u220f i \u2208 s, f i).degree \u2264 \u2211 i \u2208 s, (f i).degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/BigOperators.lean#L143-L144"},"Polynomial.jacobson_bot_of_integral_localization":{"signature":"theorem Polynomial.jacobson_bot_of_integral_localization {S : Type u_2} [CommRing S] [IsDomain S] {R : Type u_5} [CommRing R] [IsDomain R] [IsJacobsonRing R] (R\u2098 : Type u_6) (S\u2098 : Type u_7) [CommRing R\u2098] [CommRing S\u2098] (\u03c6 : R \u2192+* S) (h\u03c6 : Function.Injective \u21d1\u03c6) (x : R) (hx : x \u2260 0) [Algebra R R\u2098] [IsLocalization.Away x R\u2098] [Algebra S S\u2098] [IsLocalization (Submonoid.map \u03c6 (Submonoid.powers x)) S\u2098] (h\u03c6' : (IsLocalization.map S\u2098 \u03c6 \u22ef).IsIntegral) :\u22a5.jacobson = \u22a5","doc":"If f : R \u2192 S descends to an integral map in the localization at x, and R is a Jacobson ring, then the intersection of all maximal ideals in S is trivial","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Jacobson/Ring.lean#L323-L367"},"Polynomial.mem_closure_X_union_C":{"signature":"theorem Polynomial.mem_closure_X_union_C {R : Type u_1} [Ring R] (p : Polynomial R) :p \u2208 Subring.closure (insert X {f : Polynomial R | f.degree \u2264 0})","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Jacobson/Ring.lean#L253-L266"},"Polynomial.isIntegral_isLocalization_polynomial_quotient":{"signature":"theorem Polynomial.isIntegral_isLocalization_polynomial_quotient {R : Type u_1} [CommRing R] {R\u2098 : Type u_3} {S\u2098 : Type u_4} [CommRing R\u2098] [CommRing S\u2098] (P : Ideal (Polynomial R)) (pX : Polynomial R) (hpX : pX \u2208 P) [Algebra (R \u29f8 Ideal.comap C P) R\u2098] [IsLocalization.Away (map (Ideal.Quotient.mk (Ideal.comap C P)) pX).leadingCoeff R\u2098] [Algebra (Polynomial R \u29f8 P) S\u2098] [IsLocalization (Submonoid.map (Ideal.quotientMap P C \u22ef) (Submonoid.powers (map (Ideal.Quotient.mk (Ideal.comap C P)) pX).leadingCoeff)) S\u2098] :(IsLocalization.map S\u2098 (Ideal.quotientMap P C \u22ef) \u22ef).IsIntegral","doc":"If I is a prime ideal of R[X] and pX \u2208 I is a non-constant polynomial, then the map R \u2192+* R[x]/I descends to an integral map when localizing at pX.leadingCoeff. In particular X is integral because it satisfies pX, and constants are trivially integral, so integrality of the entire extension follows by closure under addition and multiplication.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Jacobson/Ring.lean#L271-L321"},"Polynomial.isJacobsonRing_polynomial_of_isJacobsonRing":{"signature":"theorem Polynomial.isJacobsonRing_polynomial_of_isJacobsonRing {R : Type u_1} [CommRing R] (hR : IsJacobsonRing R) :IsJacobsonRing (Polynomial R)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Jacobson/Ring.lean#L391-L417"},"Polynomial.isJacobsonRing_polynomial_iff_isJacobsonRing":{"signature":"theorem Polynomial.isJacobsonRing_polynomial_iff_isJacobsonRing {R : Type u_1} [CommRing R] :IsJacobsonRing (Polynomial R) \u2194 IsJacobsonRing R","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Jacobson/Ring.lean#L419-L423"},"Polynomial.isMaximal_comap_C_of_isMaximal":{"signature":"theorem Polynomial.isMaximal_comap_C_of_isMaximal {R : Type u_1} [CommRing R] (P : Ideal (Polynomial R)) [hP : P.IsMaximal] [IsJacobsonRing R] [Nontrivial R] (hP' : \u2200 (x : R), C x \u2208 P \u2192 x = 0) :(Ideal.comap C P).IsMaximal","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Jacobson/Ring.lean#L435-L476"},"Polynomial.quotient_mk_comp_C_isIntegral_of_isJacobsonRing":{"signature":"theorem Polynomial.quotient_mk_comp_C_isIntegral_of_isJacobsonRing {R : Type u_1} [CommRing R] (P : Ideal (Polynomial R)) [hP : P.IsMaximal] [IsJacobsonRing R] :((Ideal.Quotient.mk P).comp C).IsIntegral","doc":"If R is a Jacobson ring, and P is a maximal ideal of R[X], then R \u2192 R[X]/P is an integral map.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Jacobson/Ring.lean#L517-L541"},"Polynomial.isMaximal_comap_C_of_isJacobsonRing":{"signature":"theorem Polynomial.isMaximal_comap_C_of_isJacobsonRing {R : Type u_1} [CommRing R] (P : Ideal (Polynomial R)) [hP : P.IsMaximal] [IsJacobsonRing R] :(Ideal.comap C P).IsMaximal","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Jacobson/Ring.lean#L543-L547"},"Polynomial.transcendental_X":{"signature":"theorem Polynomial.transcendental_X (R : Type u) [CommRing R] :Transcendental R X","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Algebraic/Basic.lean#L52-L53"},"Polynomial.monomial_one_eq_iff":{"signature":"theorem Polynomial.monomial_one_eq_iff {R : Type u} [Semiring R] [Nontrivial R] {i j : \u2115} :(monomial i) 1 = (monomial j) 1 \u2194 i = j","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Monomial.lean#L27-L30"},"Polynomial.infinite":{"signature":"instance Polynomial.infinite {R : Type u} [Semiring R] [Nontrivial R] :Infinite (Polynomial R)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Monomial.lean#L32-L33"},"Polynomial.map_normalize":{"signature":"theorem Polynomial.map_normalize {R : Type u} {S : Type v} [Field R] (p : Polynomial R) [DecidableEq R] [Field S] [DecidableEq S] (f : R \u2192+* S) :map f (normalize p) = normalize (map f p)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L713-L718"},"Polynomial.toFinsuppIsoLinear":{"signature":"def Polynomial.toFinsuppIsoLinear (R : Type u) [Semiring R] :Polynomial R \u2243\u2097[R] AddMonoidAlgebra R \u2115","doc":"Linear isomorphism between R[X] and R[\u2115]. This is just an implementation detail, but it can be useful to transfer results from Finsupp to polynomials.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L337-L342"},"Polynomial.instFree":{"signature":"instance Polynomial.instFree {R : Type u_1} [Semiring R] :Module.Free R (Polynomial R)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/LinearAlgebra/Finsupp/VectorSpace.lean#L240-L240"},"Polynomial.exists_degree_le_of_mem_span":{"signature":"theorem Polynomial.exists_degree_le_of_mem_span {R : Type u} [Semiring R] {s : Set (Polynomial R)} {p : Polynomial R} (hs : s.Nonempty) (hp : p \u2208 Submodule.span R s) :\u2203 p' \u2208 s, p.degree \u2264 p'.degree","doc":"For every polynomial p in the span of a set s : Set R[X], there exists a polynomial of p' \u2208 s with higher degree. See also Polynomial.exists_degree_le_of_mem_span_of_finite.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Basic.lean#L190-L205"},"Polynomial.exists_degree_le_of_mem_span_of_finite":{"signature":"theorem Polynomial.exists_degree_le_of_mem_span_of_finite {R : Type u} [Semiring R] {s : Set (Polynomial R)} (s_fin : s.Finite) (hs : s.Nonempty) :\u2203 p' \u2208 s, \u2200 p \u2208 Submodule.span R s, p.degree \u2264 p'.degree","doc":"A stronger version of Polynomial.exists_degree_le_of_mem_span under the assumption that the set s : R[X] is finite. There exists a polynomial p' \u2208 s whose degree dominates the degree of every element of p \u2208 span R s.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Basic.lean#L207-L215"},"Polynomial.span_le_degreeLE_of_finite":{"signature":"theorem Polynomial.span_le_degreeLE_of_finite {R : Type u} [Semiring R] {s : Set (Polynomial R)} (s_fin : s.Finite) :\u2203 (n : \u2115), Submodule.span R s \u2264 degreeLE R \u2191n","doc":"The span of every finite set of polynomials is contained in a degreeLE n for some n.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Basic.lean#L217-L225"},"Polynomial.not_finite":{"signature":"theorem Polynomial.not_finite {R : Type u} [Semiring R] [Nontrivial R] :\u00acModule.Finite R (Polynomial R)","doc":"If R is a nontrivial ring, the polynomials R[X] are not finite as an R-module. When R is a field, this is equivalent to R[X] being an infinite-dimensional vector space over R.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Basic.lean#L233-L244"},"Polynomial.leadingCoeff_normalize":{"signature":"theorem Polynomial.leadingCoeff_normalize {R : Type u} [CommRing R] [NoZeroDivisors R] [NormalizationMonoid R] (p : Polynomial R) :(normalize p).leadingCoeff = normalize p.leadingCoeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L229-L231"},"Polynomial.monic_normalize":{"signature":"theorem Polynomial.monic_normalize {R : Type u} [Field R] {p : Polynomial R} [DecidableEq R] (hp0 : p \u2260 0) :(normalize p).Monic","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L547-L550"},"Polynomial.normalize_eq_self_iff_monic":{"signature":"theorem Polynomial.normalize_eq_self_iff_monic {R : Type u} [Field R] [DecidableEq R] {p : Polynomial R} (hp : p \u2260 0) :normalize p = p \u2194 p.Monic","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L552-L554"},"Polynomial.mem_normalizedFactors_iff":{"signature":"theorem Polynomial.mem_normalizedFactors_iff {R : Type u} [Field R] {p q : Polynomial R} [DecidableEq R] (hq : q \u2260 0) :p \u2208 UniqueFactorizationMonoid.normalizedFactors q \u2194 Irreducible p \u2227 p.Monic \u2227 p \u2223 q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L706-L710"},"Polynomial.degree_mul_leadingCoeff_self_inv":{"signature":"theorem Polynomial.degree_mul_leadingCoeff_self_inv {K : Type u_1} [DivisionRing K] (p : Polynomial K) :(p * C p.leadingCoeff\u207b\u00b9).degree = p.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L467-L471"},"Polynomial.natDegree_mul_leadingCoeff_self_inv":{"signature":"theorem Polynomial.natDegree_mul_leadingCoeff_self_inv {K : Type u_1} [DivisionRing K] (p : Polynomial K) :(p * C p.leadingCoeff\u207b\u00b9).natDegree = p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L473-L475"},"Polynomial.natDegree_of_dvd_cyclotomic_of_irreducible":{"signature":"theorem Polynomial.natDegree_of_dvd_cyclotomic_of_irreducible {K : Type u_1} [Field K] [Fintype K] {p f n : \u2115} {P : Polynomial K} (hK : Fintype.card K = p ^ f) (hn : p.Coprime n) [hp : Fact (Nat.Prime p)] (hP : P \u2223 cyclotomic n K) (hPirr : Irreducible P) :P.natDegree = orderOf (ZMod.unitOfCoprime (p ^ f) \u22ef)","doc":"Let K be a finite field of cardinality p ^ f and let P be an irreducible factor of the n-th cyclotomic polynomial over K, where p and n are coprime. Then the degree of P is the multiplicative order of p ^ f modulo n.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Factorization.lean#L84-L99"},"Polynomial.mul_scaleRoots":{"signature":"theorem Polynomial.mul_scaleRoots {R : Type u_1} [CommSemiring R] (p q : Polynomial R) (r : R) :r ^ (p.natDegree + q.natDegree - (p * q).natDegree) \u2022 (p * q).scaleRoots r = p.scaleRoots r * q.scaleRoots r","doc":"Multiplication and scaleRoots commute up to a power of r. The factor disappears if we assume that the product of the leading coeffs does not vanish. See Polynomial.mul_scaleRoots'.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ScaleRoots.lean#L187-L211"},"Polynomial.mul_scaleRoots'":{"signature":"theorem Polynomial.mul_scaleRoots' {R : Type u_1} [CommSemiring R] (p q : Polynomial R) (r : R) (h : p.leadingCoeff * q.leadingCoeff \u2260 0) :(p * q).scaleRoots r = p.scaleRoots r * q.scaleRoots r","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ScaleRoots.lean#L213-L215"},"Polynomial.mul_scaleRoots_of_noZeroDivisors":{"signature":"theorem Polynomial.mul_scaleRoots_of_noZeroDivisors {R : Type u_1} [CommSemiring R] (p q : Polynomial R) (r : R) [NoZeroDivisors R] :(p * q).scaleRoots r = p.scaleRoots r * q.scaleRoots r","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ScaleRoots.lean#L217-L222"},"Polynomial.scaleRoots_dvd":{"signature":"theorem Polynomial.scaleRoots_dvd {R : Type u_1} [CommSemiring R] (p q : Polynomial R) {r : R} [NoZeroDivisors R] (hpq : p \u2223 q) :p.scaleRoots r \u2223 q.scaleRoots r","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ScaleRoots.lean#L263-L267"},"Polynomial.prime_C_iff":{"signature":"theorem Polynomial.prime_C_iff {R : Type u} [CommRing R] {r : R} :Prime (C r) \u2194 Prime r","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Basic.lean#L741-L748"},"Polynomial.integralNormalization_degree":{"signature":"theorem Polynomial.integralNormalization_degree {R : Type u} [Semiring R] {p : Polynomial R} :p.integralNormalization.degree = p.degree","doc":"Alias of Polynomial.degree_integralNormalization.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/IntegralNormalization.lean#L128-L129"},"Polynomial.natDegree_add_le_of_degree_le":{"signature":"theorem Polynomial.natDegree_add_le_of_degree_le {R : Type u} [Semiring R] {p q : Polynomial R} {n : \u2115} (hp : p.natDegree \u2264 n) (hq : q.natDegree \u2264 n) :(p + q).natDegree \u2264 n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L344-L346"},"Polynomial.coprime_of_root_cyclotomic":{"signature":"theorem Polynomial.coprime_of_root_cyclotomic {n : \u2115} (hpos : 0 < n) {p : \u2115} [hprime : Fact (Nat.Prime p)] {a : \u2115} (hroot : (cyclotomic n (ZMod p)).IsRoot ((Nat.castRingHom (ZMod p)) a)) :a.Coprime p","doc":"If (a : \u2115) is a root of cyclotomic n (ZMod p), where p is a prime, then a and p are coprime.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean#L563-L577"},"Polynomial.IsDistinguishedAt":{"signature":"structure Polynomial.IsDistinguishedAt {R : Type u_1} [CommRing R] (f : Polynomial R) (I : Ideal R) extends f.IsWeaklyEisensteinAt I :Prop","doc":"Given an ideal I of a commutative ring R, we say that a polynomial f : R[X] is Distinguished at I if f is monic and IsWeaklyEisensteinAt I. i.e. f is of the form x\u207f + a\u2081x\u207f\u207b\u00b9 + \u22ef + a\u2099 with a\u1d62 \u2208 I for all i.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Eisenstein/Distinguished.lean#L26-L33"},"Polynomial.coe_sub":{"signature":"theorem Polynomial.coe_sub {R : Type u_1} [CommRing R] (p q : Polynomial R) :\u2191(p - q) = \u2191p - \u2191q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/PowerSeries/Basic.lean#L951-L953"},"Polynomial.polynomial_map_coe":{"signature":"theorem Polynomial.polynomial_map_coe {U : Type u_2} {V : Type u_3} [CommSemiring U] [CommSemiring V] {\u03c6 : U \u2192+* V} {f : Polynomial U} :\u2191(map \u03c6 f) = (PowerSeries.map \u03c6) \u2191f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/PowerSeries/Basic.lean#L850-L854"},"Polynomial.coe_pow":{"signature":"theorem Polynomial.coe_pow {R : Type u_1} [Semiring R] {\u03c6 : Polynomial R} (n : \u2115) :\u2191(\u03c6 ^ n) = \u2191\u03c6 ^ n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/PowerSeries/Basic.lean#L892-L894"},"Polynomial.algebra":{"signature":"def Polynomial.algebra (R : Type u_1) (A : Type u_2) [CommSemiring R] [Semiring A] [Algebra R A] :Algebra (Polynomial R) (Polynomial A)","doc":"If A is an R-algebra, then A[X] is an R[X] algebra. This gives a diamond for Algebra R[X] R[X][X], so this is not a global instance.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/PolynomialAlgebra.lean#L203-L207"},"Polynomial.aevalAeval":{"signature":"abbrev Polynomial.aevalAeval {R : Type u_1} {A : Type u_2} [CommSemiring R] [CommSemiring A] [Algebra R A] (x y : A) :Polynomial (Polynomial R) \u2192\u2090[R] A","doc":"aevalAeval x y is the R-algebra evaluation morphism sending a two-variable polynomial p : R[X][Y] to p(x,y).","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Bivariate.lean#L212-L216"},"Polynomial.taylor":{"signature":"def Polynomial.taylor {R : Type u_1} [Semiring R] (r : R) :Polynomial R \u2192\u2097[R] Polynomial R","doc":"The Taylor expansion of a polynomial f at r.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Taylor.lean#L37-L41"},"Polynomial.taylor_eval":{"signature":"theorem Polynomial.taylor_eval {R : Type u_1} [CommSemiring R] (r : R) (f : Polynomial R) (s : R) :eval s ((taylor r) f) = eval (s + r) f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Taylor.lean#L160-L161"},"Polynomial.mono_map_natDegree_eq":{"signature":"theorem Polynomial.mono_map_natDegree_eq {R : Type u_1} [Semiring R] {S : Type u_2} {F : Type u_3} [Semiring S] [FunLike F (Polynomial R) (Polynomial S)] [AddMonoidHomClass F (Polynomial R) (Polynomial S)] {\u03c6 : F} {p : Polynomial R} (k : \u2115) (fu : \u2115 \u2192 \u2115) (fu0 : \u2200 {n : \u2115}, n \u2264 k \u2192 fu n = 0) (fc : \u2200 {n m : \u2115}, k \u2264 n \u2192 n < m \u2192 fu n < fu m) (\u03c6_k : \u2200 {f : Polynomial R}, f.natDegree < k \u2192 \u03c6 f = 0) (\u03c6_mon_nat : \u2200 (n : \u2115) (c : R), c \u2260 0 \u2192 (\u03c6 ((monomial n) c)).natDegree = fu n) :(\u03c6 p).natDegree = fu p.natDegree","doc":"Let \u03c6 : R[x] \u2192 S[x] be an additive map, k : \u2115 a bound, and fu : \u2115 \u2192 \u2115 a \"sufficiently monotone\" map. Assume also that","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/EraseLead.lean#L347-L370"},"Polynomial.map_natDegree_eq_sub":{"signature":"theorem Polynomial.map_natDegree_eq_sub {R : Type u_1} [Semiring R] {S : Type u_2} {F : Type u_3} [Semiring S] [FunLike F (Polynomial R) (Polynomial S)] [AddMonoidHomClass F (Polynomial R) (Polynomial S)] {\u03c6 : F} {p : Polynomial R} {k : \u2115} (\u03c6_k : \u2200 (f : Polynomial R), f.natDegree < k \u2192 \u03c6 f = 0) (\u03c6_mon : \u2200 (n : \u2115) (c : R), c \u2260 0 \u2192 (\u03c6 ((monomial n) c)).natDegree = n - k) :(\u03c6 p).natDegree = p.natDegree - k","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/EraseLead.lean#L372-L379"},"Polynomial.map_natDegree_eq_natDegree":{"signature":"theorem Polynomial.map_natDegree_eq_natDegree {R : Type u_1} [Semiring R] {S : Type u_2} {F : Type u_3} [Semiring S] [FunLike F (Polynomial R) (Polynomial S)] [AddMonoidHomClass F (Polynomial R) (Polynomial S)] {\u03c6 : F} (p : Polynomial R) (\u03c6_mon_nat : \u2200 (n : \u2115) (c : R), c \u2260 0 \u2192 (\u03c6 ((monomial n) c)).natDegree = n) :(\u03c6 p).natDegree = p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/EraseLead.lean#L381-L386"},"Polynomial.taylor_monomial":{"signature":"theorem Polynomial.taylor_monomial {R : Type u_1} [Semiring R] (r : R) (i : \u2115) (k : R) :(taylor r) ((monomial i) k) = C k * (X + C r) ^ i","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Taylor.lean#L63-L65"},"Polynomial.eq_natDegree_of_le_mem_support":{"signature":"theorem Polynomial.eq_natDegree_of_le_mem_support {R : Type u} {n : \u2115} [Semiring R] {p : Polynomial R} (pn : p.natDegree \u2264 n) (ns : n \u2208 p.support) :p.natDegree = n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L102-L104"},"Polynomial.natDegree_C_mul_of_mul_ne_zero":{"signature":"theorem Polynomial.natDegree_C_mul_of_mul_ne_zero {R : Type u} {a : R} [Semiring R] {p : Polynomial R} (h : a * p.leadingCoeff \u2260 0) :(C a * p).natDegree = p.natDegree","doc":"Although not explicitly stated, the assumptions of lemma natDegree_C_mul_of_mul_ne_zero force the polynomial p to be non-zero, via p.leadingCoeff \u2260 0.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L131-L138"},"Polynomial.leadingCoeff_pow_X_add_C":{"signature":"theorem Polynomial.leadingCoeff_pow_X_add_C {R : Type u} [Semiring R] (r : R) (i : \u2115) :((X + C r) ^ i).leadingCoeff = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L666-L669"},"Polynomial.natDegree_pow_X_add_C":{"signature":"theorem Polynomial.natDegree_pow_X_add_C {R : Type u} [Semiring R] [Nontrivial R] (n : \u2115) (r : R) :((X + C r) ^ n).natDegree = n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Monic.lean#L210-L212"},"Polynomial.natDegree_taylor":{"signature":"theorem Polynomial.natDegree_taylor {R : Type u_1} [Semiring R] (p : Polynomial R) (r : R) :((taylor r) p).natDegree = p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Taylor.lean#L92-L97"},"Polynomial.lsum":{"signature":"def Polynomial.lsum {R : Type u_1} {A : Type u_2} {M : Type u_3} [Semiring R] [Semiring A] [AddCommMonoid M] [Module R A] [Module R M] (f : \u2115 \u2192 A \u2192\u2097[R] M) :Polynomial A \u2192\u2097[R] M","doc":"Polynomial.sum as a linear map.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L67-L75"},"Polynomial.hasseDeriv":{"signature":"def Polynomial.hasseDeriv {R : Type u_1} [Semiring R] (k : \u2115) :Polynomial R \u2192\u2097[R] Polynomial R","doc":"The kth Hasse derivative of a polynomial \u2211 a_i X^i is \u2211 (i.choose k) a_i X^(i-k). It satisfies k! * (hasse_deriv k f) = derivative^[k] f.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/HasseDeriv.lean#L57-L60"},"Polynomial.leval":{"signature":"def Polynomial.leval {R : Type u_1} [Semiring R] (r : R) :Polynomial R \u2192\u2097[R] R","doc":"Polynomial.eval as linear map","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/SMul.lean#L62-L67"},"Polynomial.addSubmonoid_closure_setOf_eq_monomial":{"signature":"theorem Polynomial.addSubmonoid_closure_setOf_eq_monomial {R : Type u} [Semiring R] :AddSubmonoid.closure {p : Polynomial R | \u2203 (n : \u2115) (a : R), p = (monomial n) a} = \u22a4","doc":"Monomials generate the additive monoid of polynomials.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L709-L717"},"Polynomial.addHom_ext":{"signature":"theorem Polynomial.addHom_ext {R : Type u} [Semiring R] {M : Type u_1} [AddZeroClass M] {f g : Polynomial R \u2192+ M} (h : \u2200 (n : \u2115) (a : R), f ((monomial n) a) = g ((monomial n) a)) :f = g","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L719-L723"},"Polynomial.lhom_ext'":{"signature":"theorem Polynomial.lhom_ext' {R : Type u} [Semiring R] {M : Type u_1} [AddCommMonoid M] [Module R M] {f g : Polynomial R \u2192\u2097[R] M} (h : \u2200 (n : \u2115), f \u2218\u2097 monomial n = g \u2218\u2097 monomial n) :f = g","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L730-L733"},"Polynomial.hasseDeriv_apply":{"signature":"theorem Polynomial.hasseDeriv_apply {R : Type u_1} [Semiring R] (k : \u2115) (f : Polynomial R) :(hasseDeriv k) f = f.sum fun (i : \u2115) (r : R) => (monomial (i - k)) (\u2191(i.choose k) * r)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/HasseDeriv.lean#L62-L65"},"Polynomial.hasseDeriv_coeff":{"signature":"theorem Polynomial.hasseDeriv_coeff {R : Type u_1} [Semiring R] (k : \u2115) (f : Polynomial R) (n : \u2115) :((hasseDeriv k) f).coeff n = \u2191((n + k).choose k) * f.coeff (n + k)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/HasseDeriv.lean#L67-L80"},"Polynomial.hasseDeriv_monomial":{"signature":"theorem Polynomial.hasseDeriv_monomial {R : Type u_1} [Semiring R] (k n : \u2115) (r : R) :(hasseDeriv k) ((monomial n) r) = (monomial (n - k)) (\u2191(n.choose k) * r)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/HasseDeriv.lean#L104-L115"},"Polynomial.taylor_coeff":{"signature":"theorem Polynomial.taylor_coeff {R : Type u_1} [Semiring R] (r : R) (f : Polynomial R) (n : \u2115) :((taylor r) f).coeff n = eval r ((hasseDeriv n) f)","doc":"The kth coefficient of Polynomial.taylor r f is (Polynomial.hasseDeriv k f).eval r.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Taylor.lean#L67-L76"},"Polynomial.hasseDeriv_one'":{"signature":"theorem Polynomial.hasseDeriv_one' {R : Type u_1} [Semiring R] (f : Polynomial R) :(hasseDeriv 1) f = derivative f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/HasseDeriv.lean#L96-L98"},"Polynomial.hasseDeriv_one":{"signature":"theorem Polynomial.hasseDeriv_one {R : Type u_1} [Semiring R] :hasseDeriv 1 = derivative","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/HasseDeriv.lean#L100-L102"},"Polynomial.taylor_coeff_one":{"signature":"theorem Polynomial.taylor_coeff_one {R : Type u_1} [Semiring R] (r : R) (f : Polynomial R) :((taylor r) f).coeff 1 = eval r (derivative f)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Taylor.lean#L82-L84"},"Polynomial.hasseDeriv_zero'":{"signature":"theorem Polynomial.hasseDeriv_zero' {R : Type u_1} [Semiring R] (f : Polynomial R) :(hasseDeriv 0) f = f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/HasseDeriv.lean#L82-L84"},"Polynomial.hasseDeriv_zero":{"signature":"theorem Polynomial.hasseDeriv_zero {R : Type u_1} [Semiring R] :hasseDeriv 0 = LinearMap.id","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/HasseDeriv.lean#L86-L88"},"Polynomial.taylor_coeff_zero":{"signature":"theorem Polynomial.taylor_coeff_zero {R : Type u_1} [Semiring R] (r : R) (f : Polynomial R) :((taylor r) f).coeff 0 = eval r f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Taylor.lean#L78-L80"},"Polynomial.eval_add_of_sq_eq_zero":{"signature":"theorem Polynomial.eval_add_of_sq_eq_zero {R : Type u_1} [CommSemiring R] (p : Polynomial R) (x y : R) (hy : y ^ 2 = 0) :eval (x + y) p = eval x p + eval x (derivative p) * y","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Taylor.lean#L163-L168"},"Polynomial.aeval_add_of_sq_eq_zero":{"signature":"theorem Polynomial.aeval_add_of_sq_eq_zero {R : Type u_1} [CommSemiring R] {S : Type u_2} [CommRing S] [Algebra R S] (p : Polynomial R) (x y : S) (hy : y ^ 2 = 0) :(aeval (x + y)) p = (aeval x) p + (aeval x) (derivative p) * y","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Taylor.lean#L170-L173"},"Polynomial.eq_X_sub_C_of_splits_of_single_root":{"signature":"theorem Polynomial.eq_X_sub_C_of_splits_of_single_root {R : Type u_1} [CommRing R] {f : Polynomial R} [IsDomain R] (hf : f.Splits) {x : R} (hr : f.roots = {x}) :f = C f.leadingCoeff * (X - C x)","doc":"Alias of Polynomial.Splits.eq_X_sub_C_of_single_root.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L820-L821"},"Polynomial.algEquivCMulXAddC":{"signature":"def Polynomial.algEquivCMulXAddC {R : Type u_3} [CommRing R] (a b : R) [Invertible a] :Polynomial R \u2243\u2090[R] Polynomial R","doc":"The automorphism of the polynomial algebra given by p(X) \u21a6 p(a * X + b), with inverse p(X) \u21a6 p(a\u207b\u00b9 * (X - b)).","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L339-L344"},"Polynomial.dvd_comp_C_mul_X_add_C_iff":{"signature":"theorem Polynomial.dvd_comp_C_mul_X_add_C_iff {R : Type u} [CommRing R] (p q : Polynomial R) (a b : R) [Invertible a] :p \u2223 q.comp (C a * X + C b) \u2194 p.comp (C \u215fa * (X - C b)) \u2223 q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L662-L665"},"Polynomial.IsUnitTrinomial":{"signature":"def Polynomial.IsUnitTrinomial (p : Polynomial \u2124) :Prop","doc":"A unit trinomial is a trinomial with unit coefficients.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/UnitTrinomial.lean#L121-L123"},"Polynomial.support_trinomial":{"signature":"theorem Polynomial.support_trinomial {R : Type u} [Semiring R] {k m n : \u2115} (hkm : k < m) (hmn : m < n) {x y z : R} (hx : x \u2260 0) (hy : y \u2260 0) (hz : z \u2260 0) :(C x * X ^ k + C y * X ^ m + C z * X ^ n).support = {k, m, n}","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L202-L209"},"Polynomial.card_support_trinomial":{"signature":"theorem Polynomial.card_support_trinomial {R : Type u} [Semiring R] {k m n : \u2115} (hkm : k < m) (hmn : m < n) {x y z : R} (hx : x \u2260 0) (hy : y \u2260 0) (hz : z \u2260 0) :(C x * X ^ k + C y * X ^ m + C z * X ^ n).support.card = 3","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L215-L220"},"Polynomial.aeval_mem_adjoin_singleton":{"signature":"theorem Polynomial.aeval_mem_adjoin_singleton (R : Type u) {A : Type z} [CommSemiring R] [Semiring A] [Algebra R A] {p : Polynomial R} (x : A) :(aeval x) p \u2208 Algebra.adjoin R {x}","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Adjoin/Polynomial.lean#L53-L55"},"Polynomial.leadingCoeffHom_apply":{"signature":"theorem Polynomial.leadingCoeffHom_apply {R : Type u} [Semiring R] [NoZeroDivisors R] (p : Polynomial R) :leadingCoeffHom p = p.leadingCoeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L708-L710"},"Polynomial.leadingCoeff_multiset_prod":{"signature":"theorem Polynomial.leadingCoeff_multiset_prod {R : Type u} [CommSemiring R] [NoZeroDivisors R] (t : Multiset (Polynomial R)) :t.prod.leadingCoeff = (Multiset.map (fun (f : Polynomial R) => f.leadingCoeff) t).prod","doc":"The leading coefficient of a product of polynomials is equal to the product of the leading coefficients. See Polynomial.leadingCoeff_multiset_prod' (with a ') for a version for commutative semirings, where additionally, the product of the leading coefficients must be nonzero.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/BigOperators.lean#L353-L362"},"Polynomial.leadingCoeff_prod":{"signature":"theorem Polynomial.leadingCoeff_prod {R : Type u} {\u03b9 : Type w} (s : Finset \u03b9) [CommSemiring R] [NoZeroDivisors R] (f : \u03b9 \u2192 Polynomial R) :(\u220f i \u2208 s, f i).leadingCoeff = \u220f i \u2208 s, (f i).leadingCoeff","doc":"The leading coefficient of a product of polynomials is equal to the product of the leading coefficients. See Polynomial.leadingCoeff_prod' (with a ') for a version for commutative semirings, where additionally, the product of the leading coefficients must be nonzero.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/BigOperators.lean#L364-L371"},"Polynomial.degree_smul_of_smul_regular":{"signature":"theorem Polynomial.degree_smul_of_smul_regular {R : Type u} [Semiring R] {S : Type u_1} [SMulZeroClass S R] {k : S} (p : Polynomial R) (h : IsSMulRegular R k) :(k \u2022 p).degree = p.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Monic.lean#L528-L539"},"Polynomial.natDegree_smul_of_smul_regular":{"signature":"theorem Polynomial.natDegree_smul_of_smul_regular {R : Type u} [Semiring R] {S : Type u_1} [SMulZeroClass S R] {k : S} (p : Polynomial R) (h : IsSMulRegular R k) :(k \u2022 p).natDegree = p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Monic.lean#L541-L549"},"Polynomial.leadingCoeff_smul_of_smul_regular":{"signature":"theorem Polynomial.leadingCoeff_smul_of_smul_regular {R : Type u} [Semiring R] {S : Type u_1} [SMulZeroClass S R] {k : S} (p : Polynomial R) (h : IsSMulRegular R k) :(k \u2022 p).leadingCoeff = k \u2022 p.leadingCoeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Monic.lean#L551-L554"},"Polynomial.monic_of_isUnit_leadingCoeff_inv_smul":{"signature":"theorem Polynomial.monic_of_isUnit_leadingCoeff_inv_smul {R : Type u} [Semiring R] {p : Polynomial R} (h : IsUnit p.leadingCoeff) :(h.unit\u207b\u00b9 \u2022 p).Monic","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Monic.lean#L556-L559"},"Polynomial.isUnit_leadingCoeff_mul_left_eq_zero_iff":{"signature":"theorem Polynomial.isUnit_leadingCoeff_mul_left_eq_zero_iff {R : Type u} [Semiring R] {p : Polynomial R} (h : IsUnit p.leadingCoeff) {q : Polynomial R} :q * p = 0 \u2194 q = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Monic.lean#L576-L586"},"Polynomial.separable_cyclotomic":{"signature":"theorem Polynomial.separable_cyclotomic (n : \u2115) (K : Type u_2) [Field K] [NeZero \u2191n] :(cyclotomic n K).Separable","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean#L653-L655"},"Polynomial.natDegree_linear_le":{"signature":"theorem Polynomial.natDegree_linear_le {R : Type u} {a b : R} [Semiring R] :(C a * X + C b).natDegree \u2264 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/SmallDegree.lean#L79-L80"},"Polynomial.reflect_map":{"signature":"theorem Polynomial.reflect_map {R : Type u_1} [Semiring R] {S : Type u_2} [Semiring S] (f : R \u2192+* S) (p : Polynomial R) (n : \u2115) :reflect n (map f p) = map f (reflect n p)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Reverse.lean#L140-L142"},"Polynomial.reflect_one":{"signature":"theorem Polynomial.reflect_one {R : Type u_1} [Semiring R] (n : \u2115) :reflect n 1 = X ^ n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Reverse.lean#L144-L146"},"Polynomial.monic_of_injective":{"signature":"theorem Polynomial.monic_of_injective {R : Type u} {S : Type v} [Semiring R] [Semiring S] {f : R \u2192+* S} (hf : Function.Injective \u21d1f) {p : Polynomial R} (hp : (map f p).Monic) :p.Monic","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Monic.lean#L418-L420"},"Polynomial.monic_map_iff":{"signature":"theorem Polynomial.monic_map_iff {R : Type u} {S : Type v} [Ring R] [IsSimpleRing R] [Semiring S] [Nontrivial S] {f : R \u2192+* S} {p : Polynomial R} :(map f p).Monic \u2194 p.Monic","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L289-L290"},"Polynomial.instEquivLikeGalSplittingField":{"signature":"instance Polynomial.instEquivLikeGalSplittingField {F : Type u_1} [Field F] (p : Polynomial F) :EquivLike p.Gal p.SplittingField p.SplittingField","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/PolynomialGaloisGroup.lean#L57-L57"},"Polynomial.derivative_natCast_mul":{"signature":"theorem Polynomial.derivative_natCast_mul {R : Type u} [Semiring R] {n : \u2115} {f : Polynomial R} :derivative (\u2191n * f) = \u2191n * derivative f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L286-L288"},"Polynomial.eq_zero_of_dvd_of_natDegree_lt":{"signature":"theorem Polynomial.eq_zero_of_dvd_of_natDegree_lt {R : Type u} [Semiring R] [NoZeroDivisors R] {p q : Polynomial R} (h\u2081 : p \u2223 q) (h\u2082 : q.natDegree < p.natDegree) :q = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Domain.lean#L74-L77"},"Polynomial.not_dvd_of_natDegree_lt":{"signature":"theorem Polynomial.not_dvd_of_natDegree_lt {R : Type u} [Semiring R] [NoZeroDivisors R] {p q : Polynomial R} (h0 : q \u2260 0) (hl : q.natDegree < p.natDegree) :\u00acp \u2223 q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Domain.lean#L83-L86"},"Polynomial.derivation_C":{"signature":"theorem Polynomial.derivation_C {R : Type u_1} {A : Type u_2} [CommSemiring R] [AddCommMonoid A] [Module R A] [Module (Polynomial R) A] (D : Derivation R (Polynomial R) A) (a : R) :D (C a) = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivation.lean#L43-L45"},"Polynomial.Sequence":{"signature":"structure Polynomial.Sequence (R : Type u_1) [Semiring R] :Type u_1","doc":"A sequence of polynomials such that the polynomial at index i has degree i.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Sequence.lean#L43-L48"},"Polynomial.divX_one":{"signature":"theorem Polynomial.divX_one {R : Type u} [Semiring R] :divX 1 = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Inductions.lean#L67-L70"},"Polynomial.divX_X_pow":{"signature":"theorem Polynomial.divX_X_pow {R : Type u} {n : \u2115} [Semiring R] :(X ^ n).divX = if n = 0 then 0 else X ^ (n - 1)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Inductions.lean#L77-L81"},"Polynomial.resultant_self":{"signature":"theorem Polynomial.resultant_self {R : Type u_1} [CommRing R] (f : Polynomial R) :f.resultant f = 0 ^ f.natDegree","doc":"Res(f, f) = 0 unless deg f = 0. Also see resultant_self_eq_zero.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L591-L604"},"Polynomial.isEquivalent_atTop_lead":{"signature":"theorem Polynomial.isEquivalent_atTop_lead {\ud835\udd5c : Type u_1} [NormedField \ud835\udd5c] [LinearOrder \ud835\udd5c] [IsStrictOrderedRing \ud835\udd5c] (P : Polynomial \ud835\udd5c) [OrderTopology \ud835\udd5c] :Asymptotics.IsEquivalent Filter.atTop (fun (x : \ud835\udd5c) => eval x P) fun (x : \ud835\udd5c) => P.leadingCoeff * x ^ P.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/Basic.lean#L43-L55"},"Polynomial.isEquivalent_atTop_div":{"signature":"theorem Polynomial.isEquivalent_atTop_div {\ud835\udd5c : Type u_1} [NormedField \ud835\udd5c] [LinearOrder \ud835\udd5c] [IsStrictOrderedRing \ud835\udd5c] (P Q : Polynomial \ud835\udd5c) [OrderTopology \ud835\udd5c] :Asymptotics.IsEquivalent Filter.atTop (fun (x : \ud835\udd5c) => eval x P / eval x Q) fun (x : \ud835\udd5c) => P.leadingCoeff / Q.leadingCoeff * x ^ (\u2191P.natDegree - \u2191Q.natDegree)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/Basic.lean#L116-L126"},"Polynomial.div_tendsto_zero_of_degree_lt":{"signature":"theorem Polynomial.div_tendsto_zero_of_degree_lt {\ud835\udd5c : Type u_1} [NormedField \ud835\udd5c] [LinearOrder \ud835\udd5c] [IsStrictOrderedRing \ud835\udd5c] (P Q : Polynomial \ud835\udd5c) [OrderTopology \ud835\udd5c] (hdeg : P.degree < Q.degree) :Filter.Tendsto (fun (x : \ud835\udd5c) => eval x P / eval x Q) Filter.atTop (nhds 0)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/Basic.lean#L128-L136"},"Polynomial.div_tendsto_zero_iff_degree_lt":{"signature":"theorem Polynomial.div_tendsto_zero_iff_degree_lt {\ud835\udd5c : Type u_1} [NormedField \ud835\udd5c] [LinearOrder \ud835\udd5c] [IsStrictOrderedRing \ud835\udd5c] (P Q : Polynomial \ud835\udd5c) [OrderTopology \ud835\udd5c] (hQ : Q \u2260 0) :Filter.Tendsto (fun (x : \ud835\udd5c) => eval x P / eval x Q) Filter.atTop (nhds 0) \u2194 P.degree < Q.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/Basic.lean#L138-L152"},"Polynomial.trailingCoeff_eq_coeff_zero":{"signature":"theorem Polynomial.trailingCoeff_eq_coeff_zero {R : Type u} [Semiring R] {p : Polynomial R} (h : p.coeff 0 \u2260 0) :p.trailingCoeff = p.coeff 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L154-L155"},"Polynomial.le_degree_of_mem_supp":{"signature":"theorem Polynomial.le_degree_of_mem_supp {R : Type u} [Semiring R] {p : Polynomial R} (a : \u2115) :a \u2208 p.support \u2192 \u2191a \u2264 p.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Support.lean#L112-L113"},"Polynomial.degree_eraseLead_lt":{"signature":"theorem Polynomial.degree_eraseLead_lt {R : Type u_1} [Semiring R] {f : Polynomial R} (hf : f \u2260 0) :f.eraseLead.degree < f.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/EraseLead.lean#L188-L189"},"Polynomial.eraseLead_coeff_natDegree":{"signature":"theorem Polynomial.eraseLead_coeff_natDegree {R : Type u_1} [Semiring R] {f : Polynomial R} :f.eraseLead.coeff f.natDegree = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/EraseLead.lean#L51-L52"},"Polynomial.eraseLead_add_of_degree_lt_left":{"signature":"theorem Polynomial.eraseLead_add_of_degree_lt_left {R : Type u_1} [Semiring R] {p q : Polynomial R} (pq : q.degree < p.degree) :(p + q).eraseLead = p.eraseLead + q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/EraseLead.lean#L157-L165"},"Polynomial.eraseLead_X_pow":{"signature":"theorem Polynomial.eraseLead_X_pow {R : Type u_1} [Semiring R] (n : \u2115) :(X ^ n).eraseLead = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/EraseLead.lean#L146-L148"},"Polynomial.monicEquivDegreeLT":{"signature":"def Polynomial.monicEquivDegreeLT {R : Type u} [Semiring R] [Nontrivial R] (n : \u2115) :{ p : Polynomial R // p.Monic \u2227 p.natDegree = n } \u2243 \u21a5(degreeLT R n)","doc":"The equivalence between monic polynomials of degree n and polynomials of degree less than n, formed by adding a term X ^ n.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Basic.lean#L162-L188"},"Polynomial.cyclotomic_injective":{"signature":"theorem Polynomial.cyclotomic_injective {R : Type u_1} [CommRing R] [CharZero R] :Function.Injective fun (n : \u2115) => cyclotomic n R","doc":"Over a ring R of characteristic zero, fun n => cyclotomic n R is injective.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Roots.lean#L132-L151"},"Polynomial.cyclotomic_eq_minpoly_rat":{"signature":"theorem Polynomial.cyclotomic_eq_minpoly_rat {n : \u2115} {K : Type u_2} [Field K] {\u03bc : K} (h : IsPrimitiveRoot \u03bc n) (hpos : 0 < n) [CharZero K] :cyclotomic n \u211a = minpoly \u211a \u03bc","doc":"cyclotomic n \u211a is the minimal polynomial of a primitive n-th root of unity \u03bc.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Roots.lean#L177-L181"},"Polynomial.cyclotomic_expand_eq_cyclotomic_mul":{"signature":"theorem Polynomial.cyclotomic_expand_eq_cyclotomic_mul {p n : \u2115} (hp : Nat.Prime p) (hdiv : \u00acp \u2223 n) (R : Type u_1) [CommRing R] :(expand R p) (cyclotomic n R) = cyclotomic (n * p) R * cyclotomic n R","doc":"If p is a prime such that \u00ac p \u2223 n, then expand R p (cyclotomic n R) = (cyclotomic (n * p) R) * (cyclotomic n R).","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Expand.lean#L35-L74"},"Polynomial.cyclotomic_mul_prime_eq_pow_of_not_dvd":{"signature":"theorem Polynomial.cyclotomic_mul_prime_eq_pow_of_not_dvd (R : Type u_1) {p n : \u2115} [hp : Fact (Nat.Prime p)] [Ring R] [CharP R p] (hn : \u00acp \u2223 n) :cyclotomic (n * p) R = cyclotomic n R ^ (p - 1)","doc":"If R is of characteristic p and \u00acp \u2223 n, then cyclotomic (n * p) R = (cyclotomic n R) ^ (p - 1).","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Expand.lean#L130-L142"},"Polynomial.isMonicOfDegree_iff":{"signature":"theorem Polynomial.isMonicOfDegree_iff {R : Type u_1} [Semiring R] [Nontrivial R] (p : Polynomial R) (n : \u2115) :p.IsMonicOfDegree n \u2194 p.natDegree \u2264 n \u2227 p.coeff n = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/IsMonicOfDegree.lean#L55-L60"},"Polynomial.coeff_add_eq_left_of_lt":{"signature":"theorem Polynomial.coeff_add_eq_left_of_lt {R : Type u} {n : \u2115} [Semiring R] {p q : Polynomial R} (qn : q.natDegree < n) :(p + q).coeff n = p.coeff n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L171-L173"},"Polynomial.roots_eq_of_degree_le_card_of_ne_zero":{"signature":"theorem Polynomial.roots_eq_of_degree_le_card_of_ne_zero {R : Type u} [CommRing R] [IsDomain R] {p : Polynomial R} {S : Finset R} (hS : \u2200 x \u2208 S, eval x p = 0) (hcard : p.degree \u2264 \u2191S.card) (hp : p \u2260 0) :p.roots = S.val","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L301-L305"},"Polynomial.roots_eq_of_degree_eq_card":{"signature":"theorem Polynomial.roots_eq_of_degree_eq_card {R : Type u} [CommRing R] [IsDomain R] {p : Polynomial R} {S : Finset R} (hS : \u2200 x \u2208 S, eval x p = 0) (hcard : \u2191S.card = p.degree) :p.roots = S.val","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L307-L309"},"Polynomial.C_ofNat":{"signature":"theorem Polynomial.C_ofNat {R : Type u} [Semiring R] (n : \u2115) [n.AtLeastTwo] :C (OfNat.ofNat n) = OfNat.ofNat n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L453-L454"},"Polynomial.degree_mul_X":{"signature":"theorem Polynomial.degree_mul_X {R : Type u} [Semiring R] [Nontrivial R] {p : Polynomial R} :(p * X).degree = p.degree + 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L536-L537"},"Polynomial.natDegree_of_mem_normalizedFactors_cyclotomic":{"signature":"theorem Polynomial.natDegree_of_mem_normalizedFactors_cyclotomic {K : Type u_1} [Field K] [Fintype K] {p f n : \u2115} {P : Polynomial K} (hK : Fintype.card K = p ^ f) (hn : p.Coprime n) [hp : Fact (Nat.Prime p)] [DecidableEq K] (hP : P \u2208 UniqueFactorizationMonoid.normalizedFactors (cyclotomic n K)) :P.natDegree = orderOf (ZMod.unitOfCoprime (p ^ f) \u22ef)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Factorization.lean#L133-L137"},"Polynomial.degree_coe_units":{"signature":"theorem Polynomial.degree_coe_units {R : Type u} [Semiring R] [NoZeroDivisors R] [Nontrivial R] (u : (Polynomial R)\u02e3) :(\u2191u).degree = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Units.lean#L42-L44"},"Polynomial.degree_eq_degree_of_associated":{"signature":"theorem Polynomial.degree_eq_degree_of_associated {R : Type u} [CommRing R] [IsDomain R] {p q : Polynomial R} (h : Associated p q) :p.degree = q.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/RingDivision.lean#L199-L201"},"Polynomial.natDegree_multiset_prod":{"signature":"theorem Polynomial.natDegree_multiset_prod {R : Type u} [CommSemiring R] [NoZeroDivisors R] (t : Multiset (Polynomial R)) (h : 0 \u2209 t) :t.prod.natDegree = (Multiset.map natDegree t).sum","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/BigOperators.lean#L333-L339"},"Polynomial.squarefree_cyclotomic":{"signature":"theorem Polynomial.squarefree_cyclotomic (n : \u2115) (K : Type u_2) [Field K] [NeZero \u2191n] :Squarefree (cyclotomic n K)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean#L657-L659"},"Polynomial.normalizedFactors_cyclotomic_card":{"signature":"theorem Polynomial.normalizedFactors_cyclotomic_card {K : Type u_1} [Field K] [Fintype K] {p f n : \u2115} (hK : Fintype.card K = p ^ f) (hn : p.Coprime n) [hp : Fact (Nat.Prime p)] [DecidableEq K] :(UniqueFactorizationMonoid.normalizedFactors (cyclotomic n K)).toFinset.card = n.totient / orderOf (ZMod.unitOfCoprime (p ^ f) \u22ef)","doc":"Let K be a finite field of cardinality p ^ f and let P be an irreducible factor of the n-th cyclotomic polynomial over K, where p and n are coprime. This result computes the number of distinct irreducible factors of cyclotomic n K.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Factorization.lean#L139-L172"},"Polynomial.X_pow_sub_one_mul_prod_cyclotomic_eq_X_pow_sub_one_of_dvd":{"signature":"theorem Polynomial.X_pow_sub_one_mul_prod_cyclotomic_eq_X_pow_sub_one_of_dvd (R : Type u_1) [CommRing R] {d n : \u2115} (hdvd : d \u2223 n) (hn : n \u2260 0) :(X ^ d - 1) * \u220f x \u2208 n.divisors \\ d.divisors, cyclotomic x R = X ^ n - 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean#L396-L402"},"Polynomial.X_pow_sub_one_mul_cyclotomic_dvd_X_pow_sub_one_of_dvd":{"signature":"theorem Polynomial.X_pow_sub_one_mul_cyclotomic_dvd_X_pow_sub_one_of_dvd (R : Type u_1) [CommRing R] {d n : \u2115} (h : d \u2208 n.properDivisors) :(X ^ d - 1) * cyclotomic n R \u2223 X ^ n - 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean#L404-L413"},"Polynomial.eval_dvd":{"signature":"theorem Polynomial.eval_dvd {R : Type u} [CommSemiring R] {p q : Polynomial R} {x : R} :p \u2223 q \u2192 eval x p \u2223 eval x q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L681-L683"},"Polynomial.degreeLT_succ_eq_degreeLE":{"signature":"theorem Polynomial.degreeLT_succ_eq_degreeLE {R : Type u} [Semiring R] {n : \u2115} :degreeLT R (n + 1) = degreeLE R \u2191n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Basic.lean#L155-L160"},"Polynomial.span_of_finite_le_degreeLT":{"signature":"theorem Polynomial.span_of_finite_le_degreeLT {R : Type u} [Semiring R] {s : Set (Polynomial R)} (s_fin : s.Finite) :\u2203 (n : \u2115), Submodule.span R s \u2264 degreeLT R n","doc":"The span of every finite set of polynomials is contained in a degreeLT n for some n.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Basic.lean#L227-L231"},"Polynomial.sumIDeriv_apply_of_le":{"signature":"theorem Polynomial.sumIDeriv_apply_of_le {R : Type u_1} [Semiring R] {p : Polynomial R} {n : \u2115} (hn : p.natDegree \u2264 n) :sumIDeriv p = \u2211 i \u2208 Finset.range (n + 1), (\u21d1derivative)^[i] p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/SumIteratedDerivative.lean#L75-L78"},"Polynomial.natDegree_derivative_le":{"signature":"theorem Polynomial.natDegree_derivative_le {R : Type u} [Semiring R] (p : Polynomial R) :(derivative p).natDegree \u2264 p.natDegree - 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L180-L183"},"Polynomial.sumIDeriv_derivative":{"signature":"theorem Polynomial.sumIDeriv_derivative {R : Type u_1} [Semiring R] (p : Polynomial R) :sumIDeriv (derivative p) = derivative (sumIDeriv p)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/SumIteratedDerivative.lean#L97-L100"},"Polynomial.smeval_at_natCast":{"signature":"theorem Polynomial.smeval_at_natCast {S : Type u_2} [NonAssocSemiring S] [Pow S \u2115] [NatPowAssoc S] (q : Polynomial \u2115) (n : \u2115) :q.smeval \u2191n = \u2191(q.smeval n)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Smeval.lean#L189-L196"},"Polynomial.mul_X_sub_intCast_comp":{"signature":"theorem Polynomial.mul_X_sub_intCast_comp {R : Type u} [Ring R] {p q : Polynomial R} {n : \u2115} :(p * (X - \u2191n)).comp q = p.comp q * (q - \u2191n)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L775-L777"},"Polynomial.divX_zero":{"signature":"theorem Polynomial.divX_zero {R : Type u} [Semiring R] :divX 0 = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Inductions.lean#L64-L65"},"Polynomial.not_isUnit_of_natDegree_pos_of_isReduced":{"signature":"theorem Polynomial.not_isUnit_of_natDegree_pos_of_isReduced {R : Type u_1} [CommRing R] [IsReduced R] (p : Polynomial R) (hpl : 0 < p.natDegree) :\u00acIsUnit p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Nilpotent.lean#L177-L185"},"Polynomial.continuous_aeval":{"signature":"theorem Polynomial.continuous_aeval {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] [TopologicalSpace A] [IsTopologicalSemiring A] (p : Polynomial R) :Continuous fun (x : A) => (aeval x) p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Topology/Algebra/Polynomial.lean#L79-L81"},"Polynomial.continuousOn_aeval":{"signature":"theorem Polynomial.continuousOn_aeval {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] [TopologicalSpace A] [IsTopologicalSemiring A] (p : Polynomial R) {s : Set A} :ContinuousOn (fun (x : A) => (aeval x) p) s","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Topology/Algebra/Polynomial.lean#L92-L94"},"Polynomial.restriction":{"signature":"def Polynomial.restriction {R : Type u} [Ring R] (p : Polynomial R) :Polynomial \u21a5(Subring.closure \u2191p.coeffs)","doc":"Given a polynomial, return the polynomial whose coefficients are in the ring closure of the original coefficients.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Basic.lean#L293-L302"},"Polynomial.natDegree_coe_units":{"signature":"theorem Polynomial.natDegree_coe_units {R : Type u} [Semiring R] [NoZeroDivisors R] (u : (Polynomial R)\u02e3) :(\u2191u).natDegree = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Units.lean#L64-L66"},"Polynomial.coeff_coe_units_zero_ne_zero":{"signature":"theorem Polynomial.coeff_coe_units_zero_ne_zero {R : Type u} [Semiring R] [NoZeroDivisors R] [Nontrivial R] (u : (Polynomial R)\u02e3) :(\u2191u).coeff 0 \u2260 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Units.lean#L68-L71"},"Polynomial.coeff_inv_units":{"signature":"theorem Polynomial.coeff_inv_units {R : Type u} [Field R] (u : (Polynomial R)\u02e3) (n : \u2115) :((\u2191u).coeff n)\u207b\u00b9 = (\u2191u\u207b\u00b9).coeff n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L538-L545"},"Polynomial.mahlerMeasure_X_add_C":{"signature":"theorem Polynomial.mahlerMeasure_X_add_C (z : \u2102) :(X + C z).mahlerMeasure = max 1 \u2016z\u2016","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/MahlerMeasure.lean#L187-L189"},"Polynomial.mahlerMeasure_C_mul_X_add_C":{"signature":"theorem Polynomial.mahlerMeasure_C_mul_X_add_C {a : \u2102} (ha : a \u2260 0) (b : \u2102) :(C a * X + C b).mahlerMeasure = max \u2016a\u2016 \u2016b\u2016","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/MahlerMeasure.lean#L191-L197"},"Polynomial.mahlerMeasure_of_degree_eq_one":{"signature":"theorem Polynomial.mahlerMeasure_of_degree_eq_one {p : Polynomial \u2102} (h : p.degree = 1) :p.mahlerMeasure = max \u2016p.coeff 1\u2016 \u2016p.coeff 0\u2016","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/MahlerMeasure.lean#L199-L202"},"Polynomial.map_mem_nthRootsFinset":{"signature":"theorem Polynomial.map_mem_nthRootsFinset {R : Type u} {n : \u2115} [CommRing R] [IsDomain R] {S : Type u_1} {F : Type u_2} [CommRing S] [IsDomain S] [FunLike F R S] [MonoidHomClass F R S] {a x : R} (hx : x \u2208 nthRootsFinset n a) (f : F) :f x \u2208 nthRootsFinset n (f a)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L379-L385"},"Polynomial.map_mem_nthRootsFinset_one":{"signature":"theorem Polynomial.map_mem_nthRootsFinset_one {R : Type u} {n : \u2115} [CommRing R] [IsDomain R] {S : Type u_1} {F : Type u_2} [CommRing S] [IsDomain S] [FunLike F R S] [RingHomClass F R S] {x : R} (hx : x \u2208 nthRootsFinset n 1) (f : F) :f x \u2208 nthRootsFinset n 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L387-L391"},"Polynomial.roots_X_pow_char_pow_sub_C":{"signature":"theorem Polynomial.roots_X_pow_char_pow_sub_C {R : Type u_1} [CommRing R] [IsDomain R] {p n : \u2115} [ExpChar R p] [PerfectRing R p] {y : R} :(X ^ p ^ n - C y).roots = p ^ n \u2022 {(iterateFrobeniusEquiv R p n).symm y}","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Perfect.lean#L354-L357"},"Polynomial.logMahlerMeasure_def":{"signature":"theorem Polynomial.logMahlerMeasure_def (p : Polynomial \u2102) :p.logMahlerMeasure = Real.circleAverage (fun (x : \u2102) => Real.log \u2016eval x p\u2016) 0 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/MahlerMeasure.lean#L47-L48"},"Polynomial.trailingCoeff_zero":{"signature":"theorem Polynomial.trailingCoeff_zero {R : Type u} [Semiring R] :trailingCoeff 0 = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L81-L83"},"Polynomial.coeff_mul_natTrailingDegree_add_natTrailingDegree":{"signature":"theorem Polynomial.coeff_mul_natTrailingDegree_add_natTrailingDegree {R : Type u} [Semiring R] {p q : Polynomial R} :(p * q).coeff (p.natTrailingDegree + q.natTrailingDegree) = p.trailingCoeff * q.trailingCoeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L308-L321"},"Polynomial.le_trailingDegree_mul":{"signature":"theorem Polynomial.le_trailingDegree_mul {R : Type u} [Semiring R] {p q : Polynomial R} :p.trailingDegree + q.trailingDegree \u2264 (p * q).trailingDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L289-L296"},"Polynomial.trailingDegree_mul'":{"signature":"theorem Polynomial.trailingDegree_mul' {R : Type u} [Semiring R] {p q : Polynomial R} (h : p.trailingCoeff * q.trailingCoeff \u2260 0) :(p * q).trailingDegree = p.trailingDegree + q.trailingDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L323-L331"},"Polynomial.natTrailingDegree_mul'":{"signature":"theorem Polynomial.natTrailingDegree_mul' {R : Type u} [Semiring R] {p q : Polynomial R} (h : p.trailingCoeff * q.trailingCoeff \u2260 0) :(p * q).natTrailingDegree = p.natTrailingDegree + q.natTrailingDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L333-L340"},"Polynomial.eval_zero_map":{"signature":"theorem Polynomial.eval_zero_map {R : Type u} {S : Type v} [Semiring R] [Semiring S] (f : R \u2192+* S) (p : Polynomial R) :eval 0 (map f p) = f (eval 0 p)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Coeff.lean#L142-L144"},"Polynomial.bernoulli_eval_zero":{"signature":"theorem Polynomial.bernoulli_eval_zero (n : \u2115) :eval 0 (bernoulli n) = _root_.bernoulli n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/NumberTheory/BernoulliPolynomials.lean#L81-L88"},"Polynomial.bernoulli_eval_one":{"signature":"theorem Polynomial.bernoulli_eval_one (n : \u2115) :eval 1 (bernoulli n) = bernoulli' n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/NumberTheory/BernoulliPolynomials.lean#L90-L97"},"Polynomial.bernoulli_zero":{"signature":"theorem Polynomial.bernoulli_zero :bernoulli 0 = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/NumberTheory/BernoulliPolynomials.lean#L74-L75"},"Polynomial.derivative_bernoulli_add_one":{"signature":"theorem Polynomial.derivative_bernoulli_add_one (k : \u2115) :derivative (bernoulli (k + 1)) = (\u2191k + 1) * bernoulli k","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/NumberTheory/BernoulliPolynomials.lean#L111-L122"},"Polynomial.derivative_bernoulli":{"signature":"theorem Polynomial.derivative_bernoulli (k : \u2115) :derivative (bernoulli k) = \u2191k * bernoulli (k - 1)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/NumberTheory/BernoulliPolynomials.lean#L124-L128"},"Polynomial.derivative_C_mul_X_sq":{"signature":"theorem Polynomial.derivative_C_mul_X_sq {R : Type u} [Semiring R] (a : R) :derivative (C a * X ^ 2) = C (a * 2) * X","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L96-L97"},"Polynomial.C_mul_X_eq_monomial":{"signature":"theorem Polynomial.C_mul_X_eq_monomial {R : Type u} {a : R} [Semiring R] :C a * X = (monomial 1) a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L669-L669"},"Polynomial.toFinsupp_C_mul_X":{"signature":"theorem Polynomial.toFinsupp_C_mul_X {R : Type u} [Semiring R] (a : R) :(C a * X).toFinsupp = AddMonoidAlgebra.single 1 a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L671-L673"},"Polynomial.one_le_prod_max_one_norm_roots":{"signature":"theorem Polynomial.one_le_prod_max_one_norm_roots (p : Polynomial \u2102) :1 \u2264 (Multiset.map (fun (a : \u2102) => max 1 \u2016a\u2016) p.roots).prod","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/MahlerMeasure.lean#L234-L235"},"Polynomial.natDegree_X_pow_le":{"signature":"theorem Polynomial.natDegree_X_pow_le {R : Type u_1} [Semiring R] (n : \u2115) :(X ^ n).natDegree \u2264 n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L527-L529"},"Polynomial.isMonicOfDegree_X_pow":{"signature":"theorem Polynomial.isMonicOfDegree_X_pow (R : Type u_1) [Semiring R] [Nontrivial R] (n : \u2115) :(X ^ n).IsMonicOfDegree n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/IsMonicOfDegree.lean#L170-L171"},"Polynomial.isIntegral_coeff_prod":{"signature":"theorem Polynomial.isIntegral_coeff_prod {R : Type u_1} {S : Type u_2} {\u03b9 : Type u_3} [CommRing R] [CommRing S] [Algebra R S] (s : Finset \u03b9) (p : \u03b9 \u2192 Polynomial S) (H : \u2200 i \u2208 s, \u2200 (j : \u2115), IsIntegral R ((p i).coeff j)) (j : \u2115) :IsIntegral R ((s.prod p).coeff j)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/IsIntegral.lean#L34-L42"},"Polynomial.ofNat_comp":{"signature":"theorem Polynomial.ofNat_comp {R : Type u} [Semiring R] {p : Polynomial R} (n : \u2115) [n.AtLeastTwo] :(OfNat.ofNat n).comp p = \u2191n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L393-L395"},"Polynomial.tendsto_abv_aeval_atTop":{"signature":"theorem Polynomial.tendsto_abv_aeval_atTop {R : Type u_1} {A : Type u_2} {k : Type u_3} {\u03b1 : Type u_4} [CommSemiring R] [Ring A] [Algebra R A] [Field k] [LinearOrder k] [IsStrictOrderedRing k] (abv : A \u2192 k) [IsAbsoluteValue abv] (p : Polynomial R) (hd : 0 < p.degree) (h\u2080 : (algebraMap R A) p.leadingCoeff \u2260 0) {l : Filter \u03b1} {z : \u03b1 \u2192 A} (hz : Filter.Tendsto (abv \u2218 z) l Filter.atTop) :Filter.Tendsto (fun (x : \u03b1) => abv ((aeval (z x)) p)) l Filter.atTop","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Topology/Algebra/Polynomial.lean#L123-L128"},"Polynomial.mkDerivation_X":{"signature":"theorem Polynomial.mkDerivation_X (R : Type u_1) {A : Type u_2} [CommSemiring R] [AddCommMonoid A] [Module R A] [Module (Polynomial R) A] [IsScalarTower R (Polynomial R) A] (a : A) :((mkDerivation R) a) X = a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivation.lean#L71-L72"},"Polynomial.mkDerivationEquiv":{"signature":"def Polynomial.mkDerivationEquiv (R : Type u_1) {A : Type u_2} [CommSemiring R] [AddCommMonoid A] [Module R A] [Module (Polynomial R) A] [IsScalarTower R (Polynomial R) A] :A \u2243\u2097[R] Derivation R (Polynomial R) A","doc":"Polynomial.mkDerivation as a linear equivalence.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivation.lean#L82-L90"},"Polynomial.evalEval_mul":{"signature":"theorem Polynomial.evalEval_mul {R : Type u_1} [CommSemiring R] (x y : R) (p q : Polynomial (Polynomial R)) :evalEval x y (p * q) = evalEval x y p * evalEval x y q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Bivariate.lean#L118-L121"},"Polynomial.toLaurent_inj":{"signature":"theorem Polynomial.toLaurent_inj {R : Type u_1} [Semiring R] (f g : Polynomial R) :toLaurent f = toLaurent g \u2194 f = g","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Laurent.lean#L334-L336"},"Polynomial.eq_C_of_derivative_eq_zero":{"signature":"theorem Polynomial.eq_C_of_derivative_eq_zero {R : Type u} [Semiring R] [IsAddTorsionFree R] {f : Polynomial R} (h : derivative f = 0) :f = C (f.coeff 0)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L242-L244"},"Polynomial.continuousOn":{"signature":"theorem Polynomial.continuousOn {R : Type u_1} [Semiring R] [TopologicalSpace R] [IsTopologicalSemiring R] (p : Polynomial R) {s : Set R} :ContinuousOn (fun (x : R) => eval x p) s","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Topology/Algebra/Polynomial.lean#L68-L70"},"Polynomial.deriv":{"signature":"theorem Polynomial.deriv {\ud835\udd5c : Type u} [NontriviallyNormedField \ud835\udd5c] {x : \ud835\udd5c} (p : Polynomial \ud835\udd5c) :deriv (fun (x : \ud835\udd5c) => eval x p) x = eval x (derivative p)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Calculus/Deriv/Polynomial.lean#L106-L108"},"Polynomial.card_roots_toFinset_le_card_roots_derivative_diff_roots_succ":{"signature":"theorem Polynomial.card_roots_toFinset_le_card_roots_derivative_diff_roots_succ (p : Polynomial \u211d) :p.roots.toFinset.card \u2264 ((derivative p).roots.toFinset \\ p.roots.toFinset).card + 1","doc":"The number of roots of a real polynomial p is at most the number of roots of its derivative that are not roots of p plus one.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Calculus/LocalExtr/Polynomial.lean#L36-L48"},"Polynomial.card_roots_toFinset_le_derivative":{"signature":"theorem Polynomial.card_roots_toFinset_le_derivative (p : Polynomial \u211d) :p.roots.toFinset.card \u2264 (derivative p).roots.toFinset.card + 1","doc":"The number of roots of a real polynomial is at most the number of roots of its derivative plus one.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Calculus/LocalExtr/Polynomial.lean#L50-L55"},"Polynomial.card_rootSet_le_derivative":{"signature":"theorem Polynomial.card_rootSet_le_derivative {F : Type u_1} [CommRing F] [Algebra F \u211d] (p : Polynomial F) :Fintype.card \u2191(p.rootSet \u211d) \u2264 Fintype.card \u2191((derivative p).rootSet \u211d) + 1","doc":"The number of real roots of a polynomial is at most the number of roots of its derivative plus one.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Calculus/LocalExtr/Polynomial.lean#L87-L92"},"Polynomial.mapEquiv_symm_apply":{"signature":"theorem Polynomial.mapEquiv_symm_apply {R : Type u} {S : Type v} [Semiring R] [Semiring S] (e : R \u2243+* S) (a : Polynomial S) :(mapEquiv e).symm a = map (\u2191e.symm) a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Degree.lean#L130-L130"},"Polynomial.degree_mul_C_of_isUnit":{"signature":"theorem Polynomial.degree_mul_C_of_isUnit {R : Type u} {a : R} [Semiring R] (ha : IsUnit a) (p : Polynomial R) :(p * C a).degree = p.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L390-L396"},"Polynomial.natDegree_mul_C_of_isUnit":{"signature":"theorem Polynomial.natDegree_mul_C_of_isUnit {R : Type u} {a : R} [Semiring R] (ha : IsUnit a) (p : Polynomial R) :(p * C a).natDegree = p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L401-L402"},"Polynomial.mirror_eq_zero":{"signature":"theorem Polynomial.mirror_eq_zero {R : Type u_1} [Semiring R] {p : Polynomial R} :p.mirror = 0 \u2194 p = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Mirror.lean#L128-L130"},"Polynomial.natDegree_mul_mirror":{"signature":"theorem Polynomial.natDegree_mul_mirror {R : Type u_1} [Semiring R] (p : Polynomial R) [NoZeroDivisors R] :(p * p.mirror).natDegree = 2 * p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Mirror.lean#L155-L158"},"Polynomial.constantCoeff_apply":{"signature":"theorem Polynomial.constantCoeff_apply {R : Type u} [Semiring R] (p : Polynomial R) :constantCoeff p = p.coeff 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L125-L125"},"Polynomial.cauchyBound":{"signature":"noncomputable def Polynomial.cauchyBound {K : Type u_1} [NormedDivisionRing K] (p : Polynomial K) :NNReal","doc":"Cauchy's bound on the roots of a given polynomial. See IsRoot.norm_lt_cauchyBound for the proof that the roots satisfy this bound.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/CauchyBound.lean#L29-L34"},"Polynomial.cauchyBound_C":{"signature":"theorem Polynomial.cauchyBound_C {K : Type u_1} [NormedDivisionRing K] (x : K) :(C x).cauchyBound = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/CauchyBound.lean#L44-L46"},"Polynomial.natSepDegree_mul":{"signature":"theorem Polynomial.natSepDegree_mul {F : Type u} [Field F] (f g : Polynomial F) :(f * g).natSepDegree \u2264 f.natSepDegree + g.natSepDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/SeparableDegree.lean#L422-L428"},"Polynomial.faithfulSMul":{"signature":"instance Polynomial.faithfulSMul {R : Type u} [Semiring R] {S : Type u_1} [SMulZeroClass S R] [FaithfulSMul S R] :FaithfulSMul S (Polynomial R)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L280-L282"},"Polynomial.map_injective_iff":{"signature":"theorem Polynomial.map_injective_iff {R : Type u} {S : Type v} [Semiring R] [Semiring S] (f : R \u2192+* S) :Function.Injective (map f) \u2194 Function.Injective \u21d1f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Coeff.lean#L106-L107"},"Polynomial.splits_of_natDegree_eq_one":{"signature":"theorem Polynomial.splits_of_natDegree_eq_one {R : Type u_1} [DivisionSemiring R] {f : Polynomial R} (hf : f.natDegree = 1) :f.Splits","doc":"Alias of Polynomial.Splits.of_natDegree_eq_one.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L675-L676"},"Polynomial.separable_X_pow_sub_C_of_irreducible":{"signature":"theorem Polynomial.separable_X_pow_sub_C_of_irreducible {K : Type u} [Field K] {n : \u2115} (h\u03b6 : (primitiveRoots n K).Nonempty) (a : K) (H : Irreducible (X ^ n - C a)) :(X ^ n - C a).Separable","doc":"Also see Polynomial.separable_X_pow_sub_C_unit","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/KummerExtension.lean#L170-L184"},"Polynomial.comp_eq_zero_iff":{"signature":"theorem Polynomial.comp_eq_zero_iff {R : Type u} [Semiring R] [NoZeroDivisors R] {p q : Polynomial R} :p.comp q = 0 \u2194 p = 0 \u2228 eval (q.coeff 0) p = 0 \u2227 q = C (q.coeff 0)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L421-L430"},"Polynomial.compRingHom":{"signature":"def Polynomial.compRingHom {R : Type u} [CommSemiring R] :Polynomial R \u2192 Polynomial R \u2192+* Polynomial R","doc":"comp p, regarded as a ring homomorphism from R[X] to itself.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L624-L626"},"Polynomial.multiset_prod_comp":{"signature":"theorem Polynomial.multiset_prod_comp {R : Type u} [CommSemiring R] (s : Multiset (Polynomial R)) (q : Polynomial R) :s.prod.comp q = (Multiset.map (fun (p : Polynomial R) => p.comp q) s).prod","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L669-L671"},"Polynomial.leadingCoeff_X_pow_sub_C":{"signature":"theorem Polynomial.leadingCoeff_X_pow_sub_C {R : Type u} [Ring R] {n : \u2115} (hn : 0 < n) {r : R} :(X ^ n - C r).leadingCoeff = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L745-L748"},"Polynomial.leadingCoeff_X_pow_sub_one":{"signature":"theorem Polynomial.leadingCoeff_X_pow_sub_one {R : Type u} [Ring R] {n : \u2115} (hn : 0 < n) :(X ^ n - 1).leadingCoeff = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L750-L752"},"Polynomial.instAlgEquivClassGalSplittingField":{"signature":"instance Polynomial.instAlgEquivClassGalSplittingField {F : Type u_1} [Field F] (p : Polynomial F) :AlgEquivClass p.Gal F p.SplittingField p.SplittingField","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/PolynomialGaloisGroup.lean#L57-L57"},"Polynomial.cardPowDegree":{"signature":"noncomputable def Polynomial.cardPowDegree {Fq : Type u_1} [Field Fq] [Fintype Fq] :AbsoluteValue (Polynomial Fq) \u2124","doc":"cardPowDegree is the absolute value on \ud835\udd3d_q[t] sending f to q ^ degree f. cardPowDegree 0 is defined to be 0.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/CardPowDegree.lean#L41-L71"},"Polynomial.exists_eq_polynomial":{"signature":"theorem Polynomial.exists_eq_polynomial {Fq : Type u_1} [Fintype Fq] [Semiring Fq] {d m : \u2115} (hm : Fintype.card Fq ^ d \u2264 m) (b : Polynomial Fq) (hb : b.natDegree \u2264 d) (A : Fin m.succ \u2192 Polynomial Fq) (hA : \u2200 (i : Fin m.succ), (A i).degree < b.degree) :\u2203 (i\u2080 : Fin m.succ) (i\u2081 : Fin m.succ), i\u2080 \u2260 i\u2081 \u2227 A i\u2081 = A i\u2080","doc":"If A is a family of enough low-degree polynomials over a finite semiring, there is a pair of equal elements in A.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/NumberTheory/ClassNumber/AdmissibleCardPowDegree.lean#L34-L56"},"Polynomial.exists_approx_polynomial_aux":{"signature":"theorem Polynomial.exists_approx_polynomial_aux {Fq : Type u_1} [Fintype Fq] [Ring Fq] {d m : \u2115} (hm : Fintype.card Fq ^ d \u2264 m) (b : Polynomial Fq) (A : Fin m.succ \u2192 Polynomial Fq) (hA : \u2200 (i : Fin m.succ), (A i).degree < b.degree) :\u2203 (i\u2080 : Fin m.succ) (i\u2081 : Fin m.succ), i\u2080 \u2260 i\u2081 \u2227 (A i\u2081 - A i\u2080).degree < \u2191(b.natDegree - d)","doc":"If A is a family of enough low-degree polynomials over a finite ring, there is a pair of elements in A (with different indices but not necessarily distinct), such that their difference has small degree.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/NumberTheory/ClassNumber/AdmissibleCardPowDegree.lean#L58-L95"},"Polynomial.cardPowDegree_nonzero":{"signature":"theorem Polynomial.cardPowDegree_nonzero {Fq : Type u_1} [Field Fq] [Fintype Fq] (p : Polynomial Fq) (hp : p \u2260 0) :cardPowDegree p = \u2191(Fintype.card Fq) ^ p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/CardPowDegree.lean#L80-L83"},"Polynomial.exists_approx_polynomial":{"signature":"theorem Polynomial.exists_approx_polynomial {Fq : Type u_1} [Fintype Fq] [Field Fq] {b : Polynomial Fq} (hb : b \u2260 0) {\u03b5 : \u211d} (h\u03b5 : 0 < \u03b5) (A : Fin (Fintype.card Fq ^ \u2308-Real.log \u03b5 / Real.log \u2191(Fintype.card Fq)\u2309\u208a).succ \u2192 Polynomial Fq) :\u2203 (i\u2080 : Fin (Fintype.card Fq ^ \u2308-Real.log \u03b5 / Real.log \u2191(Fintype.card Fq)\u2309\u208a).succ) (i\u2081 : Fin (Fintype.card Fq ^ \u2308-Real.log \u03b5 / Real.log \u2191(Fintype.card Fq)\u2309\u208a).succ), i\u2080 \u2260 i\u2081 \u2227 \u2191(cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < cardPowDegree b \u2022 \u03b5","doc":"If A is a family of enough low-degree polynomials over a finite field, there is a pair of elements in A (with different indices but not necessarily distinct), such that the difference of their remainders is close together.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/NumberTheory/ClassNumber/AdmissibleCardPowDegree.lean#L99-L142"},"Polynomial.nextCoeffUp":{"signature":"def Polynomial.nextCoeffUp {R : Type u} [Semiring R] (p : Polynomial R) :R","doc":"The second-lowest coefficient, or 0 for constants","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L400-L402"},"Polynomial.nextCoeffUp_zero":{"signature":"theorem Polynomial.nextCoeffUp_zero {R : Type u} [Semiring R] :nextCoeffUp 0 = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L404-L404"},"Polynomial.fintypeSubtypeMonicDvd":{"signature":"noncomputable def Polynomial.fintypeSubtypeMonicDvd {D : Type u} [CommRing D] [UniqueFactorizationMonoid D] (f : Polynomial D) (hf : f \u2260 0) :Fintype { g : Polynomial D // g.Monic \u2227 g \u2223 f }","doc":"If D is a unique factorization domain, f is a non-zero polynomial in D[X], then f has only finitely many monic factors. (Note that its factors up to unit may be more than monic factors.) See also UniqueFactorizationMonoid.fintypeSubtypeDvd.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/UniqueFactorization.lean#L97-L111"},"Polynomial.algEquivAevalXAddC":{"signature":"def Polynomial.algEquivAevalXAddC {R : Type u_3} [CommRing R] (t : R) :Polynomial R \u2243\u2090[R] Polynomial R","doc":"The automorphism of the polynomial algebra given by p(X) \u21a6 p(X+t), with inverse p(X) \u21a6 p(X-t).","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L353-L357"},"Polynomial.comp_X_add_C_eq_zero_iff":{"signature":"theorem Polynomial.comp_X_add_C_eq_zero_iff {R : Type u} [CommRing R] {p : Polynomial R} {t : R} :p.comp (X + C t) = 0 \u2194 p = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L657-L658"},"Polynomial.comp_X_add_C_ne_zero_iff":{"signature":"theorem Polynomial.comp_X_add_C_ne_zero_iff {R : Type u} [CommRing R] {p : Polynomial R} {t : R} :p.comp (X + C t) \u2260 0 \u2194 p \u2260 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L660-L660"},"Polynomial.leval_apply":{"signature":"theorem Polynomial.leval_apply {R : Type u_1} [Semiring R] (r : R) (f : Polynomial R) :(leval r) f = eval r f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/SMul.lean#L63-L63"},"Polynomial.degree_sub_le_of_le":{"signature":"theorem Polynomial.degree_sub_le_of_le {R : Type u} [Ring R] {p q : Polynomial R} {a b : WithBot \u2115} (hp : p.degree \u2264 a) (hq : q.degree \u2264 b) :(p - q).degree \u2264 max a b","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L532-L534"},"Polynomial.bernoulli_three_eval_one_quarter":{"signature":"theorem Polynomial.bernoulli_three_eval_one_quarter :eval (1 / 4) (bernoulli 3) = 3 / 64","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/NumberTheory/BernoulliPolynomials.lean#L99-L107"},"Polynomial.degree_ne_bot":{"signature":"theorem Polynomial.degree_ne_bot {R : Type u} [Semiring R] {p : Polynomial R} :p.degree \u2260 \u22a5 \u2194 p \u2260 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L95-L95"},"Polynomial.iterate_derivative_map":{"signature":"theorem Polynomial.iterate_derivative_map {R : Type u} {S : Type v} [Semiring R] [Semiring S] (p : Polynomial R) (f : R \u2192+* S) (k : \u2115) :(\u21d1derivative)^[k] (map f p) = map f ((\u21d1derivative)^[k] p)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L278-L284"},"Polynomial.derivativeFinsupp_map":{"signature":"theorem Polynomial.derivativeFinsupp_map {R : Type u} {S : Type v} [Semiring R] [Semiring S] (p : Polynomial R) (f : R \u2192+* S) :derivativeFinsupp (map f p) = Finsupp.mapRange (fun (x : Polynomial R) => map f x) \u22ef (derivativeFinsupp p)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L432-L435"},"Polynomial.instInhabitedSplittingField":{"signature":"instance Polynomial.instInhabitedSplittingField {K : Type u_1} [Field K] (f : Polynomial K) :Inhabited f.SplittingField","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/SplittingField/Construction.lean#L218-L218"},"Polynomial.sumIDeriv_map":{"signature":"theorem Polynomial.sumIDeriv_map {R : Type u_1} {S : Type u_2} [Semiring R] [Semiring S] (p : Polynomial R) (f : R \u2192+* S) :sumIDeriv (map f p) = map f (sumIDeriv p)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/SumIteratedDerivative.lean#L89-L95"},"Polynomial.leadingCoeff_neg":{"signature":"theorem Polynomial.leadingCoeff_neg {R : Type u} [Ring R] (p : Polynomial R) :(-p).leadingCoeff = -p.leadingCoeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L295-L297"},"Polynomial.comp_neg_X_leadingCoeff_eq":{"signature":"theorem Polynomial.comp_neg_X_leadingCoeff_eq {R : Type u} [Ring R] (p : Polynomial R) :(p.comp (-X)).leadingCoeff = (-1) ^ p.natDegree * p.leadingCoeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L413-L419"},"Polynomial.aeval_neg":{"signature":"theorem Polynomial.aeval_neg {R : Type u} {A : Type z} [CommRing R] {p : Polynomial R} [Ring A] [Algebra R A] (x : A) :(aeval x) (-p) = -(aeval x) p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L640-L642"},"Polynomial.div_tendsto_atTop_of_degree_gt'":{"signature":"theorem Polynomial.div_tendsto_atTop_of_degree_gt' {\ud835\udd5c : Type u_1} [NormedField \ud835\udd5c] [LinearOrder \ud835\udd5c] [IsStrictOrderedRing \ud835\udd5c] (P Q : Polynomial \ud835\udd5c) [OrderTopology \ud835\udd5c] (hdeg : Q.degree < P.degree) (hpos : 0 < P.leadingCoeff / Q.leadingCoeff) :Filter.Tendsto (fun (x : \ud835\udd5c) => eval x P / eval x Q) Filter.atTop Filter.atTop","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/Basic.lean#L160-L170"},"Polynomial.div_tendsto_atTop_of_degree_gt":{"signature":"theorem Polynomial.div_tendsto_atTop_of_degree_gt {\ud835\udd5c : Type u_1} [NormedField \ud835\udd5c] [LinearOrder \ud835\udd5c] [IsStrictOrderedRing \ud835\udd5c] (P Q : Polynomial \ud835\udd5c) [OrderTopology \ud835\udd5c] (hdeg : Q.degree < P.degree) (hQ : Q \u2260 0) (hnng : 0 \u2264 P.leadingCoeff / Q.leadingCoeff) :Filter.Tendsto (fun (x : \ud835\udd5c) => eval x P / eval x Q) Filter.atTop Filter.atTop","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/Basic.lean#L172-L179"},"Polynomial.div_tendsto_atBot_of_degree_gt'":{"signature":"theorem Polynomial.div_tendsto_atBot_of_degree_gt' {\ud835\udd5c : Type u_1} [NormedField \ud835\udd5c] [LinearOrder \ud835\udd5c] [IsStrictOrderedRing \ud835\udd5c] (P Q : Polynomial \ud835\udd5c) [OrderTopology \ud835\udd5c] (hdeg : Q.degree < P.degree) (hneg : P.leadingCoeff / Q.leadingCoeff < 0) :Filter.Tendsto (fun (x : \ud835\udd5c) => eval x P / eval x Q) Filter.atTop Filter.atBot","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/Basic.lean#L181-L191"},"Polynomial.div_tendsto_atBot_of_degree_gt":{"signature":"theorem Polynomial.div_tendsto_atBot_of_degree_gt {\ud835\udd5c : Type u_1} [NormedField \ud835\udd5c] [LinearOrder \ud835\udd5c] [IsStrictOrderedRing \ud835\udd5c] (P Q : Polynomial \ud835\udd5c) [OrderTopology \ud835\udd5c] (hdeg : Q.degree < P.degree) (hQ : Q \u2260 0) (hnps : P.leadingCoeff / Q.leadingCoeff \u2264 0) :Filter.Tendsto (fun (x : \ud835\udd5c) => eval x P / eval x Q) Filter.atTop Filter.atBot","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/Basic.lean#L193-L200"},"Polynomial.abs_div_tendsto_atTop_of_degree_gt":{"signature":"theorem Polynomial.abs_div_tendsto_atTop_of_degree_gt {\ud835\udd5c : Type u_1} [NormedField \ud835\udd5c] [LinearOrder \ud835\udd5c] [IsStrictOrderedRing \ud835\udd5c] (P Q : Polynomial \ud835\udd5c) [OrderTopology \ud835\udd5c] (hdeg : Q.degree < P.degree) (hQ : Q \u2260 0) :Filter.Tendsto (fun (x : \ud835\udd5c) => |eval x P / eval x Q|) Filter.atTop Filter.atTop","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/Basic.lean#L202-L206"},"Polynomial.hermite_succ":{"signature":"theorem Polynomial.hermite_succ (n : \u2115) :hermite (n + 1) = X * hermite n - derivative (hermite n)","doc":"The recursion hermite (n+1) = (x - d/dx) (hermite n)","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Hermite/Basic.lean#L52-L55"},"Polynomial.coeff_hermite_succ_succ":{"signature":"theorem Polynomial.coeff_hermite_succ_succ (n k : \u2115) :(hermite (n + 1)).coeff (k + 1) = (hermite n).coeff k - (\u2191k + 2) * (hermite n).coeff (k + 2)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Hermite/Basic.lean#L78-L81"},"Polynomial.coeff_hermite_of_lt":{"signature":"theorem Polynomial.coeff_hermite_of_lt {n k : \u2115} (hnk : n < k) :(hermite n).coeff k = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Hermite/Basic.lean#L83-L90"},"Polynomial.coeff_hermite_self":{"signature":"theorem Polynomial.coeff_hermite_self (n : \u2115) :(hermite n).coeff n = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Hermite/Basic.lean#L92-L98"},"Polynomial.degree_hermite":{"signature":"theorem Polynomial.degree_hermite (n : \u2115) :(hermite n).degree = \u2191n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Hermite/Basic.lean#L100-L106"},"Polynomial.derivative_eval\u2082_C":{"signature":"theorem Polynomial.derivative_eval\u2082_C {R : Type u} [CommSemiring R] (p q : Polynomial R) :derivative (eval\u2082 C q p) = eval\u2082 C q (derivative p) * derivative q","doc":"Chain rule for formal derivative of polynomials.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L571-L579"},"Polynomial.derivative_expand":{"signature":"theorem Polynomial.derivative_expand {R : Type u} [CommSemiring R] (p : \u2115) (f : Polynomial R) :derivative ((expand R p) f) = (expand R p) (derivative f) * (\u2191p * X ^ (p - 1))","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Expand.lean#L86-L88"},"Polynomial.isUnit_or_eq_zero_of_separable_expand":{"signature":"theorem Polynomial.isUnit_or_eq_zero_of_separable_expand {F : Type u} [Field F] (p : \u2115) [HF : CharP F p] {f : Polynomial F} (n : \u2115) (hp : 0 < p) (hf : ((expand F (p ^ n)) f).Separable) :IsUnit f \u2228 n = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Separable.lean#L394-L403"},"Polynomial.derivative_prod":{"signature":"theorem Polynomial.derivative_prod {R : Type u} {\u03b9 : Type y} [CommSemiring R] [DecidableEq \u03b9] {s : Multiset \u03b9} {f : \u03b9 \u2192 Polynomial R} :derivative (Multiset.map f s).prod = (Multiset.map (fun (i : \u03b9) => (Multiset.map f (s.erase i)).prod * derivative (f i)) s).sum","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L581-L595"},"Polynomial.derivative_prod_finset":{"signature":"theorem Polynomial.derivative_prod_finset {R : Type u} {\u03b9 : Type y} [CommSemiring R] [DecidableEq \u03b9] {s : Finset \u03b9} {f : \u03b9 \u2192 Polynomial R} :derivative (\u220f b \u2208 s, f b) = \u2211 a \u2208 s, (\u220f b \u2208 s.erase a, f b) * derivative (f a)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L597-L600"},"Polynomial.isMonicOfDegree_X":{"signature":"theorem Polynomial.isMonicOfDegree_X (R : Type u_1) [Semiring R] [Nontrivial R] :X.IsMonicOfDegree 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/IsMonicOfDegree.lean#L166-L167"},"Polynomial.isMonicOfDegree_X_add_one":{"signature":"theorem Polynomial.isMonicOfDegree_X_add_one {R : Type u_1} [Semiring R] [Nontrivial R] (r : R) :(X + C r).IsMonicOfDegree 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/IsMonicOfDegree.lean#L176-L177"},"Polynomial.roots_C_mul_X_sub_C_of_IsUnit":{"signature":"theorem Polynomial.roots_C_mul_X_sub_C_of_IsUnit {R : Type u} [CommRing R] [IsDomain R] (b : R) (a : R\u02e3) :(C \u2191a * X - C b).roots = {\u2191a\u207b\u00b9 * b}","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L219-L224"},"Polynomial.roots_C_mul_X_add_C_of_IsUnit":{"signature":"theorem Polynomial.roots_C_mul_X_add_C_of_IsUnit {R : Type u} [CommRing R] [IsDomain R] (b : R) (a : R\u02e3) :(C \u2191a * X + C b).roots = {-(\u2191a\u207b\u00b9 * b)}","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L226-L229"},"Polynomial.sum_comp":{"signature":"theorem Polynomial.sum_comp {R : Type u} {\u03b9 : Type y} [Semiring R] (s : Finset \u03b9) (p : \u03b9 \u2192 Polynomial R) (q : Polynomial R) :(\u2211 i \u2208 s, p i).comp q = \u2211 i \u2208 s, (p i).comp q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L471-L472"},"Polynomial.smul_comp":{"signature":"theorem Polynomial.smul_comp {R : Type u} {S : Type v} [Semiring R] [SMulZeroClass S R] [IsScalarTower S R R] (s : S) (p q : Polynomial R) :(s \u2022 p).comp q = s \u2022 p.comp q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/SMul.lean#L73-L76"},"Polynomial.prod_comp":{"signature":"theorem Polynomial.prod_comp {R : Type u} [CommSemiring R] {\u03b9 : Type u_1} (s : Finset \u03b9) (p : \u03b9 \u2192 Polynomial R) (q : Polynomial R) :(\u220f j \u2208 s, p j).comp q = \u220f j \u2208 s, (p j).comp q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L673-L675"},"Polynomial.coeff_injective":{"signature":"theorem Polynomial.coeff_injective {R : Type u} [Semiring R] :Function.Injective coeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L570-L572"},"Polynomial.comp_C_mul_X_coeff":{"signature":"theorem Polynomial.comp_C_mul_X_coeff {R : Type u} [Semiring R] {p : Polynomial R} {r : R} {n : \u2115} :(p.comp (C r * X)).coeff n = p.coeff n * r ^ n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Degree.lean#L107-L114"},"Polynomial.comp_C_mul_X_eq_zero_iff":{"signature":"theorem Polynomial.comp_C_mul_X_eq_zero_iff {R : Type u} [Semiring R] {p : Polynomial R} {r : R} (hr : r \u2208 nonZeroDivisors R) :p.comp (C r * X) = 0 \u2194 p = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Degree.lean#L116-L120"},"Polynomial.toLaurentAlg_apply":{"signature":"theorem Polynomial.toLaurentAlg_apply {R : Type u_1} [CommSemiring R] (f : Polynomial R) :toLaurentAlg f = toLaurent f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Laurent.lean#L116-L117"},"Polynomial.toFn":{"signature":"def Polynomial.toFn {R : Type u_1} [Semiring R] (n : \u2115) :Polynomial R \u2192\u2097[R] Fin n \u2192 R","doc":"toFn n f is the vector of the first n coefficients of the polynomial f.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/OfFn.lean#L33-L34"},"Polynomial.zero_isRoot_iff_coeff_zero_eq_zero":{"signature":"theorem Polynomial.zero_isRoot_iff_coeff_zero_eq_zero {R : Type u} [Semiring R] {p : Polynomial R} :p.IsRoot 0 \u2194 p.coeff 0 = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Coeff.lean#L65-L66"},"Polynomial.zero_isRoot_of_coeff_zero_eq_zero":{"signature":"theorem Polynomial.zero_isRoot_of_coeff_zero_eq_zero {R : Type u} [Semiring R] {p : Polynomial R} :p.coeff 0 = 0 \u2192 p.IsRoot 0","doc":"Alias of the reverse direction of Polynomial.zero_isRoot_iff_coeff_zero_eq_zero.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Coeff.lean#L68-L69"},"Polynomial.exists_prod_multiset_X_sub_C_mul":{"signature":"theorem Polynomial.exists_prod_multiset_X_sub_C_mul {R : Type u} [CommRing R] [IsDomain R] (p : Polynomial R) :\u2203 (q : Polynomial R), (Multiset.map (fun (a : R) => X - C a) p.roots).prod * q = p \u2227 p.roots.card + q.natDegree = p.natDegree \u2227 q.roots = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L789-L805"},"Polynomial.eq_C_coeff_zero_iff_natDegree_eq_zero":{"signature":"theorem Polynomial.eq_C_coeff_zero_iff_natDegree_eq_zero {R : Type u} [Semiring R] {p : Polynomial R} :p = C (p.coeff 0) \u2194 p.natDegree = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L488-L489"},"Polynomial.mul_comp_neg_X":{"signature":"theorem Polynomial.mul_comp_neg_X {R : Type u_1} [Ring R] (p q : Polynomial R) :(p * q).comp (-X) = p.comp (-X) * q.comp (-X)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L454-L457"},"Polynomial.transcendental":{"signature":"theorem Polynomial.transcendental {R : Type u} [CommRing R] (f : Polynomial R) (hf : f.natDegree \u2260 0) (hf' : f.leadingCoeff \u2208 nonZeroDivisors R) :Transcendental R f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Algebraic/Basic.lean#L94-L97"},"Polynomial.trailingDegree_monomial":{"signature":"theorem Polynomial.trailingDegree_monomial {R : Type u} {a : R} {n : \u2115} [Semiring R] (ha : a \u2260 0) :((monomial n) a).trailingDegree = \u2191n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L169-L172"},"Polynomial.trailingDegree_C":{"signature":"theorem Polynomial.trailingDegree_C {R : Type u} {a : R} [Semiring R] (ha : a \u2260 0) :(C a).trailingDegree = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L186-L188"},"Polynomial.trailingDegree_one":{"signature":"theorem Polynomial.trailingDegree_one {R : Type u} [Semiring R] [Nontrivial R] :trailingDegree 1 = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L353-L355"},"Polynomial.contentIdeal":{"signature":"def Polynomial.contentIdeal {R : Type u_1} [Semiring R] (p : Polynomial R) :Ideal R","doc":"The content ideal of a polynomial p is the ideal generated by its coefficients.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ContentIdeal.lean#L51-L52"},"Polynomial.contentIdeal_def":{"signature":"theorem Polynomial.contentIdeal_def {R : Type u_1} [Semiring R] (p : Polynomial R) :p.contentIdeal = Ideal.span \u2191p.coeffs","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ContentIdeal.lean#L54-L54"},"Polynomial.contentIdeal_le_span_content":{"signature":"theorem Polynomial.contentIdeal_le_span_content {R : Type u_3} [CommRing R] [NormalizedGCDMonoid R] {p : Polynomial R} :p.contentIdeal \u2264 Ideal.span {p.content}","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ContentIdeal.lean#L194-L200"},"Polynomial.coeff_mem_contentIdeal":{"signature":"theorem Polynomial.coeff_mem_contentIdeal {R : Type u_1} [Semiring R] (p : Polynomial R) (n : \u2115) :p.coeff n \u2208 p.contentIdeal","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ContentIdeal.lean#L68-L72"},"Polynomial.degreeMonoidHom":{"signature":"def Polynomial.degreeMonoidHom {R : Type u} [Semiring R] [NoZeroDivisors R] [Nontrivial R] :Polynomial R \u2192* Multiplicative (WithBot \u2115)","doc":"degree as a monoid homomorphism between R[X] and Multiplicative (WithBot \u2115). This is useful to prove results about multiplication and degree.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L681-L686"},"Polynomial.degree_pow":{"signature":"theorem Polynomial.degree_pow {R : Type u} [Semiring R] [NoZeroDivisors R] [Nontrivial R] (p : Polynomial R) (n : \u2115) :(p ^ n).degree = n \u2022 p.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L688-L690"},"Polynomial.degree_list_sum_le_of_forall_degree_le":{"signature":"theorem Polynomial.degree_list_sum_le_of_forall_degree_le {S : Type u_1} [Semiring S] (l : List (Polynomial S)) (n : WithBot \u2115) (hl : \u2200 p \u2208 l, p.degree \u2264 n) :l.sum.degree \u2264 n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/BigOperators.lean#L81-L90"},"Polynomial.coe_mapAlgHom":{"signature":"theorem Polynomial.coe_mapAlgHom {R : Type u} {A : Type z} {B : Type u_2} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A \u2192\u2090[R] B) :\u21d1(mapAlgHom f) = map \u2191f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L171-L173"},"Polynomial.cyclotomic_mul_prime_pow_eq":{"signature":"theorem Polynomial.cyclotomic_mul_prime_pow_eq (R : Type u_1) {p m : \u2115} [Fact (Nat.Prime p)] [Ring R] [CharP R p] (hm : \u00acp \u2223 m) {k : \u2115} :0 < k \u2192 cyclotomic (p ^ k * m) R = cyclotomic m R ^ (p ^ k - p ^ (k - 1))","doc":"If R is of characteristic p and \u00acp \u2223 m, then cyclotomic (p ^ k * m) R = (cyclotomic m R) ^ (p ^ k - p ^ (k - 1)).","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Expand.lean#L155-L167"},"Polynomial.taylor_C":{"signature":"theorem Polynomial.taylor_C {R : Type u_1} [Semiring R] (r x : R) :(taylor r) (C x) = C x","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Taylor.lean#L52-L53"},"Polynomial.taylor_one":{"signature":"theorem Polynomial.taylor_one {R : Type u_1} [Semiring R] (r : R) :(taylor r) 1 = C 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Taylor.lean#L60-L61"},"Polynomial.taylor_mul":{"signature":"theorem Polynomial.taylor_mul {R : Type u_1} [CommSemiring R] (r : R) (p q : Polynomial R) :(taylor r) (p * q) = (taylor r) p * (taylor r) q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Taylor.lean#L142-L143"},"Polynomial.taylorAlgHom":{"signature":"def Polynomial.taylorAlgHom {R : Type u_1} [CommSemiring R] (r : R) :Polynomial R \u2192\u2090[R] Polynomial R","doc":"Polynomial.taylor as an AlgHom for commutative semirings","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Taylor.lean#L145-L148"},"Polynomial.coe_taylorAlgHom":{"signature":"theorem Polynomial.coe_taylorAlgHom {R : Type u_1} [CommSemiring R] (r : R) :\u2191(taylorAlgHom r) = taylor r","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Taylor.lean#L154-L155"},"Polynomial.roots_expand_pow_map_iterateFrobenius_le":{"signature":"theorem Polynomial.roots_expand_pow_map_iterateFrobenius_le {R : Type u_1} [CommRing R] [IsDomain R] (p n : \u2115) [ExpChar R p] (f : Polynomial R) :Multiset.map (\u21d1(iterateFrobenius R p n)) ((expand R (p ^ n)) f).roots \u2264 p ^ n \u2022 f.roots","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Perfect.lean#L308-L319"},"Polynomial.roots_expand_pow_image_iterateFrobenius_subset":{"signature":"theorem Polynomial.roots_expand_pow_image_iterateFrobenius_subset {R : Type u_1} [CommRing R] [IsDomain R] (p n : \u2115) [ExpChar R p] (f : Polynomial R) [DecidableEq R] :Finset.image (\u21d1(iterateFrobenius R p n)) ((expand R (p ^ n)) f).roots.toFinset \u2286 f.roots.toFinset","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Perfect.lean#L326-L330"},"Polynomial.rootsExpandPowToRoots":{"signature":"noncomputable def Polynomial.rootsExpandPowToRoots {R : Type u_1} [CommRing R] [IsDomain R] (p n : \u2115) [ExpChar R p] (f : Polynomial R) [DecidableEq R] :\u21a5((expand R (p ^ n)) f).roots.toFinset \u21aa \u21a5f.roots.toFinset","doc":"If f is a polynomial over an integral domain R of characteristic p, then there is a map from the set of roots of Polynomial.expand R (p ^ n) f to the set of roots of f. It's given by x \u21a6 x ^ (p ^ n), see rootsExpandPowToRoots_apply.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Perfect.lean#L406-L413"},"Polynomial.eq_mul_mul_of_roots_quadratic_eq_pair":{"signature":"theorem Polynomial.eq_mul_mul_of_roots_quadratic_eq_pair {R : Type u_1} [CommRing R] [IsDomain R] {a b c x1 x2 : R} (hroots : (C a * X ^ 2 + C b * X + C c).roots = {x1, x2}) :c = a * x1 * x2","doc":"Vieta's formula for quadratics.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/SmallDegreeVieta.lean#L41-L51"},"Polynomial.roots_quadratic_eq_pair_iff_of_ne_zero":{"signature":"theorem Polynomial.roots_quadratic_eq_pair_iff_of_ne_zero {R : Type u_1} [CommRing R] [IsDomain R] {a b c x1 x2 : R} (ha : a \u2260 0) :(C a * X ^ 2 + C b * X + C c).roots = {x1, x2} \u2194 b = -a * (x1 + x2) \u2227 c = a * x1 * x2","doc":"Vieta's formula for quadratics as an iff.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/SmallDegreeVieta.lean#L70-L83"},"Polynomial.aroots_quadratic_eq_pair_iff_of_ne_zero":{"signature":"theorem Polynomial.aroots_quadratic_eq_pair_iff_of_ne_zero {T : Type u_2} {S : Type u_3} [CommRing T] [CommRing S] [IsDomain S] [Algebra T S] {a b c : T} {x1 x2 : S} (ha : (algebraMap T S) a \u2260 0) :(C a * X ^ 2 + C b * X + C c).aroots S = {x1, x2} \u2194 (algebraMap T S) b = -(algebraMap T S) a * (x1 + x2) \u2227 (algebraMap T S) c = (algebraMap T S) a * x1 * x2","doc":"Vieta's formula for quadratics as an iff (aroots version).","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/SmallDegreeVieta.lean#L85-L93"},"Polynomial.tendsto_atTop_of_leadingCoeff_nonneg":{"signature":"theorem Polynomial.tendsto_atTop_of_leadingCoeff_nonneg {\ud835\udd5c : Type u_1} [NormedField \ud835\udd5c] [LinearOrder \ud835\udd5c] [IsStrictOrderedRing \ud835\udd5c] (P : Polynomial \ud835\udd5c) [OrderTopology \ud835\udd5c] (hdeg : 0 < P.degree) (hnng : 0 \u2264 P.leadingCoeff) :Filter.Tendsto (fun (x : \ud835\udd5c) => eval x P) Filter.atTop Filter.atTop","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/Basic.lean#L57-L61"},"Polynomial.tendsto_atTop_iff_leadingCoeff_nonneg":{"signature":"theorem Polynomial.tendsto_atTop_iff_leadingCoeff_nonneg {\ud835\udd5c : Type u_1} [NormedField \ud835\udd5c] [LinearOrder \ud835\udd5c] [IsStrictOrderedRing \ud835\udd5c] (P : Polynomial \ud835\udd5c) [OrderTopology \ud835\udd5c] :Filter.Tendsto (fun (x : \ud835\udd5c) => eval x P) Filter.atTop Filter.atTop \u2194 0 < P.degree \u2227 0 \u2264 P.leadingCoeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/Basic.lean#L63-L69"},"Polynomial.tendsto_atBot_iff_leadingCoeff_nonpos":{"signature":"theorem Polynomial.tendsto_atBot_iff_leadingCoeff_nonpos {\ud835\udd5c : Type u_1} [NormedField \ud835\udd5c] [LinearOrder \ud835\udd5c] [IsStrictOrderedRing \ud835\udd5c] (P : Polynomial \ud835\udd5c) [OrderTopology \ud835\udd5c] :Filter.Tendsto (fun (x : \ud835\udd5c) => eval x P) Filter.atTop Filter.atBot \u2194 0 < P.degree \u2227 P.leadingCoeff \u2264 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/Basic.lean#L71-L74"},"Polynomial.tendsto_atBot_of_leadingCoeff_nonpos":{"signature":"theorem Polynomial.tendsto_atBot_of_leadingCoeff_nonpos {\ud835\udd5c : Type u_1} [NormedField \ud835\udd5c] [LinearOrder \ud835\udd5c] [IsStrictOrderedRing \ud835\udd5c] (P : Polynomial \ud835\udd5c) [OrderTopology \ud835\udd5c] (hdeg : 0 < P.degree) (hnps : P.leadingCoeff \u2264 0) :Filter.Tendsto (fun (x : \ud835\udd5c) => eval x P) Filter.atTop Filter.atBot","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/Basic.lean#L76-L78"},"Polynomial.map_taylor":{"signature":"theorem Polynomial.map_taylor {R : Type u_2} {S : Type u_3} [Semiring R] [Semiring S] (p : Polynomial R) (r : R) (f : R \u2192+* S) :map f ((taylor r) p) = (taylor (f r)) (map f p)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Taylor.lean#L120-L122"},"Polynomial.sub_dvd_eval_sub":{"signature":"theorem Polynomial.sub_dvd_eval_sub {R : Type u} [CommRing R] (a b : R) (p : Polynomial R) :a - b \u2223 eval a p - eval b p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L615-L619"},"Polynomial.isIntegral_coeff_of_factors":{"signature":"theorem Polynomial.isIntegral_coeff_of_factors {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [Algebra R S] (p : Polynomial S) (hpmon : IsIntegral R p.leadingCoeff) (hp : p.Splits) (hpr : \u2200 (x : S), p.IsRoot x \u2192 IsIntegral R x) (i : \u2115) :IsIntegral R (p.coeff i)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/IsIntegral.lean#L44-L58"},"Polynomial.isIntegral_coeff_of_dvd":{"signature":"theorem Polynomial.isIntegral_coeff_of_dvd {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [Algebra R S] (p : Polynomial R) (q : Polynomial S) (hp : p.Monic) (hq : q.Monic) (H : q \u2223 map (algebraMap R S) p) (i : \u2115) :IsIntegral R (q.coeff i)","doc":"Stacks Tag 00H6 ((1))","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/IsIntegral.lean#L61-L72"},"Polynomial.ne_zero_of_ne_zero_of_monic":{"signature":"theorem Polynomial.ne_zero_of_ne_zero_of_monic {R : Type u} [Semiring R] {p q r : Polynomial R} (hp : p.Monic) (hne : q \u2260 r) :p \u2260 0","doc":"Alias of Polynomial.Monic.ne_zero_of_polynomial_ne.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Monic.lean#L56-L57"},"Polynomial.aeval_pi":{"signature":"theorem Polynomial.aeval_pi {R : Type u} [CommSemiring R] {I : Type u_3} {A : I \u2192 Type u_4} [(i : I) \u2192 Semiring (A i)] [(i : I) \u2192 Algebra R (A i)] (x : (i : I) \u2192 A i) :aeval x = Pi.algHom R A fun (i : I) => aeval (x i)","doc":"Polynomial evaluation on an indexed tuple is the indexed product of the evaluations on the components. Generalizes Polynomial.aeval_prod to indexed products.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L433-L438"},"Polynomial.map_rootOfSplits":{"signature":"theorem Polynomial.map_rootOfSplits {R : Type u_1} [CommRing R] {f : Polynomial R} (hf : f.Splits) (hfd : f.degree \u2260 0) :eval (rootOfSplits hf hfd) f = 0","doc":"Alias of Polynomial.eval_rootOfSplits.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L775-L776"},"Polynomial.natDegree_sum_le_of_forall_le":{"signature":"theorem Polynomial.natDegree_sum_le_of_forall_le {\u03b9 : Type w} (s : Finset \u03b9) {S : Type u_1} [Semiring S] {n : \u2115} (f : \u03b9 \u2192 Polynomial S) (h : \u2200 i \u2208 s, (f i).natDegree \u2264 n) :(\u2211 i \u2208 s, f i).natDegree \u2264 n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/BigOperators.lean#L63-L65"},"Polynomial.coeff_mem_pow_of_mem_adjoin_C_mul_X":{"signature":"theorem Polynomial.coeff_mem_pow_of_mem_adjoin_C_mul_X {R : Type u_3} [CommRing R] {I : Ideal R} {P : Polynomial R} (hP : P \u2208 Algebra.adjoin R {x : Polynomial R | \u2203 r \u2208 I, C r * X = x}) (i : \u2115) :P.coeff i \u2208 I ^ i","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/IntegralClosure/Algebra/Ideal.lean#L32-L45"},"Polynomial.exists_monic_aeval_eq_zero_forall_mem_pow_of_isIntegral":{"signature":"theorem Polynomial.exists_monic_aeval_eq_zero_forall_mem_pow_of_isIntegral {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [Algebra R S] {I : Ideal R} {x : S} (hx : IsIntegral (\u21a5(Algebra.adjoin R {x : Polynomial R | \u2203 r \u2208 I, C r * X = x})) (C x * X)) :\u2203 (p : Polynomial R), p.Monic \u2227 (aeval x) p = 0 \u2227 \u2200 (i : \u2115), p.coeff i \u2208 I ^ (p.natDegree - i)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/IntegralClosure/Algebra/Ideal.lean#L48-L71"},"Polynomial.exists_monic_aeval_eq_zero_forall_mem_pow_of_mem_map":{"signature":"theorem Polynomial.exists_monic_aeval_eq_zero_forall_mem_pow_of_mem_map {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [Algebra R S] [Algebra.IsIntegral R S] {I : Ideal R} {x : S} (hx : x \u2208 Ideal.map (algebraMap R S) I) :\u2203 (p : Polynomial R), p.Monic \u2227 (aeval x) p = 0 \u2227 \u2200 (i : \u2115), p.coeff i \u2208 I ^ (p.natDegree - i)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/IntegralClosure/Algebra/Ideal.lean#L73-L90"},"Polynomial.exists_monic_aeval_eq_zero_forall_mem_of_mem_map":{"signature":"theorem Polynomial.exists_monic_aeval_eq_zero_forall_mem_of_mem_map {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [Algebra R S] [Algebra.IsIntegral R S] {I : Ideal R} {x : S} (hx : x \u2208 Ideal.map (algebraMap R S) I) :\u2203 (p : Polynomial R), p.Monic \u2227 (aeval x) p = 0 \u2227 \u2200 (i : \u2115), i \u2260 p.natDegree \u2192 p.coeff i \u2208 I","doc":"Stacks Tag 00H5","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/IntegralClosure/Algebra/Ideal.lean#L92-L100"},"Polynomial.coeffs_empty_iff":{"signature":"theorem Polynomial.coeffs_empty_iff {R : Type u} [Semiring R] {p : Polynomial R} :p.coeffs = \u2205 \u2194 p = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L1035-L1043"},"Polynomial.coeffs_nonempty_iff":{"signature":"theorem Polynomial.coeffs_nonempty_iff {R : Type u} [Semiring R] {p : Polynomial R} :p.coeffs.Nonempty \u2194 p \u2260 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L1045-L1047"},"Polynomial.homogenize":{"signature":"noncomputable def Polynomial.homogenize {R : Type u_1} [CommSemiring R] (p : Polynomial R) (n : \u2115) :MvPolynomial (Fin 2) R","doc":"Given a polynomial p and a number n \u2265 natDegree p, returns a homogeneous bivariate polynomial q of degree n such that q(x, 1) = p(x). It is defined as \u2211 k + l = n, a_k X_0^k X_1^l, where a_k is the kth coefficient of p.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Homogenize.lean#L36-L41"},"Polynomial.homogenize_zero":{"signature":"theorem Polynomial.homogenize_zero {R : Type u_1} [CommSemiring R] (n : \u2115) :homogenize 0 n = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Homogenize.lean#L43-L45"},"Polynomial.homogenize_smul":{"signature":"theorem Polynomial.homogenize_smul {R : Type u_1} [CommSemiring R] {S : Type u_2} [Semiring S] [Module S R] (c : S) (p : Polynomial R) (n : \u2115) :(c \u2022 p).homogenize n = c \u2022 p.homogenize n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Homogenize.lean#L52-L55"},"Polynomial.homogenize_C_mul":{"signature":"theorem Polynomial.homogenize_C_mul {R : Type u_1} [CommSemiring R] (c : R) (p : Polynomial R) (n : \u2115) :(C c * p).homogenize n = MvPolynomial.C c * p.homogenize n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Homogenize.lean#L73-L76"},"Polynomial.homogenize_monomial":{"signature":"theorem Polynomial.homogenize_monomial {R : Type u_1} [CommSemiring R] {m n : \u2115} (h : m \u2264 n) (r : R) :((monomial m) r).homogenize n = (MvPolynomial.monomial fun\u2080 | 0 => m | 1 => n - m) r","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Homogenize.lean#L78-L84"},"Polynomial.homogenize_X_pow":{"signature":"theorem Polynomial.homogenize_X_pow {R : Type u_1} [CommSemiring R] {m n : \u2115} (h : m \u2264 n) :(X ^ m).homogenize n = MvPolynomial.X 0 ^ m * MvPolynomial.X 1 ^ (n - m)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Homogenize.lean#L92-L96"},"Polynomial.homogenize_add":{"signature":"theorem Polynomial.homogenize_add {R : Type u_1} [CommSemiring R] (p q : Polynomial R) (n : \u2115) :(p + q).homogenize n = p.homogenize n + q.homogenize n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Homogenize.lean#L47-L50"},"Polynomial.eval\u2082_homogenize_of_eq_one":{"signature":"theorem Polynomial.eval\u2082_homogenize_of_eq_one {R : Type u_1} [CommSemiring R] {S : Type u_2} [CommSemiring S] {p : Polynomial R} {n : \u2115} (hn : p.natDegree \u2264 n) (f : R \u2192+* S) (g : Fin 2 \u2192 S) (hg : g 1 = 1) :MvPolynomial.eval\u2082 f g (p.homogenize n) = eval\u2082 f (g 0) p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Homogenize.lean#L123-L131"},"Polynomial.aeval_homogenize_of_eq_one":{"signature":"theorem Polynomial.aeval_homogenize_of_eq_one {R : Type u_1} [CommSemiring R] {A : Type u_2} [CommSemiring A] [Algebra R A] {p : Polynomial R} {n : \u2115} (hn : p.natDegree \u2264 n) (g : Fin 2 \u2192 A) (hg : g 1 = 1) :(MvPolynomial.aeval g) (p.homogenize n) = (aeval (g 0)) p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Homogenize.lean#L133-L136"},"Polynomial.aeval_homogenize_X_one":{"signature":"theorem Polynomial.aeval_homogenize_X_one {R : Type u_1} [CommSemiring R] (p : Polynomial R) {n : \u2115} (hn : p.natDegree \u2264 n) :(MvPolynomial.aeval ![X, 1]) (p.homogenize n) = p","doc":"If deg p \u2264 n, then homogenize p n (x, 1) = p x.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Homogenize.lean#L138-L142"},"Polynomial.splits_of_natDegree_le_one_of_invertible":{"signature":"theorem Polynomial.splits_of_natDegree_le_one_of_invertible {R : Type u_1} [Semiring R] {f : Polynomial R} (hf : f.natDegree \u2264 1) (h : Invertible f.leadingCoeff) :f.Splits","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L102-L109"},"Polynomial.eval\u2082_mul_C'":{"signature":"theorem Polynomial.eval\u2082_mul_C' {R : Type u} {S : Type v} {a : R} [Semiring R] {p : Polynomial R} [Semiring S] (f : R \u2192+* S) (x : S) (h : Commute (f a) x) :eval\u2082 f x (p * C a) = eval\u2082 f x p * f a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L158-L163"},"Polynomial.hasseDeriv_comp":{"signature":"theorem Polynomial.hasseDeriv_comp {R : Type u_1} [Semiring R] (k l : \u2115) :hasseDeriv k \u2218\u2097 hasseDeriv l = (k + l).choose k \u2022 hasseDeriv (k + l)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/HasseDeriv.lean#L149-L172"},"Polynomial.span_minpoly_eq_annihilator":{"signature":"theorem Polynomial.span_minpoly_eq_annihilator (\ud835\udd5c : Type u_1) [Field \ud835\udd5c] {M : Type u_3} [AddCommGroup M] [Module \ud835\udd5c M] (f : Module.End \ud835\udd5c M) :Ideal.span {minpoly \ud835\udd5c f} = Module.annihilator (Polynomial \ud835\udd5c) (Module.AEval' f)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/LinearAlgebra/AnnihilatingPolynomial.lean#L166-L169"},"Polynomial.eraseLead_add_of_degree_lt_right":{"signature":"theorem Polynomial.eraseLead_add_of_degree_lt_right {R : Type u_1} [Semiring R] {p q : Polynomial R} (pq : p.degree < q.degree) :(p + q).eraseLead = p + q.eraseLead","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/EraseLead.lean#L171-L179"},"Polynomial.eraseLead_add_of_natDegree_lt_right":{"signature":"theorem Polynomial.eraseLead_add_of_natDegree_lt_right {R : Type u_1} [Semiring R] {p q : Polynomial R} (pq : p.natDegree < q.natDegree) :(p + q).eraseLead = p + q.eraseLead","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/EraseLead.lean#L181-L183"},"Polynomial.cyclotomic_irreducible_of_irreducible_pow":{"signature":"theorem Polynomial.cyclotomic_irreducible_of_irreducible_pow {p : \u2115} (hp : Nat.Prime p) {R : Type u_1} [CommRing R] [IsDomain R] {n : \u2115} (hn : n \u2260 0) (h : Irreducible (cyclotomic (p ^ n) R)) :Irreducible (cyclotomic p R)","doc":"If Irreducible (cyclotomic (p ^ n) R) then Irreducible (cyclotomic p R).","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Expand.lean#L122-L126"},"Polynomial.natDegree_mul_C_le":{"signature":"theorem Polynomial.natDegree_mul_C_le {R : Type u} [Semiring R] (f : Polynomial R) (a : R) :(f * C a).natDegree \u2264 f.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L99-L100"},"Polynomial.natDegree_mul_C_eq_of_mul_eq_one":{"signature":"theorem Polynomial.natDegree_mul_C_eq_of_mul_eq_one {R : Type u} {a : R} [Semiring R] {p : Polynomial R} {ai : R} (au : a * ai = 1) :(p * C a).natDegree = p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L114-L120"},"Polynomial.natDegree_mod_lt":{"signature":"theorem Polynomial.natDegree_mod_lt {k : Type y} [Field k] (p : Polynomial k) {q : Polynomial k} (hq : q.natDegree \u2260 0) :(p % q).natDegree < q.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L422-L435"},"Polynomial.dvd_pow_natDegree_of_aeval_eq_zero":{"signature":"theorem Polynomial.dvd_pow_natDegree_of_aeval_eq_zero {R : Type u} {A : Type u_1} [CommRing R] [CommRing A] [IsDomain R] [Algebra R A] [Nontrivial A] [Module.IsTorsionFree R A] {p : Polynomial R} (hp : p.Monic) (x y : R) (z : A) (h : (aeval z) p = 0) (hz : z * (algebraMap R A) x = (algebraMap R A) y) :x \u2223 y ^ p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Eisenstein/Basic.lean#L189-L193"},"Polynomial.wronskian_zero_right":{"signature":"theorem Polynomial.wronskian_zero_right {R : Type u_1} [CommRing R] (a : Polynomial R) :a.wronskian 0 = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Wronskian.lean#L58-L59"},"Polynomial.cardPowDegree_apply":{"signature":"theorem Polynomial.cardPowDegree_apply {Fq : Type u_1} [Field Fq] [Fintype Fq] [DecidableEq Fq] (p : Polynomial Fq) :cardPowDegree p = if p = 0 then 0 else \u2191(Fintype.card Fq) ^ p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/CardPowDegree.lean#L73-L75"},"Polynomial.derivative_comp":{"signature":"theorem Polynomial.derivative_comp {R : Type u} [CommSemiring R] (p q : Polynomial R) :derivative (p.comp q) = derivative q * (derivative p).comp q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L563-L569"},"Polynomial.comp_eq_aeval":{"signature":"theorem Polynomial.comp_eq_aeval {R : Type u} [CommSemiring R] {p q : Polynomial R} :p.comp q = (aeval q) p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L302-L302"},"Polynomial.eval\u2082_algebraMap_X":{"signature":"theorem Polynomial.eval\u2082_algebraMap_X {R : Type u_3} {A : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] (p : Polynomial R) (f : Polynomial R \u2192\u2090[R] A) :eval\u2082 (algebraMap R A) (f X) p = f p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L135-L141"},"Polynomial.scaleRoots_zero":{"signature":"theorem Polynomial.scaleRoots_zero {R : Type u_1} [Semiring R] (p : Polynomial R) :p.scaleRoots 0 = p.leadingCoeff \u2022 X ^ p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ScaleRoots.lean#L103-L113"},"Polynomial.scaleRoots_eval\u2082_eq_zero_of_eval\u2082_div_eq_zero":{"signature":"theorem Polynomial.scaleRoots_eval\u2082_eq_zero_of_eval\u2082_div_eq_zero {S : Type u_2} {K : Type u_4} [Semiring S] [Field K] {p : Polynomial S} {f : S \u2192+* K} (hf : Function.Injective \u21d1f) {r s : S} (hr : eval\u2082 f (f r / f s) p = 0) (hs : s \u2208 nonZeroDivisors S) :eval\u2082 f (f r) (p.scaleRoots s) = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ScaleRoots.lean#L167-L174"},"Polynomial.trailingDegree_X":{"signature":"theorem Polynomial.trailingDegree_X {R : Type u} [Semiring R] [Nontrivial R] :X.trailingDegree = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L357-L359"},"Polynomial.map_smul":{"signature":"theorem Polynomial.map_smul {R : Type u} {S : Type v} [Semiring R] {p : Polynomial R} [Semiring S] (f : R \u2192+* S) (r : R) :map f (r \u2022 p) = f r \u2022 map f p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/SMul.lean#L85-L87"},"Polynomial.natDegree_le_pred":{"signature":"theorem Polynomial.natDegree_le_pred {R : Type u} {n : \u2115} [Semiring R] {p : Polynomial R} (hf : p.natDegree \u2264 n) (hn : p.coeff n = 0) :p.natDegree \u2264 n - 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Monomial.lean#L32-L37"},"Polynomial.nextCoeff_eq_zero":{"signature":"theorem Polynomial.nextCoeff_eq_zero {R : Type u} [Semiring R] {p : Polynomial R} :p.nextCoeff = 0 \u2194 p.natDegree = 0 \u2228 0 < p.natDegree \u2227 p.coeff (p.natDegree - 1) = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L309-L311"},"Polynomial.eraseLead_C":{"signature":"theorem Polynomial.eraseLead_C {R : Type u_1} [Semiring R] (r : R) :(C r).eraseLead = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/EraseLead.lean#L138-L140"},"Polynomial.natDegree_eraseLead_le_of_nextCoeff_eq_zero":{"signature":"theorem Polynomial.natDegree_eraseLead_le_of_nextCoeff_eq_zero {R : Type u_1} [Semiring R] {f : Polynomial R} (h : f.nextCoeff = 0) :f.eraseLead.natDegree \u2264 f.natDegree - 2","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/EraseLead.lean#L229-L236"},"Polynomial.toFinsuppIso_symm_apply":{"signature":"theorem Polynomial.toFinsuppIso_symm_apply (R : Type u) [Semiring R] (toFinsupp : AddMonoidAlgebra R \u2115) :(toFinsuppIso R).symm toFinsupp = { toFinsupp := toFinsupp }","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L327-L327"},"Polynomial.mem_roots_sub_C'":{"signature":"theorem Polynomial.mem_roots_sub_C' {R : Type u} [CommRing R] [IsDomain R] {p : Polynomial R} {a x : R} :x \u2208 (p - C a).roots \u2194 p \u2260 C a \u2227 eval x p = a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L169-L170"},"Polynomial.mem_roots_sub_C":{"signature":"theorem Polynomial.mem_roots_sub_C {R : Type u} [CommRing R] [IsDomain R] {p : Polynomial R} {a x : R} (hp0 : 0 < p.degree) :x \u2208 (p - C a).roots \u2194 eval x p = a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L172-L174"},"Polynomial.degree_add_C":{"signature":"theorem Polynomial.degree_add_C {R : Type u} {a : R} [Semiring R] {p : Polynomial R} (hp : 0 < p.degree) :(p + C a).degree = p.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L200-L201"},"Polynomial.card_roots_sub_C":{"signature":"theorem Polynomial.card_roots_sub_C {R : Type u} [CommRing R] [IsDomain R] {p : Polynomial R} {a : R} (hp0 : 0 < p.degree) :\u2191(p - C a).roots.card \u2264 p.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L84-L89"},"Polynomial.card_roots_sub_C'":{"signature":"theorem Polynomial.card_roots_sub_C' {R : Type u} [CommRing R] [IsDomain R] {p : Polynomial R} {a : R} (hp0 : 0 < p.degree) :(p - C a).roots.card \u2264 p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L91-L95"},"Polynomial.monomial_mul_C":{"signature":"theorem Polynomial.monomial_mul_C {R : Type u} {a b : R} {n : \u2115} [Semiring R] :(monomial n) a * C b = (monomial n) (a * b)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L476-L478"},"Polynomial.toMvPolynomial_eq_rename_comp":{"signature":"theorem Polynomial.toMvPolynomial_eq_rename_comp {R : Type u_1} {\u03c3 : Type u_3} [CommSemiring R] (i : \u03c3) :toMvPolynomial i = (MvPolynomial.rename fun (x : Unit) => i).comp \u2191(MvPolynomial.pUnitAlgEquiv R).symm","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/MvPolynomial/Equiv.lean#L787-L791"},"Polynomial.mem_map_range":{"signature":"theorem Polynomial.mem_map_range {R : Type u_1} {S : Type u_2} [Ring R] [Ring S] (f : R \u2192+* S) {p : Polynomial S} :p \u2208 (mapRingHom f).range \u2194 \u2200 (n : \u2115), p.coeff n \u2208 f.range","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Subring.lean#L46-L48"},"Polynomial.splits_id_of_splits":{"signature":"theorem Polynomial.splits_id_of_splits {R : Type u_1} [CommRing R] {f : Polynomial R} {S : Type u_4} [CommRing S] [IsDomain S] [IsSimpleRing R] (i : R \u2192+* S) (hf : (map i f).Splits) (hi : \u2200 a \u2208 (map i f).roots, a \u2208 i.range) :f.Splits","doc":"Alias of Polynomial.Splits.of_splits_map.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L851-L852"},"Polynomial.mapAlg":{"signature":"def Polynomial.mapAlg (R : Type u) [CommSemiring R] (S : Type v) [Semiring S] [Algebra R S] :Polynomial R \u2192\u2090[R] Polynomial S","doc":"The map R[X] \u2192 S[X] as an algebra homomorphism.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L249-L252"},"Polynomial.mapAlg_eq_map":{"signature":"theorem Polynomial.mapAlg_eq_map {R : Type u} [CommSemiring R] (S : Type v) [Semiring S] [Algebra R S] (p : Polynomial R) :(mapAlg R S) p = map (algebraMap R S) p","doc":"mapAlg is the morphism induced by R \u2192 S.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L267-L270"},"Polynomial.mapAlg_comp":{"signature":"theorem Polynomial.mapAlg_comp {A : Type u_3} (B : Type u_4) (C : Type u_5) [CommSemiring A] [CommSemiring B] [Semiring C] [Algebra A B] [Algebra A C] [Algebra B C] [IsScalarTower A B C] (p : Polynomial A) :(mapAlg A C) p = (mapAlg B C) ((mapAlg A B) p)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L313-L314"},"Polynomial.coeffList":{"signature":"def Polynomial.coeffList {R : Type u_1} [Semiring R] (P : Polynomial R) :List R","doc":"The list of coefficients starting from the leading term down to the constant term.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/CoeffList.lean#L48-L50"},"Polynomial.signVariations":{"signature":"def Polynomial.signVariations {R : Type u_1} [Semiring R] [LinearOrder R] (P : Polynomial R) :\u2115","doc":"Counts the number of times that the coefficients in a polynomial change sign, with the convention that 0 can count as either sign.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/RuleOfSigns.lean#L48-L53"},"Polynomial.coeffList_zero":{"signature":"theorem Polynomial.coeffList_zero (R : Type u_1) [Semiring R] :coeffList 0 = []","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/CoeffList.lean#L55-L57"},"Polynomial.signVariations_zero":{"signature":"theorem Polynomial.signVariations_zero (R : Type u_1) [Semiring R] [LinearOrder R] :signVariations 0 = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/RuleOfSigns.lean#L56-L58"},"Polynomial.coeffList_C":{"signature":"theorem Polynomial.coeffList_C {R : Type u_1} [Semiring R] {x : R} (h : x \u2260 0) :(C x).coeffList = [x]","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/CoeffList.lean#L64-L66"},"Polynomial.withBotSucc_degree_eq_natDegree_add_one":{"signature":"theorem Polynomial.withBotSucc_degree_eq_natDegree_add_one {R : Type u} [Semiring R] {p : Polynomial R} (h : p \u2260 0) :p.degree.succ = p.natDegree + 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L249-L251"},"Polynomial.length_coeffList_eq_ite":{"signature":"theorem Polynomial.length_coeffList_eq_ite {R : Type u_1} [Semiring R] [DecidableEq R] (P : Polynomial R) :P.coeffList.length = if P = 0 then 0 else P.natDegree + 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/CoeffList.lean#L86-L89"},"Polynomial.coeffList_eq_cons_leadingCoeff":{"signature":"theorem Polynomial.coeffList_eq_cons_leadingCoeff {R : Type u_1} [Semiring R] {P : Polynomial R} (h : P \u2260 0) :\u2203 (ls : List R), P.coeffList = P.leadingCoeff :: ls","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/CoeffList.lean#L68-L70"},"Polynomial.natDegree_monomial_eq":{"signature":"theorem Polynomial.natDegree_monomial_eq {R : Type u} [Semiring R] (i : \u2115) {r : R} (r0 : r \u2260 0) :((monomial i) r).natDegree = i","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L233-L235"},"Polynomial.coeff_monomial_of_ne":{"signature":"theorem Polynomial.coeff_monomial_of_ne {R : Type u} [Semiring R] {m n : \u2115} (c : R) (h : m \u2260 n) :((monomial n) c).coeff m = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L590-L591"},"Polynomial.coeffList_monomial":{"signature":"theorem Polynomial.coeffList_monomial {R : Type u_1} [Semiring R] {x : R} (hx : x \u2260 0) (n : \u2115) :((monomial n) x).coeffList = x :: List.replicate n 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/CoeffList.lean#L99-L110"},"Polynomial.coeffList_eraseLead":{"signature":"theorem Polynomial.coeffList_eraseLead {R : Type u_1} [Semiring R] {P : Polynomial R} (h : P \u2260 0) :P.coeffList = P.leadingCoeff :: (List.replicate (P.natDegree - P.eraseLead.degree.succ) 0 ++ P.eraseLead.coeffList)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/CoeffList.lean#L114-L158"},"Polynomial.coeffList_eq_nil":{"signature":"theorem Polynomial.coeffList_eq_nil {R : Type u_1} [Semiring R] {P : Polynomial R} :P.coeffList = [] \u2194 P = 0","doc":"Only the zero polynomial has no coefficients.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/CoeffList.lean#L59-L62"},"Polynomial.signVariations_eq_eraseLead_add_ite":{"signature":"theorem Polynomial.signVariations_eq_eraseLead_add_ite {R : Type u_1} [Semiring R] [LinearOrder R] {P : Polynomial R} (h : P \u2260 0) :P.signVariations = P.eraseLead.signVariations + if SignType.sign P.leadingCoeff = -SignType.sign P.eraseLead.leadingCoeff then 1 else 0","doc":"If we drop the leading coefficient, the sign changes drop by 0 or 1 depending on whether the first two nonzero coefficients match.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/RuleOfSigns.lean#L77-L109"},"Polynomial.signVariations_le_eraseLead_succ":{"signature":"theorem Polynomial.signVariations_le_eraseLead_succ {R : Type u_1} [Semiring R] [LinearOrder R] (P : Polynomial R) :P.signVariations \u2264 P.eraseLead.signVariations + 1","doc":"We can only lose at most one sign changes if we drop the leading coefficient.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/RuleOfSigns.lean#L117-L121"},"Polynomial.valuation_monomial_eq_valuation_X_pow":{"signature":"theorem Polynomial.valuation_monomial_eq_valuation_X_pow (K : Type u_1) [Field K] {\u0393 : Type u_2} [LinearOrderedCommGroupWithZero \u0393] {v : Valuation (RatFunc K) \u0393} (hv : \u2200 (a : K), a \u2260 0 \u2192 v \u2191(C a) = 1) (n : \u2115) {a : K} (ha : a \u2260 0) :v \u2191((monomial n) a) = v RatFunc.X ^ n","doc":"If a valuation v is trivial on constants then for every n : \u2115 the valuation of (monomial n a) is equal to (v RatFunc.X) ^ n.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/RatFunc/AsPolynomial.lean#L301-L305"},"Polynomial.valuation_inv_monomial_eq_valuation_X_zpow":{"signature":"theorem Polynomial.valuation_inv_monomial_eq_valuation_X_zpow (K : Type u_1) [Field K] {\u0393 : Type u_2} [LinearOrderedCommGroupWithZero \u0393] {v : Valuation (RatFunc K) \u0393} (hv : \u2200 (a : K), a \u2260 0 \u2192 v \u2191(C a) = 1) (n : \u2115) {a : K} (ha : a \u2260 0) :v (1 / \u2191((monomial n) a)) = v RatFunc.X ^ (-\u2191n)","doc":"If a valuation v is trivial on constants then for every n : \u2115 the valuation of 1 / (monomial n a) (as an element of the field of rational functions) is equal to (v RatFunc.X) ^ (- n).","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/RatFunc/AsPolynomial.lean#L331-L336"},"Polynomial.natSepDegree_map":{"signature":"theorem Polynomial.natSepDegree_map {E : Type v} [Field E] (K : Type w) [Field K] (f : Polynomial E) (i : E \u2192+* K) :(map i f).natSepDegree = f.natSepDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/SeparableDegree.lean#L382-L386"},"Polynomial.natTrailingDegree_mem_support_of_nonzero":{"signature":"theorem Polynomial.natTrailingDegree_mem_support_of_nonzero {R : Type u} [Semiring R] {p : Polynomial R} :p \u2260 0 \u2192 p.natTrailingDegree \u2208 p.support","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L256-L257"},"Polynomial.iterate_derivative_C":{"signature":"theorem Polynomial.iterate_derivative_C {R : Type u} {a : R} [Semiring R] {k : \u2115} (h : 0 < k) :(\u21d1derivative)^[k] (C a) = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L214-L216"},"Polynomial.degree_pos_of_eval\u2082_root":{"signature":"theorem Polynomial.degree_pos_of_eval\u2082_root {R : Type u} {S : Type v} [Semiring R] [Semiring S] {p : Polynomial R} (hp : p \u2260 0) (f : R \u2192+* S) {z : S} (hz : eval\u2082 f z p = 0) (inj : \u2200 (x : R), f x = 0 \u2192 x = 0) :0 < p.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L248-L250"},"Polynomial.aeval_primPart_eq_zero":{"signature":"theorem Polynomial.aeval_primPart_eq_zero {R : Type u_1} [CommRing R] [NormalizedGCDMonoid R] {S : Type u_2} [Ring S] [IsDomain S] [Algebra R S] [Module.IsTorsionFree R S] {p : Polynomial R} {s : S} (hpzero : p \u2260 0) (hp : (aeval s) p = 0) :(aeval s) p.primPart = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Content.lean#L285-L292"},"Polynomial.div_tendsto_leadingCoeff_div_of_degree_eq":{"signature":"theorem Polynomial.div_tendsto_leadingCoeff_div_of_degree_eq {\ud835\udd5c : Type u_1} [NormedField \ud835\udd5c] [LinearOrder \ud835\udd5c] [IsStrictOrderedRing \ud835\udd5c] (P Q : Polynomial \ud835\udd5c) [OrderTopology \ud835\udd5c] (hdeg : P.degree = Q.degree) :Filter.Tendsto (fun (x : \ud835\udd5c) => eval x P / eval x Q) Filter.atTop (nhds (P.leadingCoeff / Q.leadingCoeff))","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/Basic.lean#L154-L158"},"Polynomial.dickson_one":{"signature":"theorem Polynomial.dickson_one {R : Type u_1} [CommRing R] (k : \u2115) (a : R) :dickson k a 1 = X","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Dickson.lean#L72-L74"},"Polynomial.dickson_two_one_eq_chebyshev_S":{"signature":"theorem Polynomial.dickson_two_one_eq_chebyshev_S (R : Type u_1) [CommRing R] (n : \u2115) :dickson 2 1 n = Chebyshev.S R \u2191n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Dickson.lean#L161-L171"},"Polynomial.card_support_eq_one_of_eraseLead_eq_zero":{"signature":"theorem Polynomial.card_support_eq_one_of_eraseLead_eq_zero {R : Type u_1} [Semiring R] {f : Polynomial R} (h\u2080 : f \u2260 0) (h\u2081 : f.eraseLead = 0) :f.support.card = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/EraseLead.lean#L121-L123"},"Polynomial.card_support_le_one_of_eraseLead_eq_zero":{"signature":"theorem Polynomial.card_support_le_one_of_eraseLead_eq_zero {R : Type u_1} [Semiring R] {f : Polynomial R} (h : f.eraseLead = 0) :f.support.card \u2264 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/EraseLead.lean#L125-L128"},"Polynomial.taylor_apply":{"signature":"theorem Polynomial.taylor_apply {R : Type u_1} [Semiring R] (r : R) (f : Polynomial R) :(taylor r) f = f.comp (X + C r)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Taylor.lean#L43-L44"},"Polynomial.taylor_zero":{"signature":"theorem Polynomial.taylor_zero {R : Type u_1} [Semiring R] (f : Polynomial R) :(taylor 0) f = f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Taylor.lean#L55-L55"},"Polynomial.taylor_zero'":{"signature":"theorem Polynomial.taylor_zero' {R : Type u_1} [Semiring R] :taylor 0 = LinearMap.id","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Taylor.lean#L57-L58"},"Polynomial.natDegree_hasseDeriv_le":{"signature":"theorem Polynomial.natDegree_hasseDeriv_le {R : Type u_1} [Semiring R] (p : Polynomial R) (n : \u2115) :((hasseDeriv n) p).natDegree \u2264 p.natDegree - n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/HasseDeriv.lean#L174-L186"},"Polynomial.hasseDeriv_natDegree_eq_C":{"signature":"theorem Polynomial.hasseDeriv_natDegree_eq_C {R : Type u_1} [Semiring R] (f : Polynomial R) :(hasseDeriv f.natDegree) f = C f.leadingCoeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/HasseDeriv.lean#L188-L191"},"Polynomial.coeff_taylor_natDegree":{"signature":"theorem Polynomial.coeff_taylor_natDegree {R : Type u_1} [Semiring R] (r : R) (f : Polynomial R) :((taylor r) f).coeff f.natDegree = f.leadingCoeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Taylor.lean#L86-L90"},"Polynomial.leadingCoeff_taylor":{"signature":"theorem Polynomial.leadingCoeff_taylor {R : Type u_1} [Semiring R] (r : R) (f : Polynomial R) :((taylor r) f).leadingCoeff = f.leadingCoeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Taylor.lean#L99-L101"},"Polynomial.taylor_eq_zero":{"signature":"theorem Polynomial.taylor_eq_zero {R : Type u_1} [Semiring R] (r : R) (f : Polynomial R) :(taylor r) f = 0 \u2194 f = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Taylor.lean#L103-L105"},"Polynomial.taylor_injective":{"signature":"theorem Polynomial.taylor_injective {R : Type u_1} [Ring R] (r : R) :Function.Injective \u21d1(taylor r)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Taylor.lean#L130-L131"},"Polynomial.taylor_taylor":{"signature":"theorem Polynomial.taylor_taylor {R : Type u_1} [CommSemiring R] (f : Polynomial R) (r s : R) :(taylor r) ((taylor s) f) = (taylor (r + s)) f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Taylor.lean#L157-L158"},"Polynomial.map_mapRingHom_eval_map":{"signature":"theorem Polynomial.map_mapRingHom_eval_map {R : Type u_1} {S : Type u_2} [Semiring R] [Semiring S] (f : R \u2192+* S) (p : Polynomial (Polynomial R)) (q : Polynomial R) :eval (map f q) (map (mapRingHom f) p) = map f (eval q p)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Bivariate.lean#L166-L167"},"Polynomial.map_mapRingHom_eval_map_eval":{"signature":"theorem Polynomial.map_mapRingHom_eval_map_eval {R : Type u_1} {S : Type u_2} [Semiring R] [Semiring S] (f : R \u2192+* S) (p : Polynomial (Polynomial R)) (q : Polynomial R) (r : R) :eval (f r) (eval (map f q) (map (mapRingHom f) p)) = f (eval r (eval q p))","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Bivariate.lean#L169-L171"},"Polynomial.map_mapRingHom_evalEval":{"signature":"theorem Polynomial.map_mapRingHom_evalEval {R : Type u_1} {S : Type u_2} [Semiring R] [Semiring S] (f : R \u2192+* S) (p : Polynomial (Polynomial R)) (x y : R) :evalEval (f x) (f y) (map (mapRingHom f) p) = f (evalEval x y p)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Bivariate.lean#L173-L175"},"Polynomial.div_eq_quo_add_rem_div":{"signature":"theorem Polynomial.div_eq_quo_add_rem_div {R : Type u_1} [CommRing R] [IsDomain R] (K : Type u_3) [Field K] [Algebra (Polynomial R) K] [IsFractionRing (Polynomial R) K] (f : Polynomial R) {g : Polynomial R} (hg : g.Monic) :\u2203 (q : Polynomial R) (r : Polynomial R), r.degree < g.degree \u2227 (algebraMap (Polynomial R) K) f / (algebraMap (Polynomial R) K) g = (algebraMap (Polynomial R) K) q + (algebraMap (Polynomial R) K) r / (algebraMap (Polynomial R) K) g","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/IntegralDomain.lean#L158-L167"},"Polynomial.natTrailingDegree_mul":{"signature":"theorem Polynomial.natTrailingDegree_mul {R : Type u} [Semiring R] {p q : Polynomial R} [NoZeroDivisors R] (hp : p \u2260 0) (hq : q \u2260 0) :(p * q).natTrailingDegree = p.natTrailingDegree + q.natTrailingDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L342-L345"},"Polynomial.mirror_mul_of_domain":{"signature":"theorem Polynomial.mirror_mul_of_domain {R : Type u_1} [Ring R] (p q : Polynomial R) [NoZeroDivisors R] :(p * q).mirror = p.mirror * q.mirror","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Mirror.lean#L177-L184"},"Polynomial.reverse_neg":{"signature":"theorem Polynomial.reverse_neg {R : Type u_1} [Ring R] (f : Polynomial R) :(-f).reverse = -f.reverse","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Reverse.lean#L357-L359"},"Polynomial.trailingDegree_neg":{"signature":"theorem Polynomial.trailingDegree_neg {R : Type u} [Ring R] (p : Polynomial R) :(-p).trailingDegree = p.trailingDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L381-L384"},"Polynomial.natTrailingDegree_neg":{"signature":"theorem Polynomial.natTrailingDegree_neg {R : Type u} [Ring R] (p : Polynomial R) :(-p).natTrailingDegree = p.natTrailingDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L386-L388"},"Polynomial.mirror_neg":{"signature":"theorem Polynomial.mirror_neg {R : Type u_1} [Ring R] (p : Polynomial R) :(-p).mirror = -p.mirror","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Mirror.lean#L172-L173"},"Polynomial.irreducible_of_mirror":{"signature":"theorem Polynomial.irreducible_of_mirror {R : Type u_1} [CommRing R] [NoZeroDivisors R] {f : Polynomial R} (h1 : \u00acIsUnit f) (h2 : \u2200 (k : Polynomial R), f * f.mirror = k * k.mirror \u2192 k = f \u2228 k = -f \u2228 k = f.mirror \u2228 k = -f.mirror) (h3 : IsRelPrime f f.mirror) :Irreducible f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Mirror.lean#L195-L220"},"Polynomial.natTrailingDegree_mul_mirror":{"signature":"theorem Polynomial.natTrailingDegree_mul_mirror {R : Type u_1} [Semiring R] (p : Polynomial R) [NoZeroDivisors R] :(p * p.mirror).natTrailingDegree = 2 * p.natTrailingDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Mirror.lean#L160-L164"},"Polynomial.coeff_mul_mirror":{"signature":"theorem Polynomial.coeff_mul_mirror {R : Type u_1} [Semiring R] (p : Polynomial R) :(p * p.mirror).coeff (p.natDegree + p.natTrailingDegree) = p.sum fun (x : \u2115) (x_1 : R) => x_1 ^ 2","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Mirror.lean#L143-L151"},"Polynomial.trinomial_support":{"signature":"theorem Polynomial.trinomial_support {R : Type u_1} [Semiring R] {k m n : \u2115} {u v w : R} (hkm : k < m) (hmn : m < n) (hu : u \u2260 0) (hv : v \u2260 0) (hw : w \u2260 0) :(trinomial k m n u v w).support = {k, m, n}","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/UnitTrinomial.lean#L113-L115"},"Polynomial.card_support_eq_three":{"signature":"theorem Polynomial.card_support_eq_three {R : Type u_1} [Semiring R] {f : Polynomial R} :f.support.card = 3 \u2194 \u2203 (k : \u2115) (m : \u2115) (n : \u2115) (_ : k < m) (_ : m < n) (x : R) (y : R) (z : R) (_ : x \u2260 0) (_ : y \u2260 0) (_ : z \u2260 0), f = C x * X ^ k + C y * X ^ m + C z * X ^ n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/EraseLead.lean#L468-L480"},"Polynomial.isUnitTrinomial_iff":{"signature":"theorem Polynomial.isUnitTrinomial_iff {p : Polynomial \u2124} :p.IsUnitTrinomial \u2194 p.support.card = 3 \u2227 \u2200 k \u2208 p.support, IsUnit (p.coeff k)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/UnitTrinomial.lean#L162-L175"},"Polynomial.isUnitTrinomial_iff'":{"signature":"theorem Polynomial.isUnitTrinomial_iff' {p : Polynomial \u2124} :p.IsUnitTrinomial \u2194 (p * p.mirror).coeff (((p * p.mirror).natDegree + (p * p.mirror).natTrailingDegree) / 2) = 3","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/UnitTrinomial.lean#L177-L196"},"Polynomial.isUnitTrinomial_iff''":{"signature":"theorem Polynomial.isUnitTrinomial_iff'' {p q : Polynomial \u2124} (h : p * p.mirror = q * q.mirror) :p.IsUnitTrinomial \u2194 q.IsUnitTrinomial","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/UnitTrinomial.lean#L198-L200"},"Polynomial.trinomial_trailingCoeff":{"signature":"theorem Polynomial.trinomial_trailingCoeff {R : Type u_1} [Semiring R] {k m n : \u2115} {u v w : R} (hkm : k < m) (hmn : m < n) (hu : u \u2260 0) :(trinomial k m n u v w).trailingCoeff = u","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/UnitTrinomial.lean#L96-L98"},"Polynomial.trinomial_leadingCoeff":{"signature":"theorem Polynomial.trinomial_leadingCoeff {R : Type u_1} [Semiring R] {k m n : \u2115} {u v w : R} (hkm : k < m) (hmn : m < n) (hw : w \u2260 0) :(trinomial k m n u v w).leadingCoeff = w","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/UnitTrinomial.lean#L92-L94"},"Polynomial.mirror_trailingCoeff":{"signature":"theorem Polynomial.mirror_trailingCoeff {R : Type u_1} [Semiring R] (p : Polynomial R) :p.mirror.trailingCoeff = p.leadingCoeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Mirror.lean#L134-L137"},"Polynomial.mirror_leadingCoeff":{"signature":"theorem Polynomial.mirror_leadingCoeff {R : Type u_1} [Semiring R] (p : Polynomial R) :p.mirror.leadingCoeff = p.trailingCoeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Mirror.lean#L139-L141"},"Polynomial.mirror_eval_one":{"signature":"theorem Polynomial.mirror_eval_one {R : Type u_1} [Semiring R] (p : Polynomial R) :eval 1 p.mirror = eval 1 p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Mirror.lean#L91-L110"},"Polynomial.mirror_inj":{"signature":"theorem Polynomial.mirror_inj {R : Type u_1} [Semiring R] {p q : Polynomial R} :p.mirror = q.mirror \u2194 p = q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Mirror.lean#L124-L126"},"Polynomial.X_pow_sub_X_sub_one_irreducible":{"signature":"theorem Polynomial.X_pow_sub_X_sub_one_irreducible {n : \u2115} (hn1 : n \u2260 1) :Irreducible (X ^ n - X - 1)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Selmer.lean#L49-L67"},"Polynomial.hilbertPoly":{"signature":"noncomputable def Polynomial.hilbertPoly {F : Type u_1} [Field F] (p : Polynomial F) (d : \u2115) :Polynomial F","doc":"Polynomial.hilbertPoly p 0 = 0; for any d : \u2115, Polynomial.hilbertPoly p (d + 1) is defined as \u2211 i \u2208 p.support, (p.coeff i) \u2022 Polynomial.preHilbertPoly F d i. If M is a graded module whose Poincar\u00e9 series can be written as p(X)/(1 - X)\u1d48 for some p : \u211a[X] with integer coefficients, then Polynomial.hilbertPoly p d is the Hilbert polynomial of M. See also Polynomial.coeff_mul_invOneSubPow_eq_hilbertPoly_eval, which says that PowerSeries.coeff F n (p * PowerSeries.invOneSubPow F d) equals (Polynomial.hilbertPoly p d).eval (n : F) for any large enough n : \u2115.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/HilbertPoly.lean#L101-L112"},"Polynomial.smul_sum":{"signature":"theorem Polynomial.smul_sum {R : Type u} [Semiring R] {S : Type u_1} {T : Type u_2} [AddCommMonoid S] [DistribSMul T S] (p : Polynomial R) (b : T) (f : \u2115 \u2192 R \u2192 S) :b \u2022 p.sum f = p.sum fun (n : \u2115) (a : R) => b \u2022 f n a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L885-L887"},"Polynomial.sum_smul_index'":{"signature":"theorem Polynomial.sum_smul_index' {R : Type u} [Semiring R] {S : Type u_1} {T : Type u_2} [DistribSMul T R] [AddCommMonoid S] (p : Polynomial R) (b : T) (f : \u2115 \u2192 R \u2192 S) (hf : \u2200 (i : \u2115), f i 0 = 0) :(b \u2022 p).sum f = p.sum fun (n : \u2115) (a : R) => f n (b \u2022 a)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L881-L883"},"Polynomial.hilbertPoly_smul":{"signature":"theorem Polynomial.hilbertPoly_smul {F : Type u_1} [Field F] (a : F) (p : Polynomial F) (d : \u2115) :(a \u2022 p).hilbertPoly d = a \u2022 p.hilbertPoly d","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/HilbertPoly.lean#L138-L147"},"Polynomial.coeffs_monomial":{"signature":"theorem Polynomial.coeffs_monomial {R : Type u} [Semiring R] (n : \u2115) {c : R} (hc : c \u2260 0) :((monomial n) c).coeffs = {c}","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L1049-L1051"},"Polynomial.contentIdeal_monomial":{"signature":"theorem Polynomial.contentIdeal_monomial {R : Type u_1} [Semiring R] (n : \u2115) (r : R) :((monomial n) r).contentIdeal = Ideal.span {r}","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ContentIdeal.lean#L74-L77"},"Polynomial.contentIdeal_C":{"signature":"theorem Polynomial.contentIdeal_C {R : Type u_1} [Semiring R] (r : R) :(C r).contentIdeal = Ideal.span {r}","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ContentIdeal.lean#L79-L82"},"Polynomial.contentIdeal_mul_le_mul_contentIdeal":{"signature":"theorem Polynomial.contentIdeal_mul_le_mul_contentIdeal {R : Type u_1} [Semiring R] (p q : Polynomial R) :(p * q).contentIdeal \u2264 p.contentIdeal * q.contentIdeal","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ContentIdeal.lean#L100-L105"},"Polynomial.contentIdeal_le_contentIdeal_of_dvd":{"signature":"theorem Polynomial.contentIdeal_le_contentIdeal_of_dvd {R : Type u_3} [CommSemiring R] {p q : Polynomial R} (hpq : p \u2223 q) :q.contentIdeal \u2264 p.contentIdeal","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ContentIdeal.lean#L111-L113"},"Polynomial.isPrimitive_of_contentIdeal_eq_top":{"signature":"theorem Polynomial.isPrimitive_of_contentIdeal_eq_top {R : Type u_3} [CommSemiring R] {p : Polynomial R} (h : p.contentIdeal = \u22a4) :p.IsPrimitive","doc":"If the coefficients of p generate the whole ring, then p is primitive.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ContentIdeal.lean#L141-L147"},"Polynomial.roots_X_add_C":{"signature":"theorem Polynomial.roots_X_add_C {R : Type u} [CommRing R] [IsDomain R] (r : R) :(X + C r).roots = {-r}","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L182-L183"},"Polynomial.eq_quadratic_of_degree_le_two":{"signature":"theorem Polynomial.eq_quadratic_of_degree_le_two {R : Type u_1} [Semiring R] {p : Polynomial R} (hp : p.degree \u2264 2) :p = C (p.coeff 2) * X ^ 2 + C (p.coeff 1) * X + C (p.coeff 0)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/SmallDegreeVieta.lean#L22-L27"},"Polynomial.preHilbertPoly_eq_choose_sub_add":{"signature":"theorem Polynomial.preHilbertPoly_eq_choose_sub_add (F : Type u_1) [Field F] [CharZero F] (d : \u2115) {k n : \u2115} (hkn : k \u2264 n) :eval (\u2191n) (preHilbertPoly F d k) = \u2191((n - k + d).choose d)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/HilbertPoly.lean#L90-L97"},"Polynomial.coeff_mul_invOneSubPow_eq_hilbertPoly_eval":{"signature":"theorem Polynomial.coeff_mul_invOneSubPow_eq_hilbertPoly_eval {F : Type u_1} [Field F] [CharZero F] {p : Polynomial F} (d : \u2115) {n : \u2115} (hn : p.natDegree < n) :(PowerSeries.coeff n) (\u2191p * \u2191(PowerSeries.invOneSubPow F d)) = eval (\u2191n) (p.hilbertPoly d)","doc":"The key property of Hilbert polynomials. If F is a field with characteristic 0, p : F[X] and d : \u2115, then for any large enough n : \u2115, (Polynomial.hilbertPoly p d).eval (n : F) equals the coefficient of X\u207f in the power series expansion of p/(1 - X)\u1d48.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/HilbertPoly.lean#L161-L190"},"Polynomial.eq_of_infinite_eval_eq":{"signature":"theorem Polynomial.eq_of_infinite_eval_eq {R : Type u} [CommRing R] [IsDomain R] (p q : Polynomial R) (h : {x : R | eval x p = eval x q}.Infinite) :p = q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L154-L158"},"Polynomial.existsUnique_hilbertPoly":{"signature":"theorem Polynomial.existsUnique_hilbertPoly {F : Type u_1} [Field F] [CharZero F] (p : Polynomial F) (d : \u2115) :\u2203! h : Polynomial F, \u2203 (N : \u2115), \u2200 n > N, (PowerSeries.coeff n) (\u2191p * \u2191(PowerSeries.invOneSubPow F d)) = eval (\u2191n) h","doc":"The polynomial satisfying the key property of Polynomial.hilbertPoly p d is unique.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/HilbertPoly.lean#L192-L206"},"Polynomial.eq_hilbertPoly_of_forall_coeff_eq_eval":{"signature":"theorem Polynomial.eq_hilbertPoly_of_forall_coeff_eq_eval {F : Type u_1} [Field F] [CharZero F] {p h : Polynomial F} {d : \u2115} (N : \u2115) (hhN : \u2200 n > N, (PowerSeries.coeff n) (\u2191p * \u2191(PowerSeries.invOneSubPow F d)) = eval (\u2191n) h) :h = p.hilbertPoly d","doc":"If h : F[X] and there exists some N : \u2115 such that for any number n : \u2115 bigger than N we have PowerSeries.coeff F n (p * invOneSubPow F d) = h.eval (n : F), then h is exactly Polynomial.hilbertPoly p d.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/HilbertPoly.lean#L208-L218"},"Polynomial.hilbertPoly_mul_one_sub_succ":{"signature":"theorem Polynomial.hilbertPoly_mul_one_sub_succ {F : Type u_1} [Field F] [CharZero F] (p : Polynomial F) (d : \u2115) :(p * (1 - X)).hilbertPoly (d + 1) = p.hilbertPoly d","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/HilbertPoly.lean#L220-L226"},"Polynomial.hilbertPoly_mul_one_sub_pow_add":{"signature":"theorem Polynomial.hilbertPoly_mul_one_sub_pow_add {F : Type u_1} [Field F] [CharZero F] (p : Polynomial F) (d e : \u2115) :(p * (1 - X) ^ e).hilbertPoly (d + e) = p.hilbertPoly d","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/HilbertPoly.lean#L228-L232"},"Polynomial.coeff_preHilbertPoly_self":{"signature":"theorem Polynomial.coeff_preHilbertPoly_self (F : Type u_1) [Field F] [CharZero F] (d k : \u2115) :(preHilbertPoly F d k).coeff d = (\u2191d.factorial)\u207b\u00b9","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/HilbertPoly.lean#L72-L84"},"Polynomial.natDegree_hilbertPoly_of_ne_zero_of_rootMultiplicity_lt":{"signature":"theorem Polynomial.natDegree_hilbertPoly_of_ne_zero_of_rootMultiplicity_lt {F : Type u_1} [Field F] [CharZero F] {p : Polynomial F} {d : \u2115} (hp : p \u2260 0) (hpd : rootMultiplicity 1 p < d) :(p.hilbertPoly d).natDegree = d - rootMultiplicity 1 p - 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/HilbertPoly.lean#L246-L265"},"Polynomial.derivative_ofNat":{"signature":"theorem Polynomial.derivative_ofNat {R : Type u} [Semiring R] (n : \u2115) [n.AtLeastTwo] :derivative (OfNat.ofNat n) = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L198-L201"},"Polynomial.derivative_intCast":{"signature":"theorem Polynomial.derivative_intCast {R : Type u} [Ring R] {n : \u2124} :derivative \u2191n = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L626-L629"},"Polynomial.iterate_derivative_X_pow_eq_natCast_mul":{"signature":"theorem Polynomial.iterate_derivative_X_pow_eq_natCast_mul {R : Type u} [CommSemiring R] (n k : \u2115) :(\u21d1derivative)^[k] (X ^ n) = \u2191(n.descFactorial k) * X ^ (n - k)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L487-L495"},"Polynomial.iterate_derivative_X_pow_eq_C_mul":{"signature":"theorem Polynomial.iterate_derivative_X_pow_eq_C_mul {R : Type u} [CommSemiring R] (n k : \u2115) :(\u21d1derivative)^[k] (X ^ n) = C \u2191(n.descFactorial k) * X ^ (n - k)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L497-L499"},"Polynomial.iterate_derivative_X_pow_eq_smul":{"signature":"theorem Polynomial.iterate_derivative_X_pow_eq_smul {R : Type u} [CommSemiring R] (n k : \u2115) :(\u21d1derivative)^[k] (X ^ n) = \u2191(n.descFactorial k) \u2022 X ^ (n - k)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L501-L503"},"Polynomial.iterate_derivative_mul_X_pow":{"signature":"theorem Polynomial.iterate_derivative_mul_X_pow {R : Type u} [CommSemiring R] (n m : \u2115) (p : Polynomial R) :(\u21d1derivative)^[n] (p * X ^ m) = \u2211 k \u2208 Finset.range (min m n).succ, (n.choose k * m.descFactorial k) \u2022 ((\u21d1derivative)^[n - k] p * X ^ (m - k))","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L522-L538"},"Polynomial.iterate_derivative_derivative_mul_X":{"signature":"theorem Polynomial.iterate_derivative_derivative_mul_X {R : Type u} [CommSemiring R] {n : \u2115} (p : Polynomial R) :(\u21d1derivative)^[n] (derivative p * X) = (\u21d1derivative)^[n + 1] p * X + n \u2022 (\u21d1derivative)^[n] p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L545-L548"},"Polynomial.iterate_derivative_derivative_mul_X_sq":{"signature":"theorem Polynomial.iterate_derivative_derivative_mul_X_sq {R : Type u} [CommSemiring R] {n : \u2115} (p : Polynomial R) :(\u21d1derivative)^[n] ((\u21d1derivative)^[2] p * X ^ 2) = (\u21d1derivative)^[n + 2] p * X ^ 2 + (2 * n) \u2022 (\u21d1derivative)^[n + 1] p * X + (n * (n - 1)) \u2022 (\u21d1derivative)^[n] p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L550-L561"},"Polynomial.iterate_derivative_intCast_mul":{"signature":"theorem Polynomial.iterate_derivative_intCast_mul {R : Type u} [Ring R] {n : \u2124} {k : \u2115} {f : Polynomial R} :(\u21d1derivative)^[k] (\u2191n * f) = \u2191n * (\u21d1derivative)^[k] f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L635-L638"},"Polynomial.iterate_derivative_natCast_mul":{"signature":"theorem Polynomial.iterate_derivative_natCast_mul {R : Type u} [Semiring R] {n k : \u2115} {f : Polynomial R} :(\u21d1derivative)^[k] (\u2191n * f) = \u2191n * (\u21d1derivative)^[k] f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L290-L293"},"Polynomial.iterate_derivative_sub":{"signature":"theorem Polynomial.iterate_derivative_sub {R : Type u} [Ring R] {k : \u2115} {f g : Polynomial R} :(\u21d1derivative)^[k] (f - g) = (\u21d1derivative)^[k] f - (\u21d1derivative)^[k] g","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L622-L624"},"Polynomial.eval_intCast":{"signature":"theorem Polynomial.eval_intCast {R : Type u} [Ring R] {n : \u2124} {x : R} :eval x \u2191n = \u2191n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L734-L736"},"Polynomial.instInhabitedSplittingFieldAux":{"signature":"instance Polynomial.instInhabitedSplittingFieldAux (n : \u2115) {K : Type u} [Field K] (f : Polynomial K) :Inhabited (SplittingFieldAux n f)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/SplittingField/Construction.lean#L133-L134"},"Polynomial.toSubring_zero":{"signature":"theorem Polynomial.toSubring_zero {R : Type u} [Ring R] (T : Subring R) :toSubring 0 T \u22ef = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Basic.lean#L404-L407"},"Polynomial.natDegree_eq_of_natDegree_add_lt_left":{"signature":"theorem Polynomial.natDegree_eq_of_natDegree_add_lt_left {R : Type u} [Semiring R] (p q : Polynomial R) (H : (p + q).natDegree < p.natDegree) :p.natDegree = q.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L230-L237"},"Polynomial.natDegree_eq_of_natDegree_add_lt_right":{"signature":"theorem Polynomial.natDegree_eq_of_natDegree_add_lt_right {R : Type u} [Semiring R] (p q : Polynomial R) (H : (p + q).natDegree < q.natDegree) :p.natDegree = q.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L239-L241"},"Polynomial.natDegree_eq_of_natDegree_add_eq_zero":{"signature":"theorem Polynomial.natDegree_eq_of_natDegree_add_eq_zero {R : Type u} [Semiring R] (p q : Polynomial R) (H : (p + q).natDegree = 0) :p.natDegree = q.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L243-L248"},"Polynomial.mem_nonZeroDivisors_of_trailingCoeff":{"signature":"theorem Polynomial.mem_nonZeroDivisors_of_trailingCoeff {R : Type u} [CommRing R] {p : Polynomial R} (h : p.trailingCoeff \u2208 nonZeroDivisors R) :p \u2208 nonZeroDivisors (Polynomial R)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/RingDivision.lean#L133-L134"},"Polynomial.degree_add_degree_leadingCoeff_inv":{"signature":"theorem Polynomial.degree_add_degree_leadingCoeff_inv {K : Type u_1} [DivisionRing K] (p : Polynomial K) :p.degree + (C p.leadingCoeff\u207b\u00b9).degree = p.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L478-L480"},"Polynomial.degree_mod_lt":{"signature":"theorem Polynomial.degree_mod_lt {R : Type u} [Field R] (p : Polynomial R) {q : Polynomial R} (hq : q \u2260 0) :(p % q).degree < q.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L437-L440"},"Polynomial.cardPowDegree_isEuclidean":{"signature":"theorem Polynomial.cardPowDegree_isEuclidean {Fq : Type u_1} [Field Fq] [Fintype Fq] :cardPowDegree.IsEuclidean","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/CardPowDegree.lean#L85-L99"},"Polynomial.cardPowDegree_anti_archimedean":{"signature":"theorem Polynomial.cardPowDegree_anti_archimedean {Fq : Type u_1} [Fintype Fq] [Field Fq] {x y z : Polynomial Fq} {a : \u2124} (hxy : cardPowDegree (x - y) < a) (hyz : cardPowDegree (y - z) < a) :cardPowDegree (x - z) < a","doc":"If x is close to y and y is close to z, then x and z are at least as close.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/NumberTheory/ClassNumber/AdmissibleCardPowDegree.lean#L144-L164"},"Polynomial.exists_partition_polynomial_aux":{"signature":"theorem Polynomial.exists_partition_polynomial_aux {Fq : Type u_1} [Fintype Fq] [Field Fq] (n : \u2115) {\u03b5 : \u211d} (h\u03b5 : 0 < \u03b5) {b : Polynomial Fq} (hb : b \u2260 0) (A : Fin n \u2192 Polynomial Fq) :\u2203 (t : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-Real.log \u03b5 / Real.log \u2191(Fintype.card Fq)\u2309\u208a)), \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < cardPowDegree b \u2022 \u03b5","doc":"A slightly stronger version of exists_partition on which we perform induction on n: for all \u03b5 > 0, we can partition the remainders of any family of polynomials A into equivalence classes, where the equivalence(!) relation is \"closer than \u03b5\".","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/NumberTheory/ClassNumber/AdmissibleCardPowDegree.lean#L166-L233"},"Polynomial.exists_partition_polynomial":{"signature":"theorem Polynomial.exists_partition_polynomial {Fq : Type u_1} [Fintype Fq] [Field Fq] (n : \u2115) {\u03b5 : \u211d} (h\u03b5 : 0 < \u03b5) {b : Polynomial Fq} (hb : b \u2260 0) (A : Fin n \u2192 Polynomial Fq) :\u2203 (t : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-Real.log \u03b5 / Real.log \u2191(Fintype.card Fq)\u2309\u208a)), \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2192 \u2191(cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < cardPowDegree b \u2022 \u03b5","doc":"For all \u03b5 > 0, we can partition the remainders of any family of polynomials A into classes, where all remainders in a class are close together.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/NumberTheory/ClassNumber/AdmissibleCardPowDegree.lean#L235-L242"},"Polynomial.cardPowDegreeIsAdmissible":{"signature":"noncomputable def Polynomial.cardPowDegreeIsAdmissible {Fq : Type u_1} [Fintype Fq] [Field Fq] :cardPowDegree.IsAdmissible","doc":"fun p => Fintype.card Fq ^ degree p is an admissible absolute value. We set q ^ degree 0 = 0.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/NumberTheory/ClassNumber/AdmissibleCardPowDegree.lean#L244-L251"},"Polynomial.support_C_mul_X_pow":{"signature":"theorem Polynomial.support_C_mul_X_pow {R : Type u} [Semiring R] (n : \u2115) {c : R} (h : c \u2260 0) :(C c * X ^ n).support = {n}","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L760-L762"},"Polynomial.card_support_eq_one":{"signature":"theorem Polynomial.card_support_eq_one {R : Type u_1} [Semiring R] {f : Polynomial R} :f.support.card = 1 \u2194 \u2203 (k : \u2115) (x : R) (_ : x \u2260 0), f = C x * X ^ k","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/EraseLead.lean#L448-L454"},"Polynomial.natDegree_finset_prod_X_sub_C_eq_card":{"signature":"theorem Polynomial.natDegree_finset_prod_X_sub_C_eq_card {R : Type u} [CommRing R] [Nontrivial R] {\u03b1 : Type u_1} (s : Finset \u03b1) (f : \u03b1 \u2192 R) :(\u220f a \u2208 s, (X - C (f a))).natDegree = s.card","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/BigOperators.lean#L294-L297"},"Polynomial.monic_prod_X_sub_C":{"signature":"theorem Polynomial.monic_prod_X_sub_C {R : Type u} [CommRing R] {\u03b1 : Type u_1} (b : \u03b1 \u2192 R) (s : Finset \u03b1) :(\u220f a \u2208 s, (X - C (b a))).Monic","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L752-L754"},"Polynomial.gaussNorm":{"signature":"def Polynomial.gaussNorm {R : Type u_1} {F : Type u_2} [Semiring R] [FunLike F R \u211d] (v : F) (c : \u211d) (p : Polynomial R) :\u211d","doc":"Given a polynomial p in R[X], a function v : R \u2192 \u211d and a real number c, the Gauss norm is defined as the supremum of the set of all values of v (p.coeff i) * c ^ i for all i in the support of p.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/GaussNorm.lean#L44-L48"},"Polynomial.gaussNorm_zero":{"signature":"theorem Polynomial.gaussNorm_zero {R : Type u_1} {F : Type u_2} [Semiring R] [FunLike F R \u211d] (v : F) (c : \u211d) :gaussNorm v c 0 = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/GaussNorm.lean#L50-L51"},"Polynomial.exists_eq_gaussNorm":{"signature":"theorem Polynomial.exists_eq_gaussNorm {R : Type u_1} {F : Type u_2} [Semiring R] [FunLike F R \u211d] (v : F) (c : \u211d) (p : Polynomial R) [ZeroHomClass F R \u211d] :\u2203 (i : \u2115), gaussNorm v c p = v (p.coeff i) * c ^ i","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/GaussNorm.lean#L53-L59"},"Polynomial.gaussNorm_coe_powerSeries":{"signature":"theorem Polynomial.gaussNorm_coe_powerSeries {R : Type u_1} {F : Type u_2} [Semiring R] [FunLike F R \u211d] (v : F) {c : \u211d} (p : Polynomial R) [ZeroHomClass F R \u211d] [NonnegHomClass F R \u211d] (hc : 0 \u2264 c) :PowerSeries.gaussNorm v c \u2191p = gaussNorm v c p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/GaussNorm.lean#L92-L108"},"Polynomial.aeval_ofReal":{"signature":"theorem Polynomial.aeval_ofReal {K : Type u_1} [RCLike K] (p : Polynomial \u211d) (x : \u211d) :(aeval \u2191x) p = \u2191(eval x p)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/RCLike/Lemmas.lean#L111-L112"},"Polynomial.degree_taylor":{"signature":"theorem Polynomial.degree_taylor {R : Type u_1} [Semiring R] (p : Polynomial R) (r : R) :((taylor r) p).degree = p.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Taylor.lean#L107-L112"},"Polynomial.ne_zero_of_degree_ge_degree":{"signature":"theorem Polynomial.ne_zero_of_degree_ge_degree {R : Type u} [Semiring R] {p q : Polynomial R} (hpq : p.degree \u2264 q.degree) (hp : p \u2260 0) :q \u2260 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L142-L145"},"Polynomial.eventually_no_roots":{"signature":"theorem Polynomial.eventually_no_roots {\ud835\udd5c : Type u_1} [NormedField \ud835\udd5c] [LinearOrder \ud835\udd5c] [IsStrictOrderedRing \ud835\udd5c] (P : Polynomial \ud835\udd5c) (hP : P \u2260 0) :\u2200\u1da0 (x : \ud835\udd5c) in Filter.atTop, \u00acP.IsRoot x","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/Basic.lean#L36-L37"},"Polynomial.isBigO_of_degree_le":{"signature":"theorem Polynomial.isBigO_of_degree_le {\ud835\udd5c : Type u_1} [NormedField \ud835\udd5c] [LinearOrder \ud835\udd5c] [IsStrictOrderedRing \ud835\udd5c] (P Q : Polynomial \ud835\udd5c) [OrderTopology \ud835\udd5c] (h : P.degree \u2264 Q.degree) :(fun (x : \ud835\udd5c) => eval x P) =O[Filter.atTop] fun (x : \ud835\udd5c) => eval x Q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/Basic.lean#L210-L219"},"Polynomial.aeval_root_of_mapAlg_eq_multiset_prod_X_sub_C":{"signature":"theorem Polynomial.aeval_root_of_mapAlg_eq_multiset_prod_X_sub_C {R : Type u_1} {A : Type u_2} [CommSemiring R] [CommRing A] [Algebra R A] (s : Multiset A) {x : A} (hx : x \u2208 s) {p : Polynomial R} (hp : (mapAlg R A) p = (Multiset.map (fun (x : A) => X - C x) s).prod) :(aeval x) p = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Tower.lean#L90-L94"},"Polynomial.monic_of_monic_mapAlg":{"signature":"theorem Polynomial.monic_of_monic_mapAlg {R : Type u} [CommSemiring R] {S : Type v} [Semiring S] [Algebra R S] [FaithfulSMul R S] {p : Polynomial R} (hp : ((mapAlg R S) p).Monic) :p.Monic","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Lifts.lean#L218-L220"},"Polynomial.cyclotomic_eval_lt_add_one_pow_totient":{"signature":"theorem Polynomial.cyclotomic_eval_lt_add_one_pow_totient {n : \u2115} {q : \u211d} (hn' : 3 \u2264 n) (hq' : 1 < q) :eval q (cyclotomic n \u211d) < (q + 1) ^ n.totient","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Eval.lean#L227-L288"},"Polynomial.natDegree_iterate_comp":{"signature":"theorem Polynomial.natDegree_iterate_comp {R : Type u} [Semiring R] {p q : Polynomial R} [NoZeroDivisors R] (k : \u2115) :(p.comp^[k] q).natDegree = p.natDegree ^ k * q.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L392-L397"},"Polynomial.one_scaleRoots":{"signature":"theorem Polynomial.one_scaleRoots {R : Type u_1} [Semiring R] (r : R) :scaleRoots 1 r = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ScaleRoots.lean#L115-L117"},"Polynomial.pow_scaleRoots'":{"signature":"theorem Polynomial.pow_scaleRoots' {R : Type u_1} [CommSemiring R] (p : Polynomial R) (r : R) (n : \u2115) (hp : p.leadingCoeff ^ n \u2260 0) :(p ^ n).scaleRoots r = p.scaleRoots r ^ n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ScaleRoots.lean#L224-L232"},"Polynomial.X_sub_C_scaleRoots":{"signature":"theorem Polynomial.X_sub_C_scaleRoots {R : Type u_1} [Ring R] (r s : R) :(X - C r).scaleRoots s = X - C (r * s)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ScaleRoots.lean#L317-L321"},"Polynomial.scaleRoots_eval_mul":{"signature":"theorem Polynomial.scaleRoots_eval_mul {R : Type u_1} [CommSemiring R] (p : Polynomial R) (r s : R) :eval (s * r) (p.scaleRoots s) = s ^ p.natDegree * eval r p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ScaleRoots.lean#L157-L159"},"Polynomial.rootMultiplicity_scaleRoots":{"signature":"theorem Polynomial.rootMultiplicity_scaleRoots {R : Type u_1} [CommRing R] (p : Polynomial R) {r a : R} (hr : IsLeftRegular r) :rootMultiplicity (r * a) (p.scaleRoots r) = rootMultiplicity a p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ScaleRoots.lean#L329-L346"},"Polynomial.support_map_subset":{"signature":"theorem Polynomial.support_map_subset {R : Type u} {S : Type v} [Semiring R] [Semiring S] (f : R \u2192+* S) (p : Polynomial R) :(map f p).support \u2286 p.support","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Coeff.lean#L205-L209"},"Polynomial.sum_bernoulli":{"signature":"theorem Polynomial.sum_bernoulli (n : \u2115) :\u2211 k \u2208 Finset.range (n + 1), \u2191((n + 1).choose k) \u2022 bernoulli k = (monomial n) (\u2191n + 1)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/NumberTheory/BernoulliPolynomials.lean#L130-L158"},"Polynomial.bernoulli_generating_function":{"signature":"theorem Polynomial.bernoulli_generating_function {A : Type u_1} [CommRing A] [Algebra \u211a A] (t : A) :(PowerSeries.mk fun (n : \u2115) => (aeval t) ((1 / \u2191n.factorial) \u2022 bernoulli n)) * (PowerSeries.exp A - 1) = PowerSeries.X * (PowerSeries.rescale t) (PowerSeries.exp A)","doc":"The theorem that $(e^X - 1) * \u2211 B\u2099(t)* X^n/n! = Xe^{tX}$","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/NumberTheory/BernoulliPolynomials.lean#L267-L304"},"Polynomial.coeff_restriction":{"signature":"theorem Polynomial.coeff_restriction {R : Type u} [Ring R] {p : Polynomial R} {n : \u2115} :\u2191(p.restriction.coeff n) = p.coeff n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Basic.lean#L304-L312"},"Polynomial.support_restriction":{"signature":"theorem Polynomial.support_restriction {R : Type u} [Ring R] (p : Polynomial R) :p.restriction.support = p.support","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Basic.lean#L317-L322"},"Polynomial.eval\u2082_restriction":{"signature":"theorem Polynomial.eval\u2082_restriction {R : Type u} {S : Type u_1} [Ring R] [Semiring S] {f : R \u2192+* S} {x : S} {p : Polynomial R} :eval\u2082 f x p = eval\u2082 (f.comp (Subring.closure \u2191p.coeffs).subtype) x p.restriction","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Basic.lean#L352-L356"},"Polynomial.fwdDiff_iter_degree_eq_factorial":{"signature":"theorem Polynomial.fwdDiff_iter_degree_eq_factorial {R : Type u_3} [CommRing R] (P : Polynomial R) :((fwdDiff 1)^[P.natDegree] fun (x : R) => eval x P) = P.leadingCoeff \u2022 \u2191P.natDegree.factorial","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Group/ForwardDiff.lean#L262-L268"},"Polynomial.aeval_conj":{"signature":"theorem Polynomial.aeval_conj {K : Type u_1} [RCLike K] (p : Polynomial \u211d) (z : K) :(aeval ((starRingEnd K) z)) p = (starRingEnd K) ((aeval z) p)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/RCLike/Lemmas.lean#L108-L109"},"Polynomial.mul_star_dvd_of_aeval_eq_zero_im_ne_zero":{"signature":"theorem Polynomial.mul_star_dvd_of_aeval_eq_zero_im_ne_zero (p : Polynomial \u211d) {z : \u2102} (h0 : (aeval z) p = 0) (hz : z.im \u2260 0) :(X - C ((starRingEnd \u2102) z)) * (X - C z) \u2223 map (algebraMap \u211d \u2102) p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Complex/Polynomial/Basic.lean#L176-L181"},"Polynomial.quadratic_dvd_of_aeval_eq_zero_im_ne_zero":{"signature":"theorem Polynomial.quadratic_dvd_of_aeval_eq_zero_im_ne_zero (p : Polynomial \u211d) {z : \u2102} (h0 : (aeval z) p = 0) (hz : z.im \u2260 0) :X ^ 2 - C (2 * z.re) * X + C (\u2016z\u2016 ^ 2) \u2223 p","doc":"If z is a non-real complex root of a real polynomial, then p is divisible by a quadratic polynomial.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Complex/Polynomial/Basic.lean#L183-L194"},"Polynomial.degree_C_mul_of_mul_ne_zero":{"signature":"theorem Polynomial.degree_C_mul_of_mul_ne_zero {R : Type u} {a : R} [Semiring R] {p : Polynomial R} (h : a * p.leadingCoeff \u2260 0) :(C a * p).degree = p.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L140-L141"},"Polynomial.integralNormalization_coeff_degree_ne":{"signature":"theorem Polynomial.integralNormalization_coeff_degree_ne {R : Type u} [Semiring R] {p : Polynomial R} {i : \u2115} (hi : p.degree \u2260 \u2191i) :p.integralNormalization.coeff i = p.coeff i * p.leadingCoeff ^ (p.natDegree - 1 - i)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/IntegralNormalization.lean#L72-L74"},"Polynomial.integralNormalization_coeff_ne_natDegree":{"signature":"theorem Polynomial.integralNormalization_coeff_ne_natDegree {R : Type u} [Semiring R] {p : Polynomial R} {i : \u2115} (hi : i \u2260 p.natDegree) :p.integralNormalization.coeff i = p.coeff i * p.leadingCoeff ^ (p.natDegree - 1 - i)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/IntegralNormalization.lean#L76-L78"},"Polynomial.trailingDegree_mul":{"signature":"theorem Polynomial.trailingDegree_mul {R : Type u} [Semiring R] [NoZeroDivisors R] {p q : Polynomial R} :(p * q).trailingDegree = p.trailingDegree + q.trailingDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/RingDivision.lean#L93-L100"},"Polynomial.comp_zero":{"signature":"theorem Polynomial.comp_zero {R : Type u} [Semiring R] {p : Polynomial R} :p.comp 0 = C (eval 0 p)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L397-L398"},"Polynomial.eval\u2082AlgHom'_apply":{"signature":"theorem Polynomial.eval\u2082AlgHom'_apply {R : Type u} {A : Type z} {B : Type u_2} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A \u2192\u2090[R] B) (b : B) (hf : \u2200 (a : A), Commute (f a) b) (p : Polynomial A) :(eval\u2082AlgHom' f b hf) p = eval\u2082 (\u2191f) b p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L157-L157"},"Polynomial.hasSMulPi'":{"signature":"noncomputable def Polynomial.hasSMulPi' (R : Type u_1) (S : Type u_2) (T : Type u_3) [CommSemiring R] [Semiring S] [Algebra R S] [SMul S T] :SMul (Polynomial R) (S \u2192 T)","doc":"This is not an instance as it forms a diamond with Pi.instSMul. See the instance_diamonds test for details.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Algebraic/Pi.lean#L32-L37"},"Polynomial.algebraPi":{"signature":"noncomputable def Polynomial.algebraPi (R : Type u_1) (S : Type u_2) (T : Type u_3) [CommSemiring R] [CommSemiring S] [CommSemiring T] [Algebra R S] [Algebra S T] :Algebra (Polynomial R) (S \u2192 T)","doc":"This is not an instance for the same reasons as Polynomial.hasSMulPi'.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Algebraic/Pi.lean#L53-L65"},"Polynomial.algebraMap_pi_eq_aeval":{"signature":"theorem Polynomial.algebraMap_pi_eq_aeval (R : Type u_1) (S : Type u_2) (T : Type u_3) [CommSemiring R] [CommSemiring S] [CommSemiring T] [Algebra R S] [Algebra S T] :\u21d1(algebraMap (Polynomial R) (S \u2192 T)) = fun (p : Polynomial R) (z : S) => (algebraMap S T) ((aeval z) p)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Algebraic/Pi.lean#L69-L72"},"Polynomial.natDegree_prod_of_monic":{"signature":"theorem Polynomial.natDegree_prod_of_monic {R : Type u} {\u03b9 : Type w} (s : Finset \u03b9) [CommSemiring R] (f : \u03b9 \u2192 Polynomial R) (h : \u2200 i \u2208 s, (f i).Monic) :(\u220f i \u2208 s, f i).natDegree = \u2211 i \u2208 s, (f i).natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/BigOperators.lean#L214-L216"},"Polynomial.count_map_roots_of_injective":{"signature":"theorem Polynomial.count_map_roots_of_injective {A : Type u_1} {B : Type u_2} [CommRing A] [CommRing B] [IsDomain A] [DecidableEq B] (p : Polynomial A) {f : A \u2192+* B} (hf : Function.Injective \u21d1f) (b : B) :Multiset.count b (Multiset.map (\u21d1f) p.roots) \u2264 rootMultiplicity b (map f p)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L893-L899"},"Polynomial.intCast_coeff_zero":{"signature":"theorem Polynomial.intCast_coeff_zero {i : \u2124} {R : Type u_1} [Ring R] :(\u2191i).coeff 0 = \u2191i","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L355-L357"},"Polynomial.intCast_inj":{"signature":"theorem Polynomial.intCast_inj {m n : \u2124} {R : Type u_1} [Ring R] [CharZero R] :\u2191m = \u2191n \u2194 m = n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L359-L366"},"Polynomial.coeff_intCast_mul":{"signature":"theorem Polynomial.coeff_intCast_mul {S : Type v} [Ring S] {p : Polynomial S} {a : \u2124} {k : \u2115} :(\u2191a * p).coeff k = \u2191a * p.coeff k","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L182-L183"},"Polynomial.map_rootOfSplits'":{"signature":"theorem Polynomial.map_rootOfSplits' {R : Type u_1} [CommRing R] {f : Polynomial R} (hf : f.Splits) (hfd : f.degree \u2260 0) :eval (rootOfSplits hf hfd) f = 0","doc":"Alias of Polynomial.eval_rootOfSplits.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L728-L729"},"Polynomial.coeff_natTrailingDegree_pred_eq_zero":{"signature":"theorem Polynomial.coeff_natTrailingDegree_pred_eq_zero {R : Type u} [Semiring R] {p : Polynomial R} {hp : 0 < \u2191p.natTrailingDegree} :p.coeff (p.natTrailingDegree - 1) = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L230-L234"},"Polynomial.natDegree_ne_zero_induction_on":{"signature":"theorem Polynomial.natDegree_ne_zero_induction_on {R : Type u} [Semiring R] {M : Polynomial R \u2192 Prop} {f : Polynomial R} (f0 : f.natDegree \u2260 0) (h_C_add : \u2200 {a : R} {p : Polynomial R}, M p \u2192 M (C a + p)) (h_add : \u2200 {p q : Polynomial R}, M p \u2192 M q \u2192 M (p + q)) (h_monomial : \u2200 {n : \u2115} {a : R}, a \u2260 0 \u2192 n \u2260 0 \u2192 M ((monomial n) a)) :M f","doc":"A property holds for all polynomials of non-zero natDegree with coefficients in a semiring R if it holds for","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Inductions.lean#L182-L214"},"Polynomial.isRoot_of_mem_roots":{"signature":"theorem Polynomial.isRoot_of_mem_roots {R : Type u} {a : R} [CommRing R] [IsDomain R] {p : Polynomial R} (h : a \u2208 p.roots) :p.IsRoot a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L116-L117"},"Polynomial.ne_zero_of_mem_roots":{"signature":"theorem Polynomial.ne_zero_of_mem_roots {R : Type u} {a : R} [CommRing R] [IsDomain R] {p : Polynomial R} (h : a \u2208 p.roots) :p \u2260 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L113-L114"},"Polynomial.coeff_X_sub_C_mul":{"signature":"theorem Polynomial.coeff_X_sub_C_mul {R : Type u} [Ring R] {p : Polynomial R} {r : R} {a : \u2115} :((X - C r) * p).coeff (a + 1) = p.coeff a - r * p.coeff (a + 1)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L126-L127"},"Polynomial.leadingCoeff_eraseLead_eq_nextCoeff":{"signature":"theorem Polynomial.leadingCoeff_eraseLead_eq_nextCoeff {R : Type u_1} [Semiring R] {f : Polynomial R} (h : f.nextCoeff \u2260 0) :f.eraseLead.leadingCoeff = f.nextCoeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/EraseLead.lean#L249-L254"},"Polynomial.nextCoeff_eq_zero_of_eraseLead_eq_zero":{"signature":"theorem Polynomial.nextCoeff_eq_zero_of_eraseLead_eq_zero {R : Type u_1} [Semiring R] {f : Polynomial R} (h : f.eraseLead = 0) :f.nextCoeff = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/EraseLead.lean#L256-L258"},"Polynomial.natDegree_eraseLead_add_one":{"signature":"theorem Polynomial.natDegree_eraseLead_add_one {R : Type u_1} [Semiring R] {f : Polynomial R} (h : f.nextCoeff \u2260 0) :f.eraseLead.natDegree + 1 = f.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/EraseLead.lean#L224-L227"},"Polynomial.X_mul_monomial":{"signature":"theorem Polynomial.X_mul_monomial {R : Type u} [Semiring R] (n : \u2115) (r : R) :X * (monomial n) r = (monomial (n + 1)) r","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L555-L557"},"Polynomial.monomial_neg":{"signature":"theorem Polynomial.monomial_neg {R : Type u} [Ring R] (n : \u2115) (a : R) :(monomial n) (-a) = -(monomial n) a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L1107-L1109"},"Polynomial.monomial_sub":{"signature":"theorem Polynomial.monomial_sub {R : Type u} {a b : R} [Ring R] (n : \u2115) :(monomial n) (a - b) = (monomial n) a - (monomial n) b","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L1111-L1112"},"Polynomial.signVariations_eraseLead_mul_X_sub_C":{"signature":"theorem Polynomial.signVariations_eraseLead_mul_X_sub_C {R : Type u_1} [Ring R] [LinearOrder R] [IsStrictOrderedRing R] {P : Polynomial R} {\u03b7 : R} (h\u03b7 : 0 < \u03b7) (hP\u2080 : 0 < P.leadingCoeff) (hc : P.nextCoeff < 0) :((X - C \u03b7) * P).eraseLead.signVariations = ((X - C \u03b7) * P.eraseLead).signVariations","doc":"If P's coefficients start with signs [+, -, ...], then multiplying by a binomial X - \u03b7 commutes with eraseLead in the number of sign changes. This is because the product of P and X - \u03b7 has the pattern [+, -, ...] as well, so then P.eraseLead starts with [-,...], and multiplying by X - \u03b7 gives [-, ...] too.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/RuleOfSigns.lean#L162-L206"},"Polynomial.coeff_mul_monomial":{"signature":"theorem Polynomial.coeff_mul_monomial {R : Type u} [Semiring R] (p : Polynomial R) (n d : \u2115) (r : R) :(p * (monomial n) r).coeff (d + n) = p.coeff d * r","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L258-L260"},"Polynomial.leadingCoeff_cons_eraseLead":{"signature":"theorem Polynomial.leadingCoeff_cons_eraseLead {R : Type u_1} [Semiring R] {P : Polynomial R} (h : P.nextCoeff \u2260 0) :P.leadingCoeff :: P.eraseLead.coeffList = P.coeffList","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/CoeffList.lean#L91-L97"},"Polynomial.succ_signVariations_X_sub_C_mul_monomial":{"signature":"theorem Polynomial.succ_signVariations_X_sub_C_mul_monomial {R : Type u_1} [Ring R] [LinearOrder R] [IsStrictOrderedRing R] {\u03b7 : R} {d : \u2115} {c : R} (hc : c \u2260 0) (h\u03b7 : 0 < \u03b7) :((monomial d) c).signVariations + 1 \u2264 ((X - C \u03b7) * (monomial d) c).signVariations","doc":"This lemma is really a specialization of succ_signVariations_le_sub_mul to monomials.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/RuleOfSigns.lean#L208-L222"},"Polynomial.signVariations_eraseLead_le":{"signature":"theorem Polynomial.signVariations_eraseLead_le {R : Type u_1} [Semiring R] [LinearOrder R] (P : Polynomial R) :P.eraseLead.signVariations \u2264 P.signVariations","doc":"We can only lose, not gain, sign changes if we drop the leading coefficient.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/RuleOfSigns.lean#L111-L115"},"Polynomial.card_support_mul_le":{"signature":"theorem Polynomial.card_support_mul_le {R : Type u} [Semiring R] {p q : Polynomial R} :(p * q).support.card \u2264 p.support.card * q.support.card","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L60-L65"},"Polynomial.natDegree_eq_one":{"signature":"theorem Polynomial.natDegree_eq_one {R : Type u} [Semiring R] {p : Polynomial R} :p.natDegree = 1 \u2194 \u2203 (a : R), a \u2260 0 \u2227 \u2203 (b : R), C a * X + C b = p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L348-L360"},"Polynomial.eraseLead_C_mul_X":{"signature":"theorem Polynomial.eraseLead_C_mul_X {R : Type u_1} [Semiring R] (r : R) :(C r * X).eraseLead = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/EraseLead.lean#L154-L155"},"Polynomial.nextCoeff_C_mul_X_add_C":{"signature":"theorem Polynomial.nextCoeff_C_mul_X_add_C {R : Type u} [Semiring R] {a : R} (ha : a \u2260 0) (c : R) :(C a * X + C c).nextCoeff = c","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L344-L346"},"Polynomial.two_le_natDegree_of_nextCoeff_eraseLead":{"signature":"theorem Polynomial.two_le_natDegree_of_nextCoeff_eraseLead {R : Type u_1} [Semiring R] {f : Polynomial R} (hlead : f.eraseLead \u2260 0) (hnext : f.nextCoeff = 0) :2 \u2264 f.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/EraseLead.lean#L238-L247"},"Polynomial.eraseLead_mul_eq_mul_eraseLead_of_nextCoeff_zero":{"signature":"theorem Polynomial.eraseLead_mul_eq_mul_eraseLead_of_nextCoeff_zero {R : Type u_2} [Ring R] [NoZeroDivisors R] [Nontrivial R] {x : R} {P : Polynomial R} (hx : x \u2260 0) (h : P.nextCoeff = 0) :((X - C x) * P).eraseLead.eraseLead = (X - C x) * P.eraseLead","doc":"If we erase the leading coefficient of a Polynomial.coeffList like [+,0,...], and then multiply by a linear term, it's equivalent to erasing the first two coefficients of the product.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/EraseLead.lean#L260-L311"},"Polynomial.signVariations_X_sub_C_mul_eraseLead_le":{"signature":"theorem Polynomial.signVariations_X_sub_C_mul_eraseLead_le {R : Type u_1} [Ring R] [LinearOrder R] [IsStrictOrderedRing R] {P : Polynomial R} {\u03b7 : R} (h : 0 < P.leadingCoeff) (h\u2082 : 0 < P.nextCoeff) :((X - C \u03b7) * P.eraseLead).signVariations \u2264 ((X - C \u03b7) * P).signVariations","doc":"If a polynomial starts with two positive coefficients, then the sign changes in the product (X - \u03b7) * P is the same as (X - \u03b7) * P.eraseLead. This lemma lets us do induction on the degree of P when P starts with matching coefficient signs. Of course this is also true when the first two coefficients of P are negative, but we just prove the case where they're positive since it's cleaner and sufficient for the later use.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/RuleOfSigns.lean#L272-L292"},"Polynomial.coeffList_neg":{"signature":"theorem Polynomial.coeffList_neg {R : Type u_1} [Ring R] (P : Polynomial R) :(-P).coeffList = List.map (fun (x : R) => -x) P.coeffList","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/CoeffList.lean#L166-L170"},"Polynomial.signVariations_neg":{"signature":"theorem Polynomial.signVariations_neg {R : Type u_1} [Ring R] [LinearOrder R] [IsOrderedRing R] (P : Polynomial R) :(-P).signVariations = P.signVariations","doc":"The number of sign changes does not change if we negate.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/RuleOfSigns.lean#L129-L142"},"Polynomial.succ_signVariations_le_X_sub_C_mul":{"signature":"theorem Polynomial.succ_signVariations_le_X_sub_C_mul {R : Type u_1} [Ring R] [LinearOrder R] [IsStrictOrderedRing R] {P : Polynomial R} {\u03b7 : R} (h\u03b7 : 0 < \u03b7) (hP : P \u2260 0) :P.signVariations + 1 \u2264 ((X - C \u03b7) * P).signVariations","doc":"Multiplying a polynomial by a linear term X - \u03b7 adds at least one sign change. This is the basis for the induction in roots_countP_pos_le_signVariations.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/RuleOfSigns.lean#L296-L370"},"Polynomial.roots_countP_pos_le_signVariations":{"signature":"theorem Polynomial.roots_countP_pos_le_signVariations {R : Type u_1} [CommRing R] [LinearOrder R] [IsStrictOrderedRing R] (P : Polynomial R) :Multiset.countP (fun (x : R) => 0 < x) P.roots \u2264 P.signVariations","doc":"Descartes' Rule of Signs: the number of positive roots is at most the number of sign variations.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/RuleOfSigns.lean#L377-L392"},"Polynomial.aeval_smul":{"signature":"theorem Polynomial.aeval_smul {R : Type u} {A : Type z} [CommSemiring R] [Semiring A] [Algebra R A] (f : Polynomial R) {G : Type u_3} [Monoid G] [MulSemiringAction G A] [SMulCommClass G R A] (g : G) (x : A) :(aeval (g \u2022 x)) f = g \u2022 (aeval x) f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L403-L405"},"Polynomial.smul_mem_rootSet":{"signature":"theorem Polynomial.smul_mem_rootSet {R : Type u} {S : Type v} [CommRing R] [IsDomain R] [CommRing S] [Algebra S R] {G : Type u_1} [Monoid G] [MulSemiringAction G R] [SMulCommClass G S R] {f : Polynomial S} (g : G) {x : R} (hx : x \u2208 f.rootSet R) :g \u2022 x \u2208 f.rootSet R","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L646-L649"},"Polynomial.instMulActionElemRootSetOfSMulCommClass":{"signature":"instance Polynomial.instMulActionElemRootSetOfSMulCommClass {R : Type u} {S : Type v} [CommRing R] [IsDomain R] [CommRing S] [Algebra S R] (G : Type u_1) [Monoid G] [MulSemiringAction G R] [SMulCommClass G S R] (f : Polynomial S) :MulAction G \u2191(f.rootSet R)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L663-L668"},"Polynomial.X_sub_C_mul_divByMonic_eq_sub_modByMonic":{"signature":"theorem Polynomial.X_sub_C_mul_divByMonic_eq_sub_modByMonic {K : Type u_1} [Ring K] (f : Polynomial K) (a : K) :(X - C a) * (f /\u2098 (X - C a)) = f - f %\u2098 (X - C a)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L630-L633"},"Polynomial.divByMonic_add_X_sub_C_mul_derivative_divByMonic_eq_derivative":{"signature":"theorem Polynomial.divByMonic_add_X_sub_C_mul_derivative_divByMonic_eq_derivative {K : Type u_1} [CommRing K] (f : Polynomial K) (a : K) :f /\u2098 (X - C a) + (X - C a) * derivative (f /\u2098 (X - C a)) = derivative f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L635-L640"},"Polynomial.X_sub_C_dvd_derivative_of_X_sub_C_dvd_divByMonic":{"signature":"theorem Polynomial.X_sub_C_dvd_derivative_of_X_sub_C_dvd_divByMonic {K : Type u_1} [Field K] (f : Polynomial K) {a : K} (hf : X - C a \u2223 f /\u2098 (X - C a)) :X - C a \u2223 derivative f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L642-L647"},"Polynomial.adjSylvester_comp_sylveserMap":{"signature":"theorem Polynomial.adjSylvester_comp_sylveserMap {m n : \u2115} {R : Type u_1} [CommRing R] (f g : Polynomial R) (hf : f.natDegree \u2264 m) (hg : g.natDegree \u2264 n) :f.adjSylvester g \u2218\u2097 f.sylvesterMap g hf hg = f.resultant g m n \u2022 LinearMap.id","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L868-L874"},"Polynomial.coeff_list_sum":{"signature":"theorem Polynomial.coeff_list_sum {R : Type u} [Semiring R] (l : List (Polynomial R)) (n : \u2115) :l.sum.coeff n = (List.map (\u21d1(lcoeff R n)) l).sum","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L93-L95"},"Polynomial.coeff_list_sum_map":{"signature":"theorem Polynomial.coeff_list_sum_map {R : Type u} [Semiring R] {\u03b9 : Type u_1} (l : List \u03b9) (f : \u03b9 \u2192 Polynomial R) (n : \u2115) :(List.map f l).sum.coeff n = (List.map (fun (a : \u03b9) => (f a).coeff n) l).sum","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L97-L99"},"Polynomial.roots_X_pow_char_sub_C":{"signature":"theorem Polynomial.roots_X_pow_char_sub_C {R : Type u_1} [CommRing R] [IsDomain R] {p : \u2115} [ExpChar R p] [PerfectRing R p] {y : R} :(X ^ p - C y).roots = p \u2022 {(frobeniusEquiv R p).symm y}","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Perfect.lean#L363-L366"},"Polynomial.reflect_support":{"signature":"theorem Polynomial.reflect_support {R : Type u_1} [Semiring R] (N : \u2115) (f : Polynomial R) :(reflect N f).support = Finset.image (\u21d1(revAt N)) f.support","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Reverse.lean#L90-L94"},"Polynomial.noZeroDivisors_iff":{"signature":"theorem Polynomial.noZeroDivisors_iff {R : Type u} [Semiring R] :NoZeroDivisors (Polynomial R) \u2194 NoZeroDivisors R","doc":"See also Polynomial.isCancelMulZero_iff: in order for R[X] to have cancellative multiplication (stronger than NoZeroDivisors in general, but equivalent if R is a ring), R must have both cancellative multiplication and cancellative addition.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L1161-L1166"},"Polynomial.aeval_root_derivative_of_splits":{"signature":"theorem Polynomial.aeval_root_derivative_of_splits {R : Type u_1} [CommRing R] {f : Polynomial R} [IsDomain R] [DecidableEq R] (hf : f.Splits) (hm : f.Monic) {x : R} (hx : x \u2208 f.roots) :eval x (derivative f) = (Multiset.map (fun (x_1 : R) => x - x_1) (f.roots.erase x)).prod","doc":"Alias of Polynomial.Splits.eval_root_derivative. Let f be a monic polynomial over that splits. Let x be a root of f. Then $f'(r) = \\prod_{a}(x-a)$, where the product in the RHS is taken over all roots of f, with the multiplicity of x reduced by one.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L874-L875"},"Polynomial.isWeaklyEisensteinAt_iff":{"signature":"theorem Polynomial.isWeaklyEisensteinAt_iff {R : Type u} [CommSemiring R] (f : Polynomial R) (\ud835\udcdf : Ideal R) :f.IsWeaklyEisensteinAt \ud835\udcdf \u2194 \u2200 {n : \u2115}, n < f.natDegree \u2192 f.coeff n \u2208 \ud835\udcdf","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Eisenstein/Basic.lean#L47-L47"},"Polynomial.algEquivAevalXAddC_symm":{"signature":"theorem Polynomial.algEquivAevalXAddC_symm {R : Type u_3} [CommRing R] (t : R) :(algEquivAevalXAddC t).symm = algEquivAevalXAddC (-t)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L364-L367"},"Polynomial.coeff_monomial_succ":{"signature":"theorem Polynomial.coeff_monomial_succ {R : Type u} {a : R} {n : \u2115} [Semiring R] :((monomial (n + 1)) a).coeff 0 = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L614-L615"},"Polynomial.not_isUnit_X":{"signature":"theorem Polynomial.not_isUnit_X {R : Type u} [Semiring R] [Nontrivial R] :\u00acIsUnit X","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L531-L534"},"Polynomial.leval_coe_eq_smeval":{"signature":"theorem Polynomial.leval_coe_eq_smeval {R : Type u_3} [Semiring R] (r : R) :\u21d1(leval r) = fun (p : Polynomial R) => p.smeval r","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Smeval.lean#L135-L138"},"Polynomial.taylor_eval_sub":{"signature":"theorem Polynomial.taylor_eval_sub {R : Type u_1} [CommRing R] (r : R) (f : Polynomial R) (s : R) :eval (s - r) ((taylor r) f) = eval s f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Taylor.lean#L196-L197"},"Polynomial.natTrailingDegree_monomial":{"signature":"theorem Polynomial.natTrailingDegree_monomial {R : Type u} {a : R} {n : \u2115} [Semiring R] (ha : a \u2260 0) :((monomial n) a).natTrailingDegree = n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L174-L176"},"Polynomial.natTrailingDegree_X_pow":{"signature":"theorem Polynomial.natTrailingDegree_X_pow {R : Type u} [Semiring R] [Nontrivial R] (n : \u2115) :(X ^ n).natTrailingDegree = n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L370-L373"},"Polynomial.ofSubring":{"signature":"def Polynomial.ofSubring {R : Type u} [Ring R] (T : Subring R) (p : Polynomial \u21a5T) :Polynomial R","doc":"Given a polynomial whose coefficients are in some subring, return the corresponding polynomial whose coefficients are in the ambient ring.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Basic.lean#L427-L430"},"Polynomial.irreducible_of_dvd_cyclotomic_of_natDegree":{"signature":"theorem Polynomial.irreducible_of_dvd_cyclotomic_of_natDegree {K : Type u_1} [Field K] [Fintype K] {p f n : \u2115} {P : Polynomial K} (hK : Fintype.card K = p ^ f) (hn : p.Coprime n) [hp : Fact (Nat.Prime p)] (hP : P \u2223 cyclotomic n K) (hPdeg : P.natDegree = orderOf (ZMod.unitOfCoprime (p ^ f) \u22ef)) :Irreducible P","doc":"Let K be a finite field of cardinality p ^ f and let P be a factor of the n-th cyclotomic polynomial over K, where p and n are coprime. If the degree of P is the multiplicative order of p ^ f modulo n then P is irreducible.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Factorization.lean#L102-L116"},"Polynomial.degree_sub_eq_right_of_degree_lt":{"signature":"theorem Polynomial.degree_sub_eq_right_of_degree_lt {R : Type u} [Ring R] {p q : Polynomial R} (h : p.degree < q.degree) :(p - q).degree = q.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L577-L579"},"Polynomial.eval\u2082_at_intCast":{"signature":"theorem Polynomial.eval\u2082_at_intCast {R : Type u} [Ring R] {p : Polynomial R} {S : Type u_1} [Ring S] (f : R \u2192+* S) (n : \u2124) :eval\u2082 f (\u2191n) p = f (eval (\u2191n) p)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L769-L773"},"Polynomial.roots_map":{"signature":"theorem Polynomial.roots_map {R : Type u_1} [CommRing R] {f : Polynomial R} [IsDomain R] {S : Type u_4} [CommRing S] [IsDomain S] [IsSimpleRing R] (hf : f.Splits) (i : R \u2192+* S) :(map i f).roots = Multiset.map (\u21d1i) f.roots","doc":"Alias of Polynomial.Splits.roots_map. Alias of Polynomial.Splits.roots_map.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L790-L791"},"Polynomial.mapAlgEquiv_toAlgHom":{"signature":"theorem Polynomial.mapAlgEquiv_toAlgHom {R : Type u} {A : Type z} {B : Type u_2} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A \u2243\u2090[R] B) :\u2191(mapAlgEquiv f) = mapAlgHom \u2191f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L222-L224"},"Polynomial.shiftedLegendre":{"signature":"noncomputable def Polynomial.shiftedLegendre (n : \u2115) :Polynomial \u2124","doc":"shiftedLegendre n is an integer polynomial for each n : \u2115, defined by: P\u2099(x) = \u2211 k \u2208 Finset.range (n + 1), (-1)\u1d4f * choose n k * choose (n + k) n * x\u1d4f These polynomials appear in combinatorics and the theory of orthogonal polynomials.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ShiftedLegendre.lean#L37-L41"},"Polynomial.coeff_shiftedLegendre":{"signature":"theorem Polynomial.coeff_shiftedLegendre (n k : \u2115) :(shiftedLegendre n).coeff k = (-1) ^ k * \u2191(n.choose k) * \u2191((n + k).choose n)","doc":"The coefficient of the shifted Legendre polynomial at k is (-1) ^ k * (n.choose k) * (n + k).choose n.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ShiftedLegendre.lean#L82-L88"},"Polynomial.degree_shiftedLegendre":{"signature":"theorem Polynomial.degree_shiftedLegendre (n : \u2115) :(shiftedLegendre n).degree = \u2191n","doc":"The degree of shiftedLegendre n is n.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ShiftedLegendre.lean#L90-L97"},"Polynomial.natDegree_shiftedLegendre":{"signature":"theorem Polynomial.natDegree_shiftedLegendre (n : \u2115) :(shiftedLegendre n).natDegree = n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ShiftedLegendre.lean#L99-L100"},"Polynomial.coeff_update":{"signature":"theorem Polynomial.coeff_update {R : Type u} [Semiring R] (p : Polynomial R) (n : \u2115) (a : R) :(p.update n a).coeff = Function.update p.coeff n a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L978-L981"},"Polynomial.natDegree_intCast":{"signature":"theorem Polynomial.natDegree_intCast {R : Type u} [Ring R] (n : \u2124) :(\u2191n).natDegree = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L289-L291"},"Polynomial.basisMonomials":{"signature":"def Polynomial.basisMonomials (R : Type u) [Semiring R] :Module.Basis \u2115 R (Polynomial R)","doc":"The monomials form a basis on R[X]. To get the rank of a polynomial ring, use this and Basis.mk_eq_rank.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basis.lean#L27-L30"},"Polynomial.mem_rootSet_of_injective":{"signature":"theorem Polynomial.mem_rootSet_of_injective {R : Type u} {S : Type v} [CommRing R] [IsDomain R] [CommRing S] {p : Polynomial S} [Algebra S R] (h : Function.Injective \u21d1(algebraMap S R)) {x : R} (hp : p \u2260 0) :x \u2208 p.rootSet R \u2194 (aeval x) p = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L634-L638"},"Polynomial.ofFn":{"signature":"def Polynomial.ofFn {R : Type u_1} [Semiring R] [DecidableEq R] (n : \u2115) :(Fin n \u2192 R) \u2192\u2097[R] Polynomial R","doc":"ofFn n v is the polynomial whose coefficients are the entries of the vector v.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/OfFn.lean#L43-L55"},"Polynomial.toFn_comp_ofFn_eq_id":{"signature":"theorem Polynomial.toFn_comp_ofFn_eq_id {R : Type u_1} [Semiring R] [DecidableEq R] (n : \u2115) (v : Fin n \u2192 R) :(toFn n) ((ofFn n) v) = v","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/OfFn.lean#L99-L100"},"Polynomial.isSplittingField_C":{"signature":"theorem Polynomial.isSplittingField_C {K : Type v} [Field K] (a : K) :IsSplittingField K K (C a)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/SplittingField/IsSplittingField.lean#L198-L200"},"Polynomial.eval\u2082_multiset_sum":{"signature":"theorem Polynomial.eval\u2082_multiset_sum {R : Type u} {S : Type v} [Semiring R] [Semiring S] (f : R \u2192+* S) (s : Multiset (Polynomial R)) (x : S) :eval\u2082 f x s.sum = (Multiset.map (eval\u2082 f x) s).sum","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L128-L130"},"Polynomial.eval_multisetSum":{"signature":"theorem Polynomial.eval_multisetSum {R : Type u} [Semiring R] (s : Multiset (Polynomial R)) (x : R) :eval x s.sum = (Multiset.map (eval x) s).sum","doc":"Polynomial evaluation commutes with Multiset.sum.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L330-L332"},"Polynomial.evalEval_intCast":{"signature":"theorem Polynomial.evalEval_intCast {R : Type u_1} [Ring R] (x y : R) (n : \u2124) :evalEval x y \u2191n = \u2191n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Bivariate.lean#L108-L110"},"Polynomial.aroots_X":{"signature":"theorem Polynomial.aroots_X {S : Type v} {T : Type w} [CommRing T] [CommRing S] [IsDomain S] [Algebra T S] :X.aroots S = {0}","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L468-L471"},"Polynomial.aroots_X_pow":{"signature":"theorem Polynomial.aroots_X_pow {S : Type v} {T : Type w} [CommRing T] [CommRing S] [IsDomain S] [Algebra T S] (n : \u2115) :(X ^ n).aroots S = n \u2022 {0}","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L510-L512"},"Polynomial.one_le_cauchyBound":{"signature":"theorem Polynomial.one_le_cauchyBound {K : Type u_1} [NormedDivisionRing K] (p : Polynomial K) :1 \u2264 p.cauchyBound","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/CauchyBound.lean#L36-L38"},"Polynomial.reverse_mul_X_pow":{"signature":"theorem Polynomial.reverse_mul_X_pow {R : Type u_1} [Semiring R] (p : Polynomial R) (n : \u2115) :(p * X ^ n).reverse = p.reverse","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Reverse.lean#L312-L315"},"Polynomial.homogenizeLM":{"signature":"noncomputable def Polynomial.homogenizeLM {R : Type u_1} [CommSemiring R] (n : \u2115) :Polynomial R \u2192\u2097[R] MvPolynomial (Fin 2) R","doc":"homogenize as a bundled linear map.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Homogenize.lean#L57-L62"},"Polynomial.homogenize_finsetSum":{"signature":"theorem Polynomial.homogenize_finsetSum {R : Type u_1} [CommSemiring R] {\u03b9 : Type u_2} (s : Finset \u03b9) (p : \u03b9 \u2192 Polynomial R) (n : \u2115) :(\u2211 i \u2208 s, p i).homogenize n = \u2211 i \u2208 s, (p i).homogenize n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Homogenize.lean#L64-L67"},"Polynomial.homogenize_eq_of_isHomogeneous":{"signature":"theorem Polynomial.homogenize_eq_of_isHomogeneous {R : Type u_1} [CommSemiring R] {p : Polynomial R} {n : \u2115} {q : MvPolynomial (Fin 2) R} (hq : q.IsHomogeneous n) (hpq : (MvPolynomial.aeval ![X, 1]) q = p) :p.homogenize n = q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Homogenize.lean#L152-L164"},"Polynomial.isHomogeneous_homogenize":{"signature":"theorem Polynomial.isHomogeneous_homogenize {R : Type u_1} [CommSemiring R] {n : \u2115} (p : Polynomial R) :(p.homogenize n).IsHomogeneous n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Homogenize.lean#L144-L150"},"Polynomial.homogenize_mul":{"signature":"theorem Polynomial.homogenize_mul {R : Type u_1} [CommSemiring R] (p q : Polynomial R) {m n : \u2115} (hm : p.natDegree \u2264 m) (hn : q.natDegree \u2264 n) :(p * q).homogenize (m + n) = p.homogenize m * q.homogenize n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Homogenize.lean#L166-L170"},"Polynomial.homogenize_dvd":{"signature":"theorem Polynomial.homogenize_dvd {R : Type u_1} [CommSemiring R] [NoZeroDivisors R] {p q : Polynomial R} (h : p \u2223 q) :p.homogenize p.natDegree \u2223 q.homogenize q.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Homogenize.lean#L182-L189"},"Polynomial.evalEval_multiset_prod":{"signature":"theorem Polynomial.evalEval_multiset_prod {R : Type u_1} [CommSemiring R] (x y : R) (l : Multiset (Polynomial (Polynomial R))) :evalEval x y l.prod = (Multiset.map (evalEval x y) l).prod","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Bivariate.lean#L131-L133"},"Polynomial.splits_of_algHom":{"signature":"theorem Polynomial.splits_of_algHom {R : Type u_1} [CommSemiring R] {f : Polynomial R} {A : Type u_2} {B : Type u_3} [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (hf : (map (algebraMap R A) f).Splits) (e : A \u2192\u2090[R] B) :(map (algebraMap R B) f).Splits","doc":"Alias of Polynomial.Splits.of_algHom.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L859-L860"},"Polynomial.taylor_X":{"signature":"theorem Polynomial.taylor_X {R : Type u_1} [Semiring R] (r : R) :(taylor r) X = X + C r","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Taylor.lean#L46-L47"},"Polynomial.ofFinsupp_zsmul":{"signature":"theorem Polynomial.ofFinsupp_zsmul {R : Type u} [Ring R] (a : \u2124) (b : AddMonoidAlgebra R \u2115) :{ toFinsupp := a \u2022 b } = a \u2022 { toFinsupp := b }","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L1072-L1075"},"Polynomial.degree_cubic_le":{"signature":"theorem Polynomial.degree_cubic_le {R : Type u} {a b c d : R} [Semiring R] :(C a * X ^ 3 + C b * X ^ 2 + C c * X + C d).degree \u2264 3","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/SmallDegree.lean#L119-L122"},"Polynomial.root_mul_left_of_isRoot":{"signature":"theorem Polynomial.root_mul_left_of_isRoot {R : Type u} {a : R} [CommSemiring R] (p : Polynomial R) {q : Polynomial R} :q.IsRoot a \u2192 (p * q).IsRoot a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L635-L636"},"Polynomial.coeff_ofNat_zero":{"signature":"theorem Polynomial.coeff_ofNat_zero {R : Type u} [Semiring R] (a : \u2115) [a.AtLeastTwo] :(OfNat.ofNat a).coeff 0 = OfNat.ofNat a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L649-L652"},"Polynomial.natTrailingDegree_le_of_trailingDegree_le":{"signature":"theorem Polynomial.natTrailingDegree_le_of_trailingDegree_le {R : Type u} [Semiring R] {p : Polynomial R} {n : \u2115} {hp : p \u2260 0} (H : \u2191n \u2264 p.trailingDegree) :n \u2264 p.natTrailingDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L161-L163"},"Polynomial.nextCoeff_eq_neg_sum_roots_mul_leadingCoeff_of_splits":{"signature":"theorem Polynomial.nextCoeff_eq_neg_sum_roots_mul_leadingCoeff_of_splits {R : Type u_1} [CommRing R] {f : Polynomial R} [IsDomain R] (hf : f.Splits) :f.nextCoeff = -f.leadingCoeff * f.roots.sum","doc":"Alias of Polynomial.Splits.nextCoeff_eq_neg_sum_roots_mul_leadingCoeff.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L890-L892"},"Polynomial.preimage_eval_singleton":{"signature":"theorem Polynomial.preimage_eval_singleton {R : Type u} {a : R} [CommRing R] [IsDomain R] {p : Polynomial R} (hp : p \u2260 C a) :(fun (x : R) => eval x p) \u207b\u00b9' {a} = (p - C a).rootSet R","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L605-L606"},"Polynomial.toPowerSeries_toMvPowerSeries":{"signature":"theorem Polynomial.toPowerSeries_toMvPowerSeries {R : Type u_2} [CommRing R] (p : Polynomial R) :\u2191p = \u2191((aeval (MvPolynomial.X ())) p)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/PowerSeries/Substitution.lean#L237-L246"},"Polynomial.ofFinsupp_one":{"signature":"theorem Polynomial.ofFinsupp_one {R : Type u} [Semiring R] :{ toFinsupp := 1 } = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L143-L145"},"Polynomial.ofFinsupp_eq_one":{"signature":"theorem Polynomial.ofFinsupp_eq_one {R : Type u} [Semiring R] {a : AddMonoidAlgebra R \u2115} :{ toFinsupp := a } = 1 \u2194 a = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L247-L248"},"Polynomial.natTrailingDegree_eq_zero_of_constantCoeff_ne_zero":{"signature":"theorem Polynomial.natTrailingDegree_eq_zero_of_constantCoeff_ne_zero {R : Type u} [Semiring R] {p : Polynomial R} (h : constantCoeff p \u2260 0) :p.natTrailingDegree = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L430-L432"},"Polynomial.sumIDeriv_X":{"signature":"theorem Polynomial.sumIDeriv_X :sumIDeriv X = X + C 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/SumIteratedDerivative.lean#L84-L87"},"Polynomial.divByMonic_eq_of_not_monic":{"signature":"theorem Polynomial.divByMonic_eq_of_not_monic {R : Type u} [Ring R] {q : Polynomial R} (p : Polynomial R) (hq : \u00acq.Monic) :p /\u2098 q = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L204-L205"},"Polynomial.degree_divByMonic_le":{"signature":"theorem Polynomial.degree_divByMonic_le {R : Type u} [Ring R] (p q : Polynomial R) :(p /\u2098 q).degree \u2264 p.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L289-L302"},"Polynomial.degree_div_le":{"signature":"theorem Polynomial.degree_div_le {R : Type u} [Field R] (p q : Polynomial R) :(p / q).degree \u2264 p.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L396-L399"},"Polynomial.smeval_neg_nat":{"signature":"theorem Polynomial.smeval_neg_nat (S : Type u_3) [NonAssocRing S] [Pow S \u2115] [NatPowAssoc S] (q : Polynomial \u2115) (n : \u2115) :q.smeval (-\u2191n) = \u2191(q.smeval (-\u2191n))","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Smeval.lean#L161-L165"},"Polynomial.natDegree_ofNat":{"signature":"theorem Polynomial.natDegree_ofNat {R : Type u} [Semiring R] (n : \u2115) [n.AtLeastTwo] :(OfNat.ofNat n).natDegree = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L182-L185"},"Polynomial.mul_X_add_natCast_comp":{"signature":"theorem Polynomial.mul_X_add_natCast_comp {R : Type u} [Semiring R] {p q : Polynomial R} {n : \u2115} :(p * (X + \u2191n)).comp q = p.comp q * (q + \u2191n)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L445-L447"},"Polynomial.wronskian_add_left":{"signature":"theorem Polynomial.wronskian_add_left {R : Type u_1} [CommRing R] (a b c : Polynomial R) :(a + b).wronskian c = a.wronskian c + b.wronskian c","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Wronskian.lean#L70-L71"},"Polynomial.natDegree_iterate_derivative":{"signature":"theorem Polynomial.natDegree_iterate_derivative {R : Type u} [Semiring R] (p : Polynomial R) (k : \u2115) :((\u21d1derivative)^[k] p).natDegree \u2264 p.natDegree - k","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L185-L191"},"Polynomial.as_sum_support_C_mul_X_pow":{"signature":"theorem Polynomial.as_sum_support_C_mul_X_pow {R : Type u} [Semiring R] (p : Polynomial R) :p = \u2211 i \u2208 p.support, C (p.coeff i) * X ^ i","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Support.lean#L58-L59"},"Polynomial.coeff_iterate_derivative":{"signature":"theorem Polynomial.coeff_iterate_derivative {R : Type u} [Semiring R] {k : \u2115} (p : Polynomial R) (m : \u2115) :((\u21d1derivative)^[k] p).coeff m = (m + k).descFactorial k \u2022 p.coeff (m + k)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L319-L333"},"Polynomial.iterate_derivative_eq_sum":{"signature":"theorem Polynomial.iterate_derivative_eq_sum {R : Type u} [Semiring R] (p : Polynomial R) (k : \u2115) :(\u21d1derivative)^[k] p = \u2211 x \u2208 ((\u21d1derivative)^[k] p).support, C ((x + k).descFactorial k \u2022 p.coeff (x + k)) * X ^ x","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L335-L340"},"Polynomial.iterate_derivative_eq_factorial_smul_sum":{"signature":"theorem Polynomial.iterate_derivative_eq_factorial_smul_sum {R : Type u} [Semiring R] (p : Polynomial R) (k : \u2115) :(\u21d1derivative)^[k] p = k.factorial \u2022 \u2211 x \u2208 ((\u21d1derivative)^[k] p).support, C ((x + k).choose k \u2022 p.coeff (x + k)) * X ^ x","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L342-L348"},"Polynomial.exists_iterate_derivative_eq_factorial_smul":{"signature":"theorem Polynomial.exists_iterate_derivative_eq_factorial_smul {R : Type u_1} [Semiring R] (p : Polynomial R) (k : \u2115) :\u2203 (gp : Polynomial R), gp.natDegree \u2264 p.natDegree - k \u2227 (\u21d1derivative)^[k] p = k.factorial \u2022 gp","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/SumIteratedDerivative.lean#L108-L115"},"Polynomial.aeval_iterate_derivative_of_ge":{"signature":"theorem Polynomial.aeval_iterate_derivative_of_ge {R : Type u_1} [CommSemiring R] (A : Type u_3) [CommRing A] [Algebra R A] (p : Polynomial R) (q : \u2115) {k : \u2115} (hk : q \u2264 k) :\u2203 (gp : Polynomial R), gp.natDegree \u2264 p.natDegree - k \u2227 \u2200 (r : A), (aeval r) ((\u21d1derivative)^[k] p) = q.factorial \u2022 (aeval r) gp","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/SumIteratedDerivative.lean#L156-L163"},"Polynomial.aeval_iterate_derivative_of_lt":{"signature":"theorem Polynomial.aeval_iterate_derivative_of_lt {R : Type u_1} [CommSemiring R] {A : Type u_3} [CommRing A] [Algebra R A] (p : Polynomial R) (q : \u2115) (r : A) {p' : Polynomial A} (hp : map (algebraMap R A) p = (X - C r) ^ q * p') {k : \u2115} (hk : k < q) :(aeval r) ((\u21d1derivative)^[k] p) = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/SumIteratedDerivative.lean#L123-L133"},"Polynomial.aeval_iterate_derivative_self":{"signature":"theorem Polynomial.aeval_iterate_derivative_self {R : Type u_1} [CommSemiring R] {A : Type u_3} [CommRing A] [Algebra R A] (p : Polynomial R) (q : \u2115) (r : A) {p' : Polynomial A} (hp : map (algebraMap R A) p = (X - C r) ^ q * p') :(aeval r) ((\u21d1derivative)^[q] p) = q.factorial \u2022 eval r p'","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/SumIteratedDerivative.lean#L135-L152"},"Polynomial.aeval_sumIDeriv_of_pos":{"signature":"theorem Polynomial.aeval_sumIDeriv_of_pos {R : Type u_1} [CommSemiring R] (A : Type u_3) [CommRing A] [Algebra R A] [Nontrivial A] [NoZeroDivisors A] (p : Polynomial R) {q : \u2115} (hq : 0 < q) (inj_amap : Function.Injective \u21d1(algebraMap R A)) :\u2203 (gp : Polynomial R), gp.natDegree \u2264 p.natDegree - q \u2227 \u2200 (r : A) {p' : Polynomial A}, map (algebraMap R A) p = (X - C r) ^ (q - 1) * p' \u2192 (aeval r) (sumIDeriv p) = (q - 1).factorial \u2022 eval r p' + q.factorial \u2022 (aeval r) gp","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/SumIteratedDerivative.lean#L196-L253"},"Polynomial.eval_sumIDeriv_of_pos":{"signature":"theorem Polynomial.eval_sumIDeriv_of_pos {R : Type u_1} [CommRing R] [Nontrivial R] [NoZeroDivisors R] (p : Polynomial R) {q : \u2115} (hq : 0 < q) :\u2203 (gp : Polynomial R), gp.natDegree \u2264 p.natDegree - q \u2227 \u2200 (r : R) {p' : Polynomial R}, p = (X - C r) ^ (q - 1) * p' \u2192 eval r (sumIDeriv p) = (q - 1).factorial \u2022 eval r p' + q.factorial \u2022 eval r gp","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/SumIteratedDerivative.lean#L257-L263"},"Polynomial.aeval_sumIDeriv":{"signature":"theorem Polynomial.aeval_sumIDeriv {R : Type u_1} [CommSemiring R] (A : Type u_3) [CommRing A] [Algebra R A] (p : Polynomial R) (q : \u2115) :\u2203 (gp : Polynomial R), gp.natDegree \u2264 p.natDegree - q \u2227 \u2200 (r : A), (X - C r) ^ q \u2223 map (algebraMap R A) p \u2192 (aeval r) (sumIDeriv p) = q.factorial \u2022 (aeval r) gp","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/SumIteratedDerivative.lean#L169-L194"},"Polynomial.aeval_sumIDeriv_eq_eval":{"signature":"theorem Polynomial.aeval_sumIDeriv_eq_eval {R : Type u_1} [CommSemiring R] (A : Type u_3) [CommRing A] [Algebra R A] (p : Polynomial R) (r : A) :(aeval r) (sumIDeriv p) = eval r (sumIDeriv (map (algebraMap R A) p))","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/SumIteratedDerivative.lean#L165-L167"},"Polynomial.derivative_monomial_succ":{"signature":"theorem Polynomial.derivative_monomial_succ {R : Type u} [Semiring R] (a : R) (n : \u2115) :derivative ((monomial (n + 1)) a) = (monomial n) (a * (\u2191n + 1))","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L84-L87"},"Polynomial.eval\u2082_primPart_eq_zero":{"signature":"theorem Polynomial.eval\u2082_primPart_eq_zero {R : Type u_1} [CommRing R] [NormalizedGCDMonoid R] {S : Type u_2} [CommSemiring S] [IsDomain S] {f : R \u2192+* S} (hinj : Function.Injective \u21d1f) {p : Polynomial R} {s : S} (hpzero : p \u2260 0) (hp : eval\u2082 f s p = 0) :eval\u2082 f s p.primPart = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Content.lean#L294-L299"},"Polynomial.roots_smul_nonzero":{"signature":"theorem Polynomial.roots_smul_nonzero {R : Type u} {a : R} [CommRing R] [IsDomain R] (p : Polynomial R) (ha : a \u2260 0) :(a \u2022 p).roots = p.roots","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L211-L213"},"Polynomial.roots_neg":{"signature":"theorem Polynomial.roots_neg {R : Type u} [CommRing R] [IsDomain R] (p : Polynomial R) :(-p).roots = p.roots","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L215-L217"},"Polynomial.aroots_neg":{"signature":"theorem Polynomial.aroots_neg {S : Type v} {T : Type w} [CommRing T] [CommRing S] [IsDomain S] [Algebra T S] (p : Polynomial T) :(-p).aroots S = p.aroots S","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L486-L489"},"Polynomial.rootSet_neg":{"signature":"theorem Polynomial.rootSet_neg {T : Type w} [CommRing T] (p : Polynomial T) (S : Type u_1) [CommRing S] [IsDomain S] [Algebra T S] :(-p).rootSet S = p.rootSet S","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L558-L561"},"Polynomial.taylorEquiv":{"signature":"noncomputable def Polynomial.taylorEquiv {R : Type u_1} [CommRing R] (r : R) :Polynomial R \u2243\u2090[R] Polynomial R","doc":"Polynomial.taylor as an AlgEquiv for commutative rings.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Taylor.lean#L182-L187"},"Polynomial.nthRootsFinset_toSet":{"signature":"theorem Polynomial.nthRootsFinset_toSet {R : Type u} [CommRing R] [IsDomain R] {n : \u2115} (h : 0 < n) (a : R) :\u2191(nthRootsFinset n a) = {r : R | r ^ n = a}","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L640-L644"},"Polynomial.splits_iff_splits":{"signature":"theorem Polynomial.splits_iff_splits {R : Type u_1} [Field R] {f : Polynomial R} :f.Splits \u2194 f = 0 \u2228 \u2200 {g : Polynomial R}, Irreducible g \u2192 g \u2223 f \u2192 g.degree = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L621-L634"},"Polynomial.mem_nonZeroDivisors_of_leadingCoeff":{"signature":"theorem Polynomial.mem_nonZeroDivisors_of_leadingCoeff {R : Type u} [CommRing R] {p : Polynomial R} (h : p.leadingCoeff \u2208 nonZeroDivisors R) :p \u2208 nonZeroDivisors (Polynomial R)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/RingDivision.lean#L130-L131"},"Polynomial.taylor_mem_degreeLT":{"signature":"theorem Polynomial.taylor_mem_degreeLT {R : Type u_1} [CommRing R] {r : R} {n : \u2115} {f : Polynomial R} :(taylor r) f \u2208 degreeLT R n \u2194 f \u2208 degreeLT R n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/DegreeLT.lean#L157-L158"},"Polynomial.comap_taylorEquiv_degreeLT":{"signature":"theorem Polynomial.comap_taylorEquiv_degreeLT {R : Type u_1} [CommRing R] {r : R} {n : \u2115} :Submodule.comap (\u2191\u2191(taylorEquiv r)) (degreeLT R n) = degreeLT R n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/DegreeLT.lean#L160-L161"},"Polynomial.map_taylorEquiv_degreeLT":{"signature":"theorem Polynomial.map_taylorEquiv_degreeLT {R : Type u_1} [CommRing R] {r : R} {n : \u2115} :Submodule.map (\u2191\u2191(taylorEquiv r)) (degreeLT R n) = degreeLT R n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/DegreeLT.lean#L163-L165"},"Polynomial.taylorLinearEquiv":{"signature":"noncomputable def Polynomial.taylorLinearEquiv {R : Type u_1} [CommRing R] (r : R) (n : \u2115) :\u21a5(degreeLT R n) \u2243\u2097[R] \u21a5(degreeLT R n)","doc":"The map taylor r induces an automorphism of the module R[X]_n of polynomials of degree < n.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/DegreeLT.lean#L167-L171"},"Polynomial.support_integralNormalization":{"signature":"theorem Polynomial.support_integralNormalization {R : Type u} [Semiring R] [IsCancelMulZero R] {f : Polynomial R} :f.integralNormalization.support = f.support","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/IntegralNormalization.lean#L188-L198"},"Polynomial.lifts_iff_coeffs_subset_range":{"signature":"theorem Polynomial.lifts_iff_coeffs_subset_range {R : Type u} [Semiring R] {S : Type v} [Semiring S] {f : R \u2192+* S} (p : Polynomial S) :p \u2208 lifts f \u2194 \u2191p.coeffs \u2286 Set.range \u21d1f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Lifts.lean#L76-L86"},"Polynomial.gaussNorm_eq_zero_iff":{"signature":"theorem Polynomial.gaussNorm_eq_zero_iff {R : Type u_1} {F : Type u_2} [Semiring R] [FunLike F R \u211d] (v : F) {c : \u211d} (p : Polynomial R) [ZeroHomClass F R \u211d] [NonnegHomClass F R \u211d] (h_eq_zero : \u2200 (x : R), v x = 0 \u2192 x = 0) (hc : 0 < c) :gaussNorm v c p = 0 \u2194 p = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/GaussNorm.lean#L110-L115"},"Polynomial.roots_expand_image_frobenius_subset":{"signature":"theorem Polynomial.roots_expand_image_frobenius_subset {R : Type u_1} [CommRing R] [IsDomain R] (p : \u2115) [ExpChar R p] (f : Polynomial R) [DecidableEq R] :Finset.image (\u21d1(frobenius R p)) ((expand R p) f).roots.toFinset \u2286 f.roots.toFinset","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Perfect.lean#L332-L336"},"Polynomial.rootsExpandToRoots":{"signature":"noncomputable def Polynomial.rootsExpandToRoots {R : Type u_1} [CommRing R] [IsDomain R] (p : \u2115) [ExpChar R p] (f : Polynomial R) [DecidableEq R] :\u21a5((expand R p) f).roots.toFinset \u21aa \u21a5f.roots.toFinset","doc":"If f is a polynomial over an integral domain R of characteristic p, then there is a map from the set of roots of Polynomial.expand R p f to the set of roots of f. It's given by x \u21a6 x ^ p, see rootsExpandToRoots_apply.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Perfect.lean#L395-L400"},"Polynomial.cyclotomic'_two":{"signature":"theorem Polynomial.cyclotomic'_two (R : Type u_2) [CommRing R] [IsDomain R] (p : \u2115) [CharP R p] (hp : p \u2260 2) :cyclotomic' 2 R = X + 1","doc":"The second modified cyclotomic polynomial is X + 1 if the characteristic of R is not 2.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean#L77-L85"},"Polynomial.roots_eq_zero_of_irreducible_of_natDegree_ne_one":{"signature":"theorem Polynomial.roots_eq_zero_of_irreducible_of_natDegree_ne_one {R : Type u} [CommRing R] [IsDomain R] {p : Polynomial R} (hirr : Irreducible p) (hdeg : p.natDegree \u2260 1) :p.roots = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L833-L838"},"Polynomial.mirror_monomial":{"signature":"theorem Polynomial.mirror_monomial {R : Type u_1} [Semiring R] (n : \u2115) (a : R) :((monomial n) a).mirror = (monomial n) a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Mirror.lean#L45-L51"},"Polynomial.mirror_X":{"signature":"theorem Polynomial.mirror_X {R : Type u_1} [Semiring R] :X.mirror = X","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Mirror.lean#L56-L57"},"Polynomial.coeff_natTrailingDegree_eq_zero_of_trailingDegree_lt":{"signature":"theorem Polynomial.coeff_natTrailingDegree_eq_zero_of_trailingDegree_lt {R : Type u} [Semiring R] {p q : Polynomial R} (h : p.trailingDegree < q.trailingDegree) :q.coeff p.natTrailingDegree = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L423-L425"},"Polynomial.iterate_comp_eval\u2082":{"signature":"theorem Polynomial.iterate_comp_eval\u2082 {R : Type u} {S : Type v} [Semiring R] {p q : Polynomial R} [CommSemiring S] (f : R \u2192+* S) (k : \u2115) (t : S) :eval\u2082 f t (p.comp^[k] q) = (fun (x : S) => eval\u2082 f x p)^[k] (eval\u2082 f t q)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Degree.lean#L208-L213"},"Polynomial.iterate_comp_eval":{"signature":"theorem Polynomial.iterate_comp_eval {R : Type u} [CommSemiring R] {p q : Polynomial R} (k : \u2115) (t : R) :eval t (p.comp^[k] q) = (fun (x : R) => eval x p)^[k] (eval t q)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Degree.lean#L221-L224"},"Polynomial.X_pow_add_C_ne_one":{"signature":"theorem Polynomial.X_pow_add_C_ne_one {R : Type u} [Nontrivial R] [Semiring R] {n : \u2115} (hn : 0 < n) (a : R) :X ^ n + C a \u2260 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L637-L638"},"Polynomial.X_add_C_ne_one":{"signature":"theorem Polynomial.X_add_C_ne_one {R : Type u} [Nontrivial R] [Semiring R] (r : R) :X + C r \u2260 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L640-L641"},"Polynomial.divX_add":{"signature":"theorem Polynomial.divX_add {R : Type u} [Semiring R] {p q : Polynomial R} :(p + q).divX = p.divX + q.divX","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Inductions.lean#L61-L62"},"Polynomial.divX_hom":{"signature":"noncomputable def Polynomial.divX_hom {R : Type u} [Semiring R] :Polynomial R \u2192+ Polynomial R","doc":"divX as an additive homomorphism.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Inductions.lean#L83-L88"},"Polynomial.divX_hom_toFun":{"signature":"theorem Polynomial.divX_hom_toFun {R : Type u} [Semiring R] {p : Polynomial R} :divX_hom p = p.divX","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Inductions.lean#L90-L90"},"Polynomial.smeval_commute_left":{"signature":"theorem Polynomial.smeval_commute_left (R : Type u_1) [Semiring R] (p : Polynomial R) {S : Type u_2} [Semiring S] [Module R S] [IsScalarTower R S S] [SMulCommClass R S S] {x y : S} (hc : Commute x y) :Commute (p.smeval x) y","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Smeval.lean#L290-L302"},"Polynomial.smeval_commute":{"signature":"theorem Polynomial.smeval_commute (R : Type u_1) [Semiring R] (p q : Polynomial R) {S : Type u_2} [Semiring S] [Module R S] [IsScalarTower R S S] [SMulCommClass R S S] {x y : S} (hc : Commute x y) :Commute (p.smeval x) (q.smeval y)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Smeval.lean#L304-L318"},"Polynomial.supp_subset_range_natDegree_succ":{"signature":"theorem Polynomial.supp_subset_range_natDegree_succ {R : Type u} [Semiring R] {p : Polynomial R} :p.support \u2286 Finset.range (p.natDegree + 1)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Support.lean#L52-L53"},"Polynomial.card_supp_le_succ_natDegree":{"signature":"theorem Polynomial.card_supp_le_succ_natDegree {R : Type u} [Semiring R] (p : Polynomial R) :p.support.card \u2264 p.natDegree + 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Support.lean#L108-L110"},"Polynomial.degree_linear":{"signature":"theorem Polynomial.degree_linear {R : Type u} {a b : R} [Semiring R] (ha : a \u2260 0) :(C a * X + C b).degree = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/SmallDegree.lean#L75-L77"},"Polynomial.cauchyBound_X_sub_C":{"signature":"theorem Polynomial.cauchyBound_X_sub_C {K : Type u_1} [NormedDivisionRing K] (x : K) :(X - C x).cauchyBound = \u2016x\u2016\u208a + 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/CauchyBound.lean#L59-L61"},"Polynomial.nextCoeff_eq_neg_sum_roots_of_monic_of_splits":{"signature":"theorem Polynomial.nextCoeff_eq_neg_sum_roots_of_monic_of_splits {R : Type u_1} [CommRing R] {f : Polynomial R} [IsDomain R] (hf : f.Splits) (hm : f.Monic) :f.nextCoeff = -f.roots.sum","doc":"Alias of Polynomial.Splits.nextCoeff_eq_neg_sum_roots_of_monic. If f is a monic polynomial that splits, then f.nextCoeff equals the negative of the sum of the roots.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L894-L895"},"Polynomial.sum_roots_eq_nextCoeff_of_monic_of_split":{"signature":"theorem Polynomial.sum_roots_eq_nextCoeff_of_monic_of_split {R : Type u_1} [CommRing R] {f : Polynomial R} [IsDomain R] (hf : f.Splits) (hm : f.Monic) :f.nextCoeff = -f.roots.sum","doc":"Alias of Polynomial.Splits.nextCoeff_eq_neg_sum_roots_of_monic. Alias of Polynomial.Splits.nextCoeff_eq_neg_sum_roots_of_monic. If f is a monic polynomial that splits, then f.nextCoeff equals the negative of the sum of the roots.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L900-L901"},"Polynomial.roots_map_of_map_ne_zero_of_card_eq_natDegree":{"signature":"theorem Polynomial.roots_map_of_map_ne_zero_of_card_eq_natDegree {A : Type u_1} {B : Type u_2} [CommRing A] [CommRing B] [IsDomain A] [IsDomain B] {p : Polynomial A} (f : A \u2192+* B) (h : map f p \u2260 0) (hroots : p.roots.card = p.natDegree) :Multiset.map (\u21d1f) p.roots = (map f p).roots","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L959-L963"},"Polynomial.coeff_zero_eq_prod_roots_of_monic_of_splits":{"signature":"theorem Polynomial.coeff_zero_eq_prod_roots_of_monic_of_splits {R : Type u_1} [CommRing R] {f : Polynomial R} [IsDomain R] (hf : f.Splits) (hm : f.Monic) :f.coeff 0 = (-1) ^ f.natDegree * f.roots.prod","doc":"Alias of Polynomial.Splits.coeff_zero_eq_prod_roots_of_monic. If f is a monic polynomial that splits, then coeff f 0 equals the product of the roots.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L887-L888"},"Polynomial.eval\u2082_X_mul":{"signature":"theorem Polynomial.eval\u2082_X_mul {R : Type u} {S : Type v} [Semiring R] {p : Polynomial R} [Semiring S] (f : R \u2192+* S) (x : S) :eval\u2082 f x (X * p) = eval\u2082 f x p * x","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L155-L156"},"Polynomial.int":{"signature":"def Polynomial.int {K : Type u_1} [Field K] (R : Subring K) (P : Polynomial K) (hP : \u2200 (n : \u2115), P.coeff n \u2208 R) :Polynomial \u21a5R","doc":"Given a polynomial in K[X] such that all coefficients belong to the subring R, Polynomial.int is the corresponding polynomial in R[X].","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Ring/Subring/IntPolynomial.lean#L29-L36"},"Polynomial.int_leadingCoeff_eq":{"signature":"theorem Polynomial.int_leadingCoeff_eq {K : Type u_1} [Field K] (R : Subring K) (P : Polynomial K) (hP : \u2200 (n : \u2115), P.coeff n \u2208 R) :\u2191(int R P hP).leadingCoeff = P.leadingCoeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Ring/Subring/IntPolynomial.lean#L45-L46"},"Polynomial.derivativeFinsupp_apply_apply":{"signature":"theorem Polynomial.derivativeFinsupp_apply_apply {R : Type u} [Semiring R] (p : Polynomial R) (x\u271d : \u2115) :(derivativeFinsupp p) x\u271d = (\u21d1derivative)^[x\u271d] p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L397-L397"},"Polynomial.derivativeFinsupp_apply_toFun":{"signature":"theorem Polynomial.derivativeFinsupp_apply_toFun {R : Type u} [Semiring R] (p : Polynomial R) (x\u271d : \u2115) :(derivativeFinsupp p) x\u271d = (\u21d1derivative)^[x\u271d] p","doc":"Alias of Polynomial.derivativeFinsupp_apply_apply.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L404-L405"},"Polynomial.factorial_smul_hasseDeriv":{"signature":"theorem Polynomial.factorial_smul_hasseDeriv {R : Type u_1} [Semiring R] (k : \u2115) :\u21d1(k.factorial \u2022 hasseDeriv k) = (\u21d1derivative)^[k]","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/HasseDeriv.lean#L128-L147"},"Polynomial.leadingCoeff_mul_prod_normalizedFactors":{"signature":"theorem Polynomial.leadingCoeff_mul_prod_normalizedFactors {R : Type u} [Field R] [DecidableEq R] (a : Polynomial R) :C a.leadingCoeff * (UniqueFactorizationMonoid.normalizedFactors a).prod = a","doc":"The normalized factors of a polynomial over a field times its leading coefficient give the polynomial.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L693-L703"},"Polynomial.eval_one_cyclotomic_not_prime_pow":{"signature":"theorem Polynomial.eval_one_cyclotomic_not_prime_pow {R : Type u_1} [Ring R] {n : \u2115} (h : \u2200 {p : \u2115}, Nat.Prime p \u2192 \u2200 (k : \u2115), p ^ k \u2260 n) :eval 1 (cyclotomic n R) = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Eval.lean#L136-L168"},"Polynomial.hasStrictDerivAt_aeval":{"signature":"theorem Polynomial.hasStrictDerivAt_aeval {\ud835\udd5c : Type u} [NontriviallyNormedField \ud835\udd5c] {R : Type u_1} [CommSemiring R] [Algebra R \ud835\udd5c] (q : Polynomial R) (x : \ud835\udd5c) :HasStrictDerivAt (fun (x : \ud835\udd5c) => (aeval x) q) ((aeval x) (derivative q)) x","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Calculus/Deriv/Polynomial.lean#L60-L63"},"Polynomial.hasDerivAt_aeval":{"signature":"theorem Polynomial.hasDerivAt_aeval {\ud835\udd5c : Type u} [NontriviallyNormedField \ud835\udd5c] {R : Type u_1} [CommSemiring R] [Algebra R \ud835\udd5c] (q : Polynomial R) (x : \ud835\udd5c) :HasDerivAt (fun (x : \ud835\udd5c) => (aeval x) q) ((aeval x) (derivative q)) x","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Calculus/Deriv/Polynomial.lean#L69-L71"},"Polynomial.differentiableAt_aeval":{"signature":"theorem Polynomial.differentiableAt_aeval {\ud835\udd5c : Type u} [NontriviallyNormedField \ud835\udd5c] {x : \ud835\udd5c} {R : Type u_1} [CommSemiring R] [Algebra R \ud835\udd5c] (q : Polynomial R) :DifferentiableAt \ud835\udd5c (fun (x : \ud835\udd5c) => (aeval x) q) x","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Calculus/Deriv/Polynomial.lean#L84-L85"},"Polynomial.differentiable_aeval":{"signature":"theorem Polynomial.differentiable_aeval {\ud835\udd5c : Type u} [NontriviallyNormedField \ud835\udd5c] {R : Type u_1} [CommSemiring R] [Algebra R \ud835\udd5c] (q : Polynomial R) :Differentiable \ud835\udd5c fun (x : \ud835\udd5c) => (aeval x) q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Calculus/Deriv/Polynomial.lean#L97-L98"},"Polynomial.opRingEquiv":{"signature":"def Polynomial.opRingEquiv (R : Type u_2) [Semiring R] :(Polynomial R)\u1d50\u1d52\u1d56 \u2243+* Polynomial R\u1d50\u1d52\u1d56","doc":"Ring isomorphism between R[X]\u1d50\u1d52\u1d56 and R\u1d50\u1d52\u1d56[X] sending each coefficient of a polynomial to the corresponding element of the opposite ring.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Opposites.lean#L31-L35"},"Polynomial.opRingEquiv_op_monomial":{"signature":"theorem Polynomial.opRingEquiv_op_monomial {R : Type u_1} [Semiring R] (n : \u2115) (r : R) :(opRingEquiv R) (MulOpposite.op ((monomial n) r)) = (monomial n) (MulOpposite.op r)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Opposites.lean#L41-L44"},"Polynomial.opRingEquiv_symm_monomial":{"signature":"theorem Polynomial.opRingEquiv_symm_monomial {R : Type u_1} [Semiring R] (n : \u2115) (r : R\u1d50\u1d52\u1d56) :(opRingEquiv R).symm ((monomial n) r) = MulOpposite.op ((monomial n) (MulOpposite.unop r))","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Opposites.lean#L62-L65"},"Polynomial.opRingEquiv_symm_C_mul_X_pow":{"signature":"theorem Polynomial.opRingEquiv_symm_C_mul_X_pow {R : Type u_1} [Semiring R] (r : R\u1d50\u1d52\u1d56) (n : \u2115) :(opRingEquiv R).symm (C r * X ^ n) = MulOpposite.op (C (MulOpposite.unop r) * X ^ n)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Opposites.lean#L75-L77"},"Polynomial.isNilpotent_pow_X_mul_C_of_isNilpotent":{"signature":"theorem Polynomial.isNilpotent_pow_X_mul_C_of_isNilpotent {R : Type u_1} {r : R} [Semiring R] (n : \u2115) (hnil : IsNilpotent r) :IsNilpotent (X ^ n * C r)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Nilpotent.lean#L43-L46"},"Polynomial.aroots_C_mul":{"signature":"theorem Polynomial.aroots_C_mul {S : Type v} {T : Type w} [CommRing T] [IsDomain T] [CommRing S] [IsDomain S] [Algebra T S] [Module.IsTorsionFree T S] {a : T} (p : Polynomial T) (ha : a \u2260 0) :(C a * p).aroots S = p.aroots S","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L491-L497"},"Polynomial.aroots_C_mul_X_pow":{"signature":"theorem Polynomial.aroots_C_mul_X_pow {S : Type v} {T : Type w} [CommRing T] [IsDomain T] [CommRing S] [IsDomain S] [Algebra T S] [Module.IsTorsionFree T S] {a : T} (ha : a \u2260 0) (n : \u2115) :(C a * X ^ n).aroots S = n \u2022 {0}","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L514-L517"},"Polynomial.aroots_monomial":{"signature":"theorem Polynomial.aroots_monomial {S : Type v} {T : Type w} [CommRing T] [IsDomain T] [CommRing S] [IsDomain S] [Algebra T S] [Module.IsTorsionFree T S] {a : T} (ha : a \u2260 0) (n : \u2115) :((monomial n) a).aroots S = n \u2022 {0}","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L519-L523"},"Polynomial.rootSet_monomial":{"signature":"theorem Polynomial.rootSet_monomial {R : Type u} {S : Type v} [Field R] [CommRing S] [IsDomain S] [Algebra R S] {n : \u2115} (hn : n \u2260 0) {a : R} (ha : a \u2260 0) :((monomial n) a).rootSet S = {0}","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L500-L504"},"Polynomial.rootSet_C_mul_X_pow":{"signature":"theorem Polynomial.rootSet_C_mul_X_pow {R : Type u} {S : Type v} [Field R] [CommRing S] [IsDomain S] [Algebra R S] {n : \u2115} (hn : n \u2260 0) {a : R} (ha : a \u2260 0) :(C a * X ^ n).rootSet S = {0}","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L506-L508"},"Polynomial.rootSet_X_pow":{"signature":"theorem Polynomial.rootSet_X_pow {R : Type u} {S : Type v} [Field R] [CommRing S] [IsDomain S] [Algebra R S] {n : \u2115} (hn : n \u2260 0) :(X ^ n).rootSet S = {0}","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L510-L513"},"Polynomial.coe_def":{"signature":"theorem Polynomial.coe_def {R : Type u_1} [Semiring R] (\u03c6 : Polynomial R) :\u2191\u03c6 = PowerSeries.mk \u03c6.coeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/PowerSeries/Basic.lean#L806-L807"},"Polynomial.toContinuousMapAlgHom":{"signature":"def Polynomial.toContinuousMapAlgHom {R : Type u_1} [CommSemiring R] [TopologicalSpace R] [IsTopologicalSemiring R] :Polynomial R \u2192\u2090[R] C(R, R)","doc":"The algebra map from R[X] to continuous functions C(R, R).","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Topology/ContinuousMap/Polynomial.lean#L89-L108"},"Polynomial.aeval_ne_zero_of_isCoprime":{"signature":"theorem Polynomial.aeval_ne_zero_of_isCoprime {S : Type v} {R : Type u_1} [CommSemiring R] [Nontrivial S] [Semiring S] [Algebra R S] {p q : Polynomial R} (h : IsCoprime p q) (s : S) :(aeval s) p \u2260 0 \u2228 (aeval s) q \u2260 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/RingDivision.lean#L255-L260"},"Polynomial.degree_pos_of_ne_zero_of_nonunit":{"signature":"theorem Polynomial.degree_pos_of_ne_zero_of_nonunit {R : Type u} [DivisionRing R] {p : Polynomial R} (hp0 : p \u2260 0) (hp : \u00acIsUnit p) :0 < p.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L256-L259"},"Polynomial.isCoprime_iff_aeval_ne_zero_of_isAlgClosed":{"signature":"theorem Polynomial.isCoprime_iff_aeval_ne_zero_of_isAlgClosed (k : Type u) [Field k] (K : Type v) [Field K] [IsAlgClosed K] [Algebra k K] (p q : Polynomial k) :IsCoprime p q \u2194 \u2200 (a : K), (aeval a) p \u2260 0 \u2228 (aeval a) q \u2260 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/IsAlgClosed/Basic.lean#L262-L271"},"Polynomial.ncard_rootSet_le":{"signature":"theorem Polynomial.ncard_rootSet_le {A : Type u_1} [CommRing A] (p : Polynomial A) (B : Type u_3) [CommRing B] [IsDomain B] [Algebra A B] :(p.rootSet B).ncard \u2264 p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L924-L928"},"Polynomial.coeToPowerSeries":{"signature":"instance Polynomial.coeToPowerSeries {R : Type u_1} [Semiring R] :Coe (Polynomial R) (PowerSeries R)","doc":"The natural inclusion from polynomials into formal power series.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/PowerSeries/Basic.lean#L802-L804"},"Polynomial.natDegree_add_coeff_mul":{"signature":"theorem Polynomial.natDegree_add_coeff_mul {R : Type u} [Semiring R] (f g : Polynomial R) :(f * g).coeff (f.natDegree + g.natDegree) = f.coeff f.natDegree * g.coeff g.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L143-L145"},"Polynomial.coeff_zero_of_isScalarTower":{"signature":"theorem Polynomial.coeff_zero_of_isScalarTower {A : Type u_3} (B : Type u_4) (C : Type u_5) [CommSemiring A] [CommSemiring B] [Semiring C] [Algebra A B] [Algebra A C] [Algebra B C] [IsScalarTower A B C] (p : Polynomial A) :(algebraMap B C) ((algebraMap A B) (p.coeff 0)) = ((mapAlg A C) p).coeff 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L316-L318"},"Polynomial.monic_mapAlg_iff":{"signature":"theorem Polynomial.monic_mapAlg_iff {R : Type u} {S : Type v} [Field R] [Semiring S] [Nontrivial S] [Algebra R S] {p : Polynomial R} :((mapAlg R S) p).Monic \u2194 p.Monic","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L720-L722"},"Polynomial.isRoot_cyclotomic_prime_pow_mul_iff_of_charP":{"signature":"theorem Polynomial.isRoot_cyclotomic_prime_pow_mul_iff_of_charP {m k p : \u2115} {R : Type u_1} [CommRing R] [IsDomain R] [hp : Fact (Nat.Prime p)] [hchar : CharP R p] {\u03bc : R} [NeZero \u2191m] :(cyclotomic (p ^ k * m) R).IsRoot \u03bc \u2194 IsPrimitiveRoot \u03bc m","doc":"If R is of characteristic p and \u00acp \u2223 m, then \u03b6 is a root of cyclotomic (p ^ k * m) R if and only if it is a primitive m-th root of unity.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Expand.lean#L169-L184"},"Polynomial.support_X_pow":{"signature":"theorem Polynomial.support_X_pow {R : Type u} [Semiring R] (H : \u00ac1 = 0) (n : \u2115) :(X ^ n).support = {n}","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L801-L803"},"Polynomial.coeff_natCast_mul":{"signature":"theorem Polynomial.coeff_natCast_mul {R : Type u} [Semiring R] {p : Polynomial R} {a k : \u2115} :(\u2191a * p).coeff k = \u2191a * p.coeff k","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L170-L171"},"Polynomial.derivativeFinsupp_C":{"signature":"theorem Polynomial.derivativeFinsupp_C {R : Type u} [Semiring R] (r : R) :derivativeFinsupp (C r) = Finsupp.single 0 (C r)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L413-L418"},"Polynomial.resultant_eq_zero_iff":{"signature":"theorem Polynomial.resultant_eq_zero_iff {K : Type u_2} [Field K] {f g : Polynomial K} :f.resultant g = 0 \u2194 (f \u2260 0 \u2228 g \u2260 0) \u2227 \u00acIsCoprime f g","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L911-L922"},"Polynomial.modByMonic_eq_mod":{"signature":"theorem Polynomial.modByMonic_eq_mod {R : Type u} [Field R] {q : Polynomial R} (p : Polynomial R) (hq : q.Monic) :p %\u2098 q = p % q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L339-L341"},"Polynomial.mod_X_sub_C_eq_C_eval":{"signature":"theorem Polynomial.mod_X_sub_C_eq_C_eval {R : Type u} [Field R] (p : Polynomial R) (a : R) :p % (X - C a) = C (eval a p)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L347-L348"},"Polynomial.homogenizeLM_apply":{"signature":"theorem Polynomial.homogenizeLM_apply {R : Type u_1} [CommSemiring R] (n : \u2115) (p : Polynomial R) :(homogenizeLM n) p = p.homogenize n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Homogenize.lean#L58-L58"},"Polynomial.eval_list_prod":{"signature":"theorem Polynomial.eval_list_prod {R : Type u} [CommSemiring R] (l : List (Polynomial R)) (x : R) :eval x l.prod = (List.map (eval x) l).prod","doc":"Polynomial evaluation commutes with List.prod","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L649-L652"},"Polynomial.evalEval_list_prod":{"signature":"theorem Polynomial.evalEval_list_prod {R : Type u_1} [CommSemiring R] (x y : R) (l : List (Polynomial (Polynomial R))) :evalEval x y l.prod = (List.map (evalEval x y) l).prod","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Bivariate.lean#L127-L129"},"Polynomial.scaleRoots_aeval_eq_zero_of_aeval_div_eq_zero":{"signature":"theorem Polynomial.scaleRoots_aeval_eq_zero_of_aeval_div_eq_zero {R : Type u_1} {K : Type u_4} [CommSemiring R] [Field K] [Algebra R K] (inj : Function.Injective \u21d1(algebraMap R K)) {p : Polynomial R} {r s : R} (hr : (aeval ((algebraMap R K) r / (algebraMap R K) s)) p = 0) (hs : s \u2208 nonZeroDivisors R) :(aeval ((algebraMap R K) r)) (p.scaleRoots s) = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ScaleRoots.lean#L176-L180"},"Polynomial.discr_C":{"signature":"theorem Polynomial.discr_C {R : Type u_1} [CommRing R] (r : R) :(C r).discr = 1","doc":"The discriminant of a constant polynomial is 1.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L938-L941"},"Polynomial.rootMultiplicity_eq_rootMultiplicity":{"signature":"theorem Polynomial.rootMultiplicity_eq_rootMultiplicity {R : Type u} [CommRing R] {p : Polynomial R} {t : R} :rootMultiplicity t p = rootMultiplicity 0 (p.comp (X + C t))","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/RingDivision.lean#L109-L116"},"Polynomial.rootMultiplicity_eq_natTrailingDegree'":{"signature":"theorem Polynomial.rootMultiplicity_eq_natTrailingDegree' {R : Type u} [CommRing R] {p : Polynomial R} :rootMultiplicity 0 p = p.natTrailingDegree","doc":"See Polynomial.rootMultiplicity_eq_natTrailingDegree for the general case.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L749-L758"},"Polynomial.rootMultiplicity_eq_natTrailingDegree":{"signature":"theorem Polynomial.rootMultiplicity_eq_natTrailingDegree {R : Type u} [CommRing R] {p : Polynomial R} {t : R} :rootMultiplicity t p = (p.comp (X + C t)).natTrailingDegree","doc":"See Polynomial.rootMultiplicity_eq_natTrailingDegree' for the special case of t = 0.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/RingDivision.lean#L118-L121"},"Polynomial.reverse_X_pow_mul":{"signature":"theorem Polynomial.reverse_X_pow_mul {R : Type u_1} [Semiring R] (p : Polynomial R) (n : \u2115) :(X ^ n * p).reverse = p.reverse","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Reverse.lean#L317-L318"},"Polynomial.eval_divByMonic_eq_trailingCoeff_comp":{"signature":"theorem Polynomial.eval_divByMonic_eq_trailingCoeff_comp {R : Type u} [CommRing R] {p : Polynomial R} {t : R} :eval t (p /\u2098 (X - C t) ^ rootMultiplicity t p) = (p.comp (X + C t)).trailingCoeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L700-L712"},"Polynomial.rootMultiplicity_mul'":{"signature":"theorem Polynomial.rootMultiplicity_mul' {R : Type u} [CommRing R] {p q : Polynomial R} {x : R} (hpq : eval x (p /\u2098 (X - C x) ^ rootMultiplicity x p) * eval x (q /\u2098 (X - C x) ^ rootMultiplicity x q) \u2260 0) :rootMultiplicity x (p * q) = rootMultiplicity x p + rootMultiplicity x q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/RingDivision.lean#L179-L184"},"Polynomial.isRoot_of_unity_of_root_cyclotomic":{"signature":"theorem Polynomial.isRoot_of_unity_of_root_cyclotomic {R : Type u_1} [CommRing R] {n : \u2115} {\u03b6 : R} {i : \u2115} (hi : i \u2208 n.divisors) (h : (cyclotomic i R).IsRoot \u03b6) :\u03b6 ^ n = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Roots.lean#L42-L51"},"Polynomial.leadingCoeff_eq_zero_iff_deg_eq_bot":{"signature":"theorem Polynomial.leadingCoeff_eq_zero_iff_deg_eq_bot {R : Type u} [Semiring R] {p : Polynomial R} :p.leadingCoeff = 0 \u2194 p.degree = \u22a5","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L365-L366"},"Polynomial.le_trailingDegree_monomial":{"signature":"theorem Polynomial.le_trailingDegree_monomial {R : Type u} {a : R} {n : \u2115} [Semiring R] :\u2191n \u2264 ((monomial n) a).trailingDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L182-L184"},"Polynomial.le_trailingDegree_C":{"signature":"theorem Polynomial.le_trailingDegree_C {R : Type u} {a : R} [Semiring R] :0 \u2264 (C a).trailingDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L190-L191"},"Polynomial.coeff_one_add_X_pow":{"signature":"theorem Polynomial.coeff_one_add_X_pow (R : Type u_1) [Semiring R] (n k : \u2115) :((1 + X) ^ n).coeff k = \u2191(n.choose k)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L302-L303"},"Polynomial.trailingDegree_eq_iff_natTrailingDegree_eq":{"signature":"theorem Polynomial.trailingDegree_eq_iff_natTrailingDegree_eq {R : Type u} [Semiring R] {p : Polynomial R} {n : \u2115} (hp : p \u2260 0) :p.trailingDegree = \u2191n \u2194 p.natTrailingDegree = n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L97-L99"},"Polynomial.coeff_natTrailingDegree_eq_zero":{"signature":"theorem Polynomial.coeff_natTrailingDegree_eq_zero {R : Type u} [Semiring R] {p : Polynomial R} :p.coeff p.natTrailingDegree = 0 \u2194 p = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L124-L132"},"Polynomial.roots_C_mul_X_sub_C":{"signature":"theorem Polynomial.roots_C_mul_X_sub_C {R : Type u} {a : R} [Field R] (b : R) (ha : a \u2260 0) :(C a * X - C b).roots = {a\u207b\u00b9 * b}","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L522-L523"},"Polynomial.coeff_monomial_mul":{"signature":"theorem Polynomial.coeff_monomial_mul {R : Type u} [Semiring R] (p : Polynomial R) (n d : \u2115) (r : R) :((monomial n) r * p).coeff (d + n) = r * p.coeff d","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L262-L264"},"Polynomial.coeff_monomial_zero_mul":{"signature":"theorem Polynomial.coeff_monomial_zero_mul {R : Type u} [Semiring R] (p : Polynomial R) (d : \u2115) (r : R) :((monomial 0) r * p).coeff d = r * p.coeff d","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L272-L274"},"Polynomial.exists_forall_norm_le":{"signature":"theorem Polynomial.exists_forall_norm_le {R : Type u_2} [NormedRing R] [IsAbsoluteValue norm] [ProperSpace R] (p : Polynomial R) :\u2203 (x : R), \u2200 (y : R), \u2016eval x p\u2016 \u2264 \u2016eval y p\u2016","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Topology/Algebra/Polynomial.lean#L136-L140"},"Polynomial.sylvesterDeriv_updateRow":{"signature":"theorem Polynomial.sylvesterDeriv_updateRow {R : Type u_1} [Semiring R] (f : Polynomial R) (hf : 0 < f.natDegree) :f.sylvesterDeriv.updateRow \u27e82 * f.natDegree - 2, \u22ef\u27e9 (f.leadingCoeff \u2022 f.sylvesterDeriv \u27e82 * f.natDegree - 2, \u22ef\u27e9) = (derivative f).sylvester f (f.natDegree - 1) f.natDegree","doc":"We can get the usual Sylvester matrix of f and f.derivative back from the modified one by multiplying the last row by the leading coefficient of f.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L94-L127"},"Polynomial.resultant_deriv":{"signature":"theorem Polynomial.resultant_deriv {R : Type u_1} [CommRing R] {f : Polynomial R} (hf : 0 < f.degree) :f.resultant (derivative f) f.natDegree (f.natDegree - 1) = (-1) ^ (f.natDegree * (f.natDegree - 1) / 2) * f.leadingCoeff * f.discr","doc":"Relation between the resultant and the discriminant. (Note this is actually false when f is a constant polynomial not equal to 1, so the assumption on the degree is genuinely needed.)","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L978-L989"},"Polynomial.separable_gcd_left":{"signature":"theorem Polynomial.separable_gcd_left {F : Type u_1} [Field F] [DecidableEq (Polynomial F)] {f : Polynomial F} (hf : f.Separable) (g : Polynomial F) :(EuclideanDomain.gcd f g).Separable","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Separable.lean#L96-L99"},"Polynomial.splits_iff_comp_splits_of_natDegree_eq_one":{"signature":"theorem Polynomial.splits_iff_comp_splits_of_natDegree_eq_one {R : Type u_1} [Field R] {f g : Polynomial R} (hg : g.natDegree = 1) :f.Splits \u2194 (f.comp g).Splits","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L568-L578"},"Polynomial.splits_iff_comp_splits_of_degree_eq_one":{"signature":"theorem Polynomial.splits_iff_comp_splits_of_degree_eq_one {R : Type u_1} [Field R] {f g : Polynomial R} (hg : g.degree = 1) :f.Splits \u2194 (f.comp g).Splits","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L580-L582"},"Polynomial.dickson_zero":{"signature":"theorem Polynomial.dickson_zero {R : Type u_1} [CommRing R] (k : \u2115) (a : R) :dickson k a 0 = 3 - \u2191k","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Dickson.lean#L68-L70"},"Polynomial.rootOfSplits'":{"signature":"def Polynomial.rootOfSplits' {R : Type u_1} [CommRing R] {f : Polynomial R} (hf : f.Splits) (hfd : f.degree \u2260 0) :R","doc":"Alias of Polynomial.rootOfSplits. Pick a root of a polynomial that splits.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L725-L726"},"Polynomial.aevalTower_comp_C":{"signature":"theorem Polynomial.aevalTower_comp_C {R : Type u} {S : Type v} {A' : Type u_1} [CommSemiring R] [CommSemiring A'] [CommSemiring S] [Algebra S R] [Algebra S A'] (g : R \u2192\u2090[S] A') (y : A') :(\u2191(aevalTower g y)).comp C = \u2191g","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L535-L537"},"Polynomial.sylvesterMap_apply_coe":{"signature":"theorem Polynomial.sylvesterMap_apply_coe {m n : \u2115} {R : Type u_1} [CommRing R] (f g : Polynomial R) (hf : f.natDegree \u2264 m) (hg : g.natDegree \u2264 n) (pq : \u21a5(degreeLT R m) \u00d7 \u21a5(degreeLT R n)) :\u2191((f.sylvesterMap g hf hg) pq) = f * \u2191pq.2 + g * \u2191pq.1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L809-L809"},"Polynomial.smeval_at_zero":{"signature":"theorem Polynomial.smeval_at_zero (R : Type u_1) [Semiring R] (p : Polynomial R) {S : Type u_2} [NonAssocSemiring S] [Module R S] [Pow S \u2115] [NatPowAssoc S] :p.smeval 0 = p.coeff 0 \u2022 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Smeval.lean#L198-L205"},"Polynomial.TrailingMonic":{"signature":"def Polynomial.TrailingMonic {R : Type u} [Semiring R] (p : Polynomial R) :Prop","doc":"a polynomial is monic_at if its trailing coefficient is 1","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L63-L65"},"Polynomial.evalEval_finset_sum":{"signature":"theorem Polynomial.evalEval_finset_sum {R : Type u_1} [Semiring R] {\u03b9 : Type u_3} (s : Finset \u03b9) (x y : R) (f : \u03b9 \u2192 Polynomial (Polynomial R)) :evalEval x y (\u2211 i \u2208 s, f i) = \u2211 i \u2208 s, evalEval x y (f i)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Bivariate.lean#L81-L83"},"Polynomial.degree_smul_of_isRightRegular_leadingCoeff":{"signature":"theorem Polynomial.degree_smul_of_isRightRegular_leadingCoeff {R : Type u} {a : R} [Semiring R] {p : Polynomial R} (ha : a \u2260 0) (hp : IsRightRegular p.leadingCoeff) :(a \u2022 p).degree = p.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L466-L470"},"Polynomial.degree_sum_eq_of_disjoint":{"signature":"theorem Polynomial.degree_sum_eq_of_disjoint {R : Type u} {S : Type v} [Semiring R] (f : S \u2192 Polynomial R) (s : Finset S) (h : {i : S | i \u2208 s \u2227 f i \u2260 0}.Pairwise (Function.onFun Ne (degree \u2218 f))) :(s.sum f).degree = s.sup fun (i : S) => (f i).degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L181-L205"},"Polynomial.degree_eq_card_roots'":{"signature":"theorem Polynomial.degree_eq_card_roots' {R : Type u_1} [CommRing R] {f : Polynomial R} [IsDomain R] (hf : f.Splits) (hf0 : f \u2260 0) :f.degree = \u2191f.roots.card","doc":"Alias of Polynomial.Splits.degree_eq_card_roots.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L734-L735"},"Polynomial.mem_lift_of_splits_of_roots_mem_range":{"signature":"theorem Polynomial.mem_lift_of_splits_of_roots_mem_range {R : Type u_1} [CommRing R] {f : Polynomial R} [IsDomain R] (hf : f.Splits) (hm : f.Monic) {S : Type u_4} [Ring S] (i : S \u2192+* R) (hr : \u2200 a \u2208 f.roots, a \u2208 i.range) :f \u2208 lifts i","doc":"Alias of Polynomial.Splits.mem_lift_of_roots_mem_range.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L823-L824"},"Polynomial.one_add_X_pow_sub_X_pow":{"signature":"theorem Polynomial.one_add_X_pow_sub_X_pow {S : Type u_1} [CommRing S] (d : \u2115) :(1 + X) ^ d - X ^ d = \u2211 i \u2208 Finset.range d, d.choose i \u2022 X ^ i","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L305-L309"},"Polynomial.hilbertPoly_succ":{"signature":"theorem Polynomial.hilbertPoly_succ {F : Type u_1} [Field F] (p : Polynomial F) (d : \u2115) :p.hilbertPoly (d + 1) = \u2211 i \u2208 p.support, p.coeff i \u2022 preHilbertPoly F d i","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/HilbertPoly.lean#L121-L122"},"Polynomial.mem_support_C_mul_X_pow":{"signature":"theorem Polynomial.mem_support_C_mul_X_pow {R : Type u} [Semiring R] {n a : \u2115} {c : R} (h : a \u2208 (C c * X ^ n).support) :a = n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Support.lean#L101-L102"},"Polynomial.degree_mul_X_pow":{"signature":"theorem Polynomial.degree_mul_X_pow {R : Type u} [Semiring R] [Nontrivial R] {p : Polynomial R} (n : \u2115) :(p * X ^ n).degree = p.degree + \u2191n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L539-L540"},"Polynomial.roots_X_pow_char_pow_sub_C_pow":{"signature":"theorem Polynomial.roots_X_pow_char_pow_sub_C_pow {R : Type u_1} [CommRing R] [IsDomain R] {p n : \u2115} [ExpChar R p] [PerfectRing R p] {y : R} {m : \u2115} :((X ^ p ^ n - C y) ^ m).roots = (m * p ^ n) \u2022 {(iterateFrobeniusEquiv R p n).symm y}","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Perfect.lean#L359-L361"},"Polynomial.det_taylorLinearEquiv_toLinearMap":{"signature":"theorem Polynomial.det_taylorLinearEquiv_toLinearMap {R : Type u_1} [CommRing R] {r : R} {n : \u2115} :LinearMap.det \u2191(taylorLinearEquiv r n) = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/DegreeLT.lean#L177-L189"},"Polynomial.det_taylorLinearEquiv":{"signature":"theorem Polynomial.det_taylorLinearEquiv {R : Type u_1} [CommRing R] {r : R} {n : \u2115} :LinearEquiv.det (taylorLinearEquiv r n) = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/DegreeLT.lean#L191-L193"},"Polynomial.monic_C_mul_of_mul_leadingCoeff_eq_one":{"signature":"theorem Polynomial.monic_C_mul_of_mul_leadingCoeff_eq_one {R : Type u} [Semiring R] {p : Polynomial R} {b : R} (hp : b * p.leadingCoeff = 1) :(C b * p).Monic","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Monic.lean#L63-L67"},"Polynomial.ker_evalRingHom":{"signature":"theorem Polynomial.ker_evalRingHom {R : Type u_1} [CommRing R] (x : R) :RingHom.ker (evalRingHom x) = Ideal.span {X - C x}","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Ideal.lean#L40-L42"},"Polynomial.quotientSpanXSubCAlgEquiv":{"signature":"noncomputable def Polynomial.quotientSpanXSubCAlgEquiv {R : Type u_1} [CommRing R] (x : R) :(Polynomial R \u29f8 Ideal.span {X - C x}) \u2243\u2090[R] R","doc":"For a commutative ring $R$, evaluating a polynomial at an element $x \\in R$ induces an isomorphism of $R$-algebras $R[X] / \\langle X - x \\rangle \\cong R$.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Quotient.lean#L30-L37"},"Polynomial.quotientSpanXSubCAlgEquiv_symm_apply":{"signature":"theorem Polynomial.quotientSpanXSubCAlgEquiv_symm_apply {R : Type u_1} [CommRing R] (x y : R) :(quotientSpanXSubCAlgEquiv x).symm y = (algebraMap R (Polynomial R \u29f8 Ideal.span {X - C x})) y","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Quotient.lean#L44-L47"},"Polynomial.coeff_ofSubring":{"signature":"theorem Polynomial.coeff_ofSubring {R : Type u} [Ring R] (T : Subring R) (p : Polynomial \u21a5T) (n : \u2115) :(ofSubring T p).coeff n = \u2191(p.coeff n)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Basic.lean#L432-L436"},"Polynomial.coeffs_ofSubring":{"signature":"theorem Polynomial.coeffs_ofSubring {R : Type u} [Ring R] (T : Subring R) {p : Polynomial \u21a5T} :\u2191(ofSubring T p).coeffs \u2286 \u2191T","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Basic.lean#L438-L446"},"Polynomial.CC":{"signature":"abbrev Polynomial.CC {R : Type u_1} [Semiring R] (r : R) :Polynomial (Polynomial R)","doc":"A constant viewed as a polynomial in two variables.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Bivariate.lean#L46-L47"},"Polynomial.coe_algebraMap_eq_CC":{"signature":"theorem Polynomial.coe_algebraMap_eq_CC {R : Type u_1} [CommSemiring R] :\u21d1(algebraMap R (Polynomial (Polynomial R))) = CC","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Bivariate.lean#L142-L142"},"Polynomial.dickson_one_one_eq_chebyshev_C":{"signature":"theorem Polynomial.dickson_one_one_eq_chebyshev_C (R : Type u_1) [CommRing R] (n : \u2115) :dickson 1 1 n = Chebyshev.C R \u2191n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Dickson.lean#L141-L151"},"Polynomial.chebyshev_T_eq_dickson_one_one":{"signature":"theorem Polynomial.chebyshev_T_eq_dickson_one_one (R : Type u_1) [CommRing R] [Invertible 2] (n : \u2115) :Chebyshev.T R \u2191n = C \u215f2 * (dickson 1 1 n).comp (2 * X)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Dickson.lean#L157-L159"},"Polynomial.coeff_update_apply":{"signature":"theorem Polynomial.coeff_update_apply {R : Type u} [Semiring R] (p : Polynomial R) (n : \u2115) (a : R) (i : \u2115) :(p.update n a).coeff i = if i = n then a else p.coeff i","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L983-L985"},"Polynomial.update_zero_eq_erase":{"signature":"theorem Polynomial.update_zero_eq_erase {R : Type u} [Semiring R] (p : Polynomial R) (n : \u2115) :p.update n 0 = erase n p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L994-L997"},"Polynomial.roots_cyclotomic_nodup":{"signature":"theorem Polynomial.roots_cyclotomic_nodup {R : Type u_1} [CommRing R] {n : \u2115} [IsDomain R] [NeZero \u2191n] :(cyclotomic n R).roots.Nodup","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Roots.lean#L105-L111"},"Polynomial.ne_zero_of_coe_le_degree":{"signature":"theorem Polynomial.ne_zero_of_coe_le_degree {R : Type u} {n : \u2115} [Semiring R] {p : Polynomial R} (hdeg : \u2191n \u2264 p.degree) :p \u2260 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/SmallDegree.lean#L62-L63"},"Polynomial.X_pow_mem_lifts":{"signature":"theorem Polynomial.X_pow_mem_lifts {R : Type u} [Semiring R] {S : Type v} [Semiring S] (f : R \u2192+* S) (n : \u2115) :X ^ n \u2208 lifts f","doc":"The polynomial X ^ n lifts.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Lifts.lean#L104-L107"},"Polynomial.coeff_update_ne":{"signature":"theorem Polynomial.coeff_update_ne {R : Type u} [Semiring R] (p : Polynomial R) {n : \u2115} (a : R) {i : \u2115} (h : i \u2260 n) :(p.update n a).coeff i = p.coeff i","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L991-L992"},"Polynomial.leading_coeff_le_mahlerMeasure":{"signature":"theorem Polynomial.leading_coeff_le_mahlerMeasure (p : Polynomial \u2102) :\u2016p.leadingCoeff\u2016 \u2264 p.mahlerMeasure","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/MahlerMeasure.lean#L237-L240"},"Polynomial.norm_leadingCoeff_eq_one_of_mahlerMeasure_eq_one":{"signature":"theorem Polynomial.norm_leadingCoeff_eq_one_of_mahlerMeasure_eq_one {p : Polynomial \u2124} (h : (map (Int.castRingHom \u2102) p).mahlerMeasure = 1) :\u2016(map (Int.castRingHom \u2102) p).leadingCoeff\u2016 = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/NumberTheory/MahlerMeasure.lean#L142-L149"},"Polynomial.abs_leadingCoeff_eq_one_of_mahlerMeasure_eq_one":{"signature":"theorem Polynomial.abs_leadingCoeff_eq_one_of_mahlerMeasure_eq_one {p : Polynomial \u2124} (h : (map (Int.castRingHom \u2102) p).mahlerMeasure = 1) :|p.leadingCoeff| = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/NumberTheory/MahlerMeasure.lean#L152-L155"},"Polynomial.isIntegral_of_mahlerMeasure_eq_one":{"signature":"theorem Polynomial.isIntegral_of_mahlerMeasure_eq_one {p : Polynomial \u2124} (h : (map (Int.castRingHom \u2102) p).mahlerMeasure = 1) {z : \u2102} (hz : z \u2208 p.aroots \u2102) :IsIntegral \u2124 z","doc":"If an integer polynomial has Mahler measure equal to 1, then all its complex roots are integral over \u2124.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/NumberTheory/MahlerMeasure.lean#L160-L166"},"Polynomial.prod_max_one_norm_roots_le_mahlerMeasure_of_one_le_leadingCoeff":{"signature":"theorem Polynomial.prod_max_one_norm_roots_le_mahlerMeasure_of_one_le_leadingCoeff {p : Polynomial \u2102} (hlc : 1 \u2264 \u2016p.leadingCoeff\u2016) :(Multiset.map (fun (a : \u2102) => max 1 \u2016a\u2016) p.roots).prod \u2264 p.mahlerMeasure","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/MahlerMeasure.lean#L242-L246"},"Polynomial.norm_root_le_one_of_mahlerMeasure_eq_one":{"signature":"theorem Polynomial.norm_root_le_one_of_mahlerMeasure_eq_one {p : Polynomial \u2124} (h : (map (Int.castRingHom \u2102) p).mahlerMeasure = 1) {z : \u2102} (hz : z \u2208 p.aroots \u2102) :\u2016z\u2016 \u2264 1","doc":"If an integer polynomial has Mahler measure equal to 1, then all its complex roots have norm at most 1.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/NumberTheory/MahlerMeasure.lean#L171-L179"},"Polynomial.pow_eq_one_of_mahlerMeasure_eq_one":{"signature":"theorem Polynomial.pow_eq_one_of_mahlerMeasure_eq_one {p : Polynomial \u2124} (h : (map (Int.castRingHom \u2102) p).mahlerMeasure = 1) {z : \u2102} (hz\u2080 : z \u2260 0) (hz : z \u2208 p.aroots \u2102) :\u2203 (n : \u2115), 0 < n \u2227 z ^ n = 1","doc":"If an integer polynomial has Mahler measure equal to 1, then all its complex nonzero roots are roots of unity.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/NumberTheory/MahlerMeasure.lean#L183-L210"},"Polynomial.isPrimitiveRoot_of_mahlerMeasure_eq_one":{"signature":"theorem Polynomial.isPrimitiveRoot_of_mahlerMeasure_eq_one {p : Polynomial \u2124} (h : (map (Int.castRingHom \u2102) p).mahlerMeasure = 1) {z : \u2102} (hz\u2080 : z \u2260 0) (hz : z \u2208 p.aroots \u2102) :\u2203 (n : \u2115), 0 < n \u2227 IsPrimitiveRoot z n","doc":"If an integer polynomial has Mahler measure equal to 1, then all its complex nonzero roots are roots of unity.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/NumberTheory/MahlerMeasure.lean#L213-L217"},"Polynomial.cyclotomic_dvd_of_mahlerMeasure_eq_one":{"signature":"theorem Polynomial.cyclotomic_dvd_of_mahlerMeasure_eq_one {p : Polynomial \u2124} (h : (map (Int.castRingHom \u2102) p).mahlerMeasure = 1) (hX : \u00acX \u2223 p) (hpdeg : p.degree \u2260 0) :\u2203 (n : \u2115), 0 < n \u2227 cyclotomic n \u2124 \u2223 p","doc":"If an integer non-constant polynomial has Mahler measure equal to 1 and is not a multiple of X, then it is divisible by a cyclotomic polynomial.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/NumberTheory/MahlerMeasure.lean#L220-L236"},"Polynomial.evalEval_prod":{"signature":"theorem Polynomial.evalEval_prod {R : Type u_1} [CommSemiring R] {\u03b9 : Type u_3} (s : Finset \u03b9) (x y : R) (p : \u03b9 \u2192 Polynomial (Polynomial R)) :evalEval x y (\u220f j \u2208 s, p j) = \u220f j \u2208 s, evalEval x y (p j)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Bivariate.lean#L123-L125"},"Polynomial.ker_modByMonicHom":{"signature":"theorem Polynomial.ker_modByMonicHom {R : Type u_1} [CommRing R] {q : Polynomial R} (hq : q.Monic) :q.modByMonicHom.ker = Submodule.restrictScalars R (Ideal.span {q})","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Ideal.lean#L44-L47"},"Polynomial.X_sub_C_dvd_sub_C_eval":{"signature":"theorem Polynomial.X_sub_C_dvd_sub_C_eval {R : Type u} {a : R} [CommRing R] {p : Polynomial R} :X - C a \u2223 p - C (eval a p)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L604-L605"},"Polynomial.mem_span_C_X_sub_C_X_sub_C_iff_eval_eval_eq_zero":{"signature":"theorem Polynomial.mem_span_C_X_sub_C_X_sub_C_iff_eval_eval_eq_zero {R : Type u_1} [CommRing R] {a : R} {b : Polynomial R} {P : Polynomial (Polynomial R)} :P \u2208 Ideal.span {C (X - C a), X - C b} \u2194 eval a (eval b P) = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Ideal.lean#L30-L38"},"Polynomial.quotientSpanCXSubCAlgEquiv":{"signature":"noncomputable def Polynomial.quotientSpanCXSubCAlgEquiv {R : Type u_1} [CommRing R] (x y : R) :(Polynomial R \u29f8 Ideal.span {C x, X - C y}) \u2243\u2090[R] R \u29f8 Ideal.span {x}","doc":"For a commutative ring $R$, evaluating a polynomial at an element $y \\in R$ induces an isomorphism of $R$-algebras $R[X] / \\langle x, X - y \\rangle \\cong R / \\langle x \\rangle$.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Quotient.lean#L49-L58"},"Polynomial.quotientSpanCXSubCXSubCAlgEquiv":{"signature":"noncomputable def Polynomial.quotientSpanCXSubCXSubCAlgEquiv {R : Type u_1} [CommRing R] {x : R} {y : Polynomial R} :(Polynomial (Polynomial R) \u29f8 Ideal.span {C (X - C x), X - C y}) \u2243\u2090[R] R","doc":"For a commutative ring $R$, evaluating a polynomial at elements $y(X) \\in R[X]$ and $x \\in R$ induces an isomorphism of $R$-algebras $R[X, Y] / \\langle X - x, Y - y(X) \\rangle \\cong R$.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Quotient.lean#L60-L65"},"Polynomial.ofFinsupp_smul":{"signature":"theorem Polynomial.ofFinsupp_smul {R : Type u} [Semiring R] {S : Type u_1} [SMulZeroClass S R] (a : S) (b : AddMonoidAlgebra R \u2115) :{ toFinsupp := a \u2022 b } = a \u2022 { toFinsupp := b }","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L168-L171"},"Polynomial.leadingCoeff_preHilbertPoly":{"signature":"theorem Polynomial.leadingCoeff_preHilbertPoly (F : Type u_1) [Field F] [CharZero F] (d k : \u2115) :(preHilbertPoly F d k).leadingCoeff = (\u2191d.factorial)\u207b\u00b9","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/HilbertPoly.lean#L86-L88"},"Polynomial.dvd_derivative_iff":{"signature":"theorem Polynomial.dvd_derivative_iff {R : Type u} [Semiring R] [NoZeroDivisors R] {P : Polynomial R} :P \u2223 derivative P \u2194 derivative P = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L737-L743"},"Polynomial.iterate_derivative_X":{"signature":"theorem Polynomial.iterate_derivative_X {R : Type u} [Semiring R] {k : \u2115} (h : 1 < k) :(\u21d1derivative)^[k] X = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L222-L224"},"Polynomial.derivativeFinsupp_X":{"signature":"theorem Polynomial.derivativeFinsupp_X {R : Type u} [Semiring R] :derivativeFinsupp X = Finsupp.single 0 X + Finsupp.single 1 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L424-L430"},"Polynomial.evalEval_neg":{"signature":"theorem Polynomial.evalEval_neg {R : Type u_1} [Ring R] (x y : R) (p : Polynomial (Polynomial R)) :evalEval x y (-p) = -evalEval x y p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Bivariate.lean#L99-L101"},"Polynomial.algEquivOfCompEqX_apply":{"signature":"theorem Polynomial.algEquivOfCompEqX_apply {R : Type u} [CommSemiring R] (p q : Polynomial R) (hpq : p.comp q = X) (hqp : q.comp p = X) (a : Polynomial R) :(p.algEquivOfCompEqX q hpq hqp) a = (aeval p) a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L324-L324"},"Polynomial.algEquivCMulXAddC_symm_eq":{"signature":"theorem Polynomial.algEquivCMulXAddC_symm_eq {R : Type u_3} [CommRing R] (a b : R) [Invertible a] :(algEquivCMulXAddC a b).symm = algEquivCMulXAddC (\u215fa) (-\u215fa * b)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L346-L351"},"Polynomial.associated_map_map":{"signature":"theorem Polynomial.associated_map_map {R : Type u} {S : Type v} [Semiring R] [Semiring S] (f : R \u2192+* S) {x y : Polynomial R} :Associated x y \u2192 Associated (map f x) (map f y)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L547-L548"},"Polynomial.splits_C":{"signature":"theorem Polynomial.splits_C {R : Type u_1} [Semiring R] (a : R) :(C a).Splits","doc":"Alias of Polynomial.Splits.C.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L660-L661"},"Polynomial.splits_of_map_degree_eq_one":{"signature":"theorem Polynomial.splits_of_map_degree_eq_one {R : Type u_1} [DivisionSemiring R] {f : Polynomial R} (hf : f.degree = 1) :f.Splits","doc":"Alias of Polynomial.Splits.of_degree_eq_one.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L663-L664"},"Polynomial.natDegree_linear":{"signature":"theorem Polynomial.natDegree_linear {R : Type u} {a b : R} [Semiring R] (ha : a \u2260 0) :(C a * X + C b).natDegree = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/SmallDegree.lean#L82-L83"},"Polynomial.resultant_X_sub_C_left":{"signature":"theorem Polynomial.resultant_X_sub_C_left {R : Type u_1} [CommRing R] (g : Polynomial R) (n : \u2115) (r : R) (hg : g.natDegree \u2264 n) :(X - C r).resultant g 1 n = eval r g","doc":"Res(X - r, g) = g(r)","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L378-L389"},"Polynomial.resultant_X_add_C_left":{"signature":"theorem Polynomial.resultant_X_add_C_left {R : Type u_1} [CommRing R] (g : Polynomial R) (n : \u2115) (r : R) (hg : g.natDegree \u2264 n) :(X + C r).resultant g 1 n = eval (-r) g","doc":"Res(X + r, g) = g(-r)","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L391-L394"},"Polynomial.coe_basisMonomials":{"signature":"theorem Polynomial.coe_basisMonomials (R : Type u) [Semiring R] :\u21d1(basisMonomials R) = fun (s : \u2115) => (monomial s) 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basis.lean#L32-L34"},"Polynomial.degree_map_eq_of_isUnit_leadingCoeff":{"signature":"theorem Polynomial.degree_map_eq_of_isUnit_leadingCoeff {R : Type u} {S : Type v} [Semiring R] {p : Polynomial R} [Semiring S] [Nontrivial S] (f : R \u2192+* S) (hp : IsUnit p.leadingCoeff) :(map f p).degree = p.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L277-L279"},"Polynomial.natDegree_map_eq_of_isUnit_leadingCoeff":{"signature":"theorem Polynomial.natDegree_map_eq_of_isUnit_leadingCoeff {R : Type u} {S : Type v} [Semiring R] {p : Polynomial R} [Semiring S] [Nontrivial S] (f : R \u2192+* S) (hp : IsUnit p.leadingCoeff) :(map f p).natDegree = p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L281-L283"},"Polynomial.cauchyBound_smul":{"signature":"theorem Polynomial.cauchyBound_smul {K : Type u_1} [NormedDivisionRing K] {x : K} (hx : x \u2260 0) (p : Polynomial K) :(x \u2022 p).cauchyBound = p.cauchyBound","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/CauchyBound.lean#L63-L69"},"Polynomial.cardPowDegree_zero":{"signature":"theorem Polynomial.cardPowDegree_zero {Fq : Type u_1} [Field Fq] [Fintype Fq] :cardPowDegree 0 = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/CardPowDegree.lean#L77-L78"},"Polynomial.content_one":{"signature":"theorem Polynomial.content_one {R : Type u_1} [CommRing R] [NormalizedGCDMonoid R] :content 1 = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Content.lean#L111-L112"},"Polynomial.content_X_mul":{"signature":"theorem Polynomial.content_X_mul {R : Type u_1} [CommRing R] [NormalizedGCDMonoid R] {p : Polynomial R} :(X * p).content = p.content","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Content.lean#L114-L134"},"Polynomial.content_X_pow":{"signature":"theorem Polynomial.content_X_pow {R : Type u_1} [CommRing R] [NormalizedGCDMonoid R] {k : \u2115} :(X ^ k).content = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Content.lean#L136-L140"},"Polynomial.reverse_C":{"signature":"theorem Polynomial.reverse_C {R : Type u_1} [Semiring R] (t : R) :(C t).reverse = C t","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Reverse.lean#L299-L301"},"Polynomial.eval\u2082_reverse_mul_pow":{"signature":"theorem Polynomial.eval\u2082_reverse_mul_pow {R : Type u_1} [Semiring R] {S : Type u_2} [CommSemiring S] (i : R \u2192+* S) (x : S) [Invertible x] (f : Polynomial R) :eval\u2082 i (\u215fx) f.reverse * x ^ f.natDegree = eval\u2082 i x f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Reverse.lean#L332-L334"},"Polynomial.mahlerMeasure_nonneg":{"signature":"theorem Polynomial.mahlerMeasure_nonneg (p : Polynomial \u2102) :0 \u2264 p.mahlerMeasure","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/MahlerMeasure.lean#L97-L98"},"Polynomial.content_eq_gcd_range_succ":{"signature":"theorem Polynomial.content_eq_gcd_range_succ {R : Type u_1} [CommRing R] [NormalizedGCDMonoid R] (p : Polynomial R) :p.content = (Finset.range p.natDegree.succ).gcd p.coeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Content.lean#L191-L193"},"Polynomial.smeval_assoc_X_pow":{"signature":"theorem Polynomial.smeval_assoc_X_pow (R : Type u_1) [Semiring R] (p : Polynomial R) {S : Type u_2} [NonAssocSemiring S] [Module R S] [Pow S \u2115] (x : S) [NatPowAssoc S] [IsScalarTower R S S] (m n : \u2115) :p.smeval x * x ^ m * x ^ n = p.smeval x * (x ^ m * x ^ n)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Smeval.lean#L250-L255"},"Polynomial.leadingCoeff_prod'":{"signature":"theorem Polynomial.leadingCoeff_prod' {R : Type u} {\u03b9 : Type w} (s : Finset \u03b9) [CommSemiring R] (f : \u03b9 \u2192 Polynomial R) (h : \u220f i \u2208 s, (f i).leadingCoeff \u2260 0) :(\u220f i \u2208 s, f i).leadingCoeff = \u220f i \u2208 s, (f i).leadingCoeff","doc":"The leading coefficient of a product of polynomials is equal to the product of the leading coefficients, provided that this product is nonzero. See Polynomial.leadingCoeff_prod (without the ') for a version for integral domains, where this condition is automatically satisfied.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/BigOperators.lean#L165-L173"},"Polynomial.resultant_mul_left":{"signature":"theorem Polynomial.resultant_mul_left {R : Type u_1} [CommRing R] (f\u2081 f\u2082 g : Polynomial R) (n : \u2115) (hn : g.natDegree \u2264 n) :(f\u2081 * f\u2082).resultant g (f\u2081.natDegree + f\u2082.natDegree) n = f\u2081.resultant g f\u2081.natDegree n * f\u2082.resultant g f\u2082.natDegree n","doc":"Res(f\u2081 * f\u2082, g) = Res(f\u2081, g) * Res(f\u2082, g).","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L583-L589"},"Polynomial.resultant_prod_left":{"signature":"theorem Polynomial.resultant_prod_left {R : Type u_1} [CommRing R] {\u03b9 : Type u_3} (s : Finset \u03b9) (f : \u03b9 \u2192 Polynomial R) (g : Polynomial R) (n : \u2115) (hf : \u220f i \u2208 s, (f i).leadingCoeff \u2260 0) (hn : g.natDegree \u2264 n) :(\u220f i \u2208 s, f i).resultant g (\u220f i \u2208 s, f i).natDegree n = \u220f i \u2208 s, (f i).resultant g (f i).natDegree n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L643-L654"},"Polynomial.resultant_pow_left":{"signature":"theorem Polynomial.resultant_pow_left {R : Type u_1} [CommRing R] (f g : Polynomial R) (m n : \u2115) (hf : f.leadingCoeff ^ m \u2260 0) (hn : g.natDegree \u2264 n) :(f ^ m).resultant g (f ^ m).natDegree n = f.resultant g f.natDegree n ^ m","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L664-L667"},"Polynomial.resultant_X_sub_C_pow_left":{"signature":"theorem Polynomial.resultant_X_sub_C_pow_left {R : Type u_1} [CommRing R] (r : R) (g : Polynomial R) (m n : \u2115) (hn : g.natDegree \u2264 n) :((X - C r) ^ m).resultant g m n = eval r g ^ m","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L674-L677"},"Polynomial.resultant_X_sub_C_pow_right":{"signature":"theorem Polynomial.resultant_X_sub_C_pow_right {R : Type u_1} [CommRing R] (f : Polynomial R) (r : R) (m n : \u2115) (hm : f.natDegree \u2264 m) :f.resultant ((X - C r) ^ n) m n = (-1) ^ (m * n) * eval r f ^ n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L679-L681"},"Polynomial.resultant_X_pow_right":{"signature":"theorem Polynomial.resultant_X_pow_right {R : Type u_1} [CommRing R] (f : Polynomial R) (m n : \u2115) (hm : f.natDegree \u2264 m) :f.resultant (X ^ n) m n = (-1) ^ (m * n) * f.coeff 0 ^ n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L687-L689"},"Polynomial.splits_of_isScalarTower":{"signature":"theorem Polynomial.splits_of_isScalarTower {R : Type u_1} [CommSemiring R] {f : Polynomial R} {A : Type u_2} (B : Type u_3) [CommSemiring A] [Semiring B] [Algebra R A] [Algebra R B] [Algebra A B] [IsScalarTower R A B] (hf : (map (algebraMap R A) f).Splits) :(map (algebraMap R B) f).Splits","doc":"Alias of Polynomial.Splits.of_isScalarTower.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L862-L863"},"Polynomial.isMonicOfDegree_X_sub_one":{"signature":"theorem Polynomial.isMonicOfDegree_X_sub_one {R : Type u_1} [Ring R] [Nontrivial R] (r : R) :(X - C r).IsMonicOfDegree 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/IsMonicOfDegree.lean#L233-L234"},"Polynomial.isMonicOfDegree_one_iff":{"signature":"theorem Polynomial.isMonicOfDegree_one_iff {R : Type u_1} [Semiring R] [Nontrivial R] {f : Polynomial R} :f.IsMonicOfDegree 1 \u2194 \u2203 (r : R), f = X + C r","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/IsMonicOfDegree.lean#L179-L185"},"Polynomial.not_quasiFiniteAt":{"signature":"theorem Polynomial.not_quasiFiniteAt {R : Type u_1} [CommRing R] (P : Ideal (Polynomial R)) [P.IsPrime] :\u00acAlgebra.QuasiFiniteAt R P","doc":"R[X] is not quasi-finite over R at any prime.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/QuasiFinite/Basic.lean#L495-L511"},"Polynomial.lhom_ext'_iff":{"signature":"theorem Polynomial.lhom_ext'_iff {R : Type u} [Semiring R] {M : Type u_1} [AddCommMonoid M] [Module R M] {f g : Polynomial R \u2192\u2097[R] M} :f = g \u2194 \u2200 (n : \u2115), f \u2218\u2097 monomial n = g \u2218\u2097 monomial n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L730-L730"},"Polynomial.list_prod_comp":{"signature":"theorem Polynomial.list_prod_comp {R : Type u} [CommSemiring R] (l : List (Polynomial R)) (q : Polynomial R) :l.prod.comp q = (List.map (fun (p : Polynomial R) => p.comp q) l).prod","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L665-L667"},"Polynomial.map_expand_pow_char":{"signature":"theorem Polynomial.map_expand_pow_char {R : Type u} [CommSemiring R] (p : \u2115) [ExpChar R p] (f : Polynomial R) (n : \u2115) :map (iterateFrobenius R p n) ((expand R (p ^ n)) f) = f ^ p ^ n","doc":"Alias of Polynomial.map_iterateFrobenius_expand.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Expand.lean#L282-L283"},"Polynomial.degree_normalize":{"signature":"theorem Polynomial.degree_normalize {R : Type u} [Field R] {p : Polynomial R} [DecidableEq R] :(normalize p).degree = p.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L605-L607"},"Polynomial.prime_of_degree_eq_one":{"signature":"theorem Polynomial.prime_of_degree_eq_one {R : Type u} [Field R] {p : Polynomial R} (hp1 : p.degree = 1) :Prime p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L609-L614"},"Polynomial.irreducible_of_degree_eq_one":{"signature":"theorem Polynomial.irreducible_of_degree_eq_one {R : Type u} [Field R] {p : Polynomial R} (hp1 : p.degree = 1) :Irreducible p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L616-L617"},"Polynomial.isCoprime_of_is_root_of_eval_derivative_ne_zero":{"signature":"theorem Polynomial.isCoprime_of_is_root_of_eval_derivative_ne_zero {K : Type u_1} [Field K] (f : Polynomial K) (a : K) (hf' : eval a (derivative f) \u2260 0) :IsCoprime (X - C a) (f /\u2098 (X - C a))","doc":"If f is a polynomial over a field, and a : K satisfies f' a \u2260 0, then f / (X - a) is coprime with X - a. Note that we do not assume f a = 0, because f / (X - a) = (f - f a) / (X - a).","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L649-L661"},"Polynomial.derivWithin":{"signature":"theorem Polynomial.derivWithin {\ud835\udd5c : Type u} [NontriviallyNormedField \ud835\udd5c] {x : \ud835\udd5c} {s : Set \ud835\udd5c} (p : Polynomial \ud835\udd5c) (hxs : UniqueDiffWithinAt \ud835\udd5c s x) :derivWithin (fun (x : \ud835\udd5c) => eval x p) s x = eval x (derivative p)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Calculus/Deriv/Polynomial.lean#L114-L117"},"Polynomial.derivative_X_add_C_sq":{"signature":"theorem Polynomial.derivative_X_add_C_sq {R : Type u} [CommSemiring R] (c : R) :derivative ((X + C c) ^ 2) = C 2 * (X + C c)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L509-L510"},"Polynomial.coeff_bernoulli":{"signature":"theorem Polynomial.coeff_bernoulli (n i : \u2115) :(bernoulli n).coeff i = if i \u2264 n then _root_.bernoulli (n - i) * \u2191(n.choose i) else 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/NumberTheory/BernoulliPolynomials.lean#L62-L67"},"Polynomial.bernoulli_comp_neg_X":{"signature":"theorem Polynomial.bernoulli_comp_neg_X (n : \u2115) :(bernoulli n).comp (-X) = (-1) ^ n \u2022 (bernoulli n + n \u2022 X ^ (n - 1))","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/NumberTheory/BernoulliPolynomials.lean#L225-L241"},"Polynomial.natDegree_mul_leadingCoeff_inv":{"signature":"theorem Polynomial.natDegree_mul_leadingCoeff_inv {K : Type u_1} [DivisionRing K] (p : Polynomial K) {q : Polynomial K} (h : q \u2260 0) :(p * C q.leadingCoeff\u207b\u00b9).natDegree = p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L463-L465"},"Polynomial.irreducible_iff_lt_natDegree_lt":{"signature":"theorem Polynomial.irreducible_iff_lt_natDegree_lt {R : Type u} [Field R] {p : Polynomial R} (hp0 : p \u2260 0) (hpu : \u00acIsUnit p) :Irreducible p \u2194 \u2200 (q : Polynomial R), q.Monic \u2192 q.natDegree \u2208 Finset.Ioc 0 (p.natDegree / 2) \u2192 \u00acq \u2223 p","doc":"To check a polynomial p over a field is irreducible, it suffices to check there are no divisors of degree 0 < d \u2264 degree p / 2. See also: Polynomial.Monic.irreducible_iff_natDegree'.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L676-L690"},"Polynomial.monic_of_degree_le_of_coeff_eq_one":{"signature":"theorem Polynomial.monic_of_degree_le_of_coeff_eq_one {R : Type u} [Semiring R] {p : Polynomial R} (n : \u2115) (pn : p.degree \u2264 \u2191n) (p1 : p.coeff n = 1) :p.Monic","doc":"Alias of Polynomial.monic_of_degree_le.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L260-L261"},"Polynomial.differentiableOn":{"signature":"theorem Polynomial.differentiableOn {\ud835\udd5c : Type u} [NontriviallyNormedField \ud835\udd5c] {s : Set \ud835\udd5c} (p : Polynomial \ud835\udd5c) :DifferentiableOn \ud835\udd5c (fun (x : \ud835\udd5c) => eval x p) s","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Calculus/Deriv/Polynomial.lean#L100-L101"},"Polynomial.pow_sub_dvd_iterate_derivative_pow":{"signature":"theorem Polynomial.pow_sub_dvd_iterate_derivative_pow {R : Type u} [CommSemiring R] (p : Polynomial R) (n m : \u2115) :p ^ (n - m) \u2223 (\u21d1derivative)^[m] (p ^ n)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L477-L478"},"Polynomial.natDegree_cubic":{"signature":"theorem Polynomial.natDegree_cubic {R : Type u} {a b c d : R} [Semiring R] (ha : a \u2260 0) :(C a * X ^ 3 + C b * X ^ 2 + C c * X + C d).natDegree = 3","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/SmallDegree.lean#L141-L142"},"Polynomial.taylorLinearEquiv_apply_coe":{"signature":"theorem Polynomial.taylorLinearEquiv_apply_coe {R : Type u_1} [CommRing R] (r : R) (n : \u2115) (a\u271d : \u21a5(degreeLT R n)) :\u2191((taylorLinearEquiv r n) a\u271d) = \u2191(((\u2191(taylorEquiv r)).submoduleMap (degreeLT R n)) a\u271d)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/DegreeLT.lean#L169-L169"},"Polynomial.discr_of_degree_eq_one":{"signature":"theorem Polynomial.discr_of_degree_eq_one {R : Type u_1} [CommRing R] {f : Polynomial R} (hf : f.degree = 1) :f.discr = 1","doc":"The discriminant of a linear polynomial is 1.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L943-L953"},"Polynomial.exists_min_root":{"signature":"theorem Polynomial.exists_min_root {R : Type u} [CommRing R] [IsDomain R] [LinearOrder R] (p : Polynomial R) (hp : p \u2260 0) :\u2203 (x\u2080 : R), \u2200 (x : R), p.IsRoot x \u2192 x\u2080 \u2264 x","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L151-L152"},"Polynomial.natDegree_sub_eq_right_of_natDegree_lt":{"signature":"theorem Polynomial.natDegree_sub_eq_right_of_natDegree_lt {R : Type u} [Ring R] {p q : Polynomial R} (h : p.natDegree < q.natDegree) :(p - q).natDegree = q.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L585-L587"},"Polynomial.degree_multiset_prod_of_monic":{"signature":"theorem Polynomial.degree_multiset_prod_of_monic {R : Type u} [CommSemiring R] (t : Multiset (Polynomial R)) [Nontrivial R] (h : \u2200 f \u2208 t, f.Monic) :t.prod.degree = (Multiset.map degree t).sum","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/BigOperators.lean#L207-L212"},"Polynomial.isNilpotent_reverse_iff":{"signature":"theorem Polynomial.isNilpotent_reverse_iff {R : Type u_1} [CommRing R] {P : Polynomial R} :IsNilpotent P.reverse \u2194 IsNilpotent P","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Nilpotent.lean#L103-L105"},"Polynomial.toLaurent_eq_zero":{"signature":"theorem Polynomial.toLaurent_eq_zero {R : Type u_1} [Semiring R] {f : Polynomial R} :toLaurent f = 0 \u2194 f = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Laurent.lean#L341-L343"},"Polynomial.coe_compRingHom":{"signature":"theorem Polynomial.coe_compRingHom {R : Type u} [CommSemiring R] (q : Polynomial R) :\u21d1q.compRingHom = fun (p : Polynomial R) => p.comp q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L628-L630"},"Polynomial.X_sub_C_pow_dvd_iff":{"signature":"theorem Polynomial.X_sub_C_pow_dvd_iff {R : Type u} [CommRing R] {p : Polynomial R} {t : R} {n : \u2115} :(X - C t) ^ n \u2223 p \u2194 X ^ n \u2223 p.comp (X + C t)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L653-L655"},"Polynomial.mem_lifts_iff_mem_alg":{"signature":"theorem Polynomial.mem_lifts_iff_mem_alg (R : Type u) [CommSemiring R] {S : Type v} [Semiring S] [Algebra R S] (p : Polynomial S) :p \u2208 lifts (algebraMap R S) \u2194 p \u2208 (mapAlg R S).range","doc":"A polynomial p lifts if and only if it is in the image of mapAlg.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Lifts.lean#L207-L210"},"Polynomial.smul_mem_lifts":{"signature":"theorem Polynomial.smul_mem_lifts {R : Type u} [CommSemiring R] {S : Type v} [Semiring S] [Algebra R S] {p : Polynomial S} (r : R) (hp : p \u2208 lifts (algebraMap R S)) :r \u2022 p \u2208 lifts (algebraMap R S)","doc":"If p lifts and (r : R) then r \u2022 p lifts.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Lifts.lean#L212-L216"},"Polynomial.bernoulli_eq_sub_sum":{"signature":"theorem Polynomial.bernoulli_eq_sub_sum (n : \u2115) :(n + 1) \u2022 bernoulli n = (n + 1) \u2022 X ^ n - \u2211 k \u2208 Finset.range n, (n + 1).choose k \u2022 bernoulli k","doc":"Another version of Polynomial.sum_bernoulli.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/NumberTheory/BernoulliPolynomials.lean#L160-L166"},"Polynomial.bernoulli_comp_one_add_X":{"signature":"theorem Polynomial.bernoulli_comp_one_add_X (n : \u2115) :(bernoulli n).comp (1 + X) = bernoulli n + n \u2022 X ^ (n - 1)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/NumberTheory/BernoulliPolynomials.lean#L191-L218"},"Polynomial.bernoulli_comp_one_sub_X":{"signature":"theorem Polynomial.bernoulli_comp_one_sub_X (n : \u2115) :(bernoulli n).comp (1 - X) = (-1) ^ n * bernoulli n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/NumberTheory/BernoulliPolynomials.lean#L247-L255"},"Polynomial.isCentralScalar":{"signature":"instance Polynomial.isCentralScalar {R : Type u} [Semiring R] {S : Type u_1} [SMulZeroClass S R] [SMulZeroClass S\u1d50\u1d52\u1d56 R] [IsCentralScalar S R] :IsCentralScalar S (Polynomial R)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L306-L310"},"Polynomial.eq_zero_of_natDegree_lt_card_of_eval_eq_zero'":{"signature":"theorem Polynomial.eq_zero_of_natDegree_lt_card_of_eval_eq_zero' {R : Type u_1} [CommRing R] [IsDomain R] (p : Polynomial R) (s : Finset R) (heval : \u2200 i \u2208 s, eval i p = 0) (hcard : p.natDegree < s.card) :p = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L718-L722"},"Polynomial.le_natTrailingDegree":{"signature":"theorem Polynomial.le_natTrailingDegree {R : Type u} {n : \u2115} [Semiring R] {p : Polynomial R} (hp : p \u2260 0) (hn : \u2200 m < n, p.coeff m = 0) :n \u2264 p.natTrailingDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L266-L269"},"Polynomial.iterate_derivative_sum":{"signature":"theorem Polynomial.iterate_derivative_sum {R : Type u} {\u03b9 : Type y} [Semiring R] (k : \u2115) (s : Finset \u03b9) (f : \u03b9 \u2192 Polynomial R) :(\u21d1derivative)^[k] (\u2211 b \u2208 s, f b) = \u2211 b \u2208 s, (\u21d1derivative)^[k] (f b)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L135-L137"},"Polynomial.roots_eq_zero_iff_isRoot_eq_bot":{"signature":"theorem Polynomial.roots_eq_zero_iff_isRoot_eq_bot {R : Type u} [CommRing R] [IsDomain R] {p : Polynomial R} (hp0 : p \u2260 0) :p.roots = 0 \u2194 p.IsRoot = \u22a5","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L119-L122"},"Polynomial.roots_eq_zero_iff_eq_zero_or_isRoot_eq_bot":{"signature":"theorem Polynomial.roots_eq_zero_iff_eq_zero_or_isRoot_eq_bot {R : Type u} [CommRing R] [IsDomain R] {p : Polynomial R} :p.roots = 0 \u2194 p = 0 \u2228 p.IsRoot = \u22a5","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L124-L126"},"Polynomial.hasseDeriv_eq_zero_of_lt_natDegree":{"signature":"theorem Polynomial.hasseDeriv_eq_zero_of_lt_natDegree {R : Type u_1} [Semiring R] (p : Polynomial R) (n : \u2115) (h : p.natDegree < n) :(hasseDeriv n) p = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/HasseDeriv.lean#L90-L94"},"Polynomial.iterate_derivative_comp_one_sub_X":{"signature":"theorem Polynomial.iterate_derivative_comp_one_sub_X {R : Type u} [CommRing R] (p : Polynomial R) (k : \u2115) :(\u21d1derivative)^[k] (p.comp (1 - X)) = (-1) ^ k * ((\u21d1derivative)^[k] p).comp (1 - X)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L649-L654"},"Polynomial.map_map_freeMonic":{"signature":"theorem Polynomial.map_map_freeMonic (R : Type u_1) (S : Type u_2) [CommRing R] [CommRing S] (n : \u2115) (f : R \u2192+* S) :map (MvPolynomial.map f) (freeMonic R n) = freeMonic S n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/UniversalFactorizationRing.lean#L73-L75"},"Polynomial.instFiniteUniversalFactorizationRing":{"signature":"instance Polynomial.instFiniteUniversalFactorizationRing {R : Type u_1} [CommRing R] {n : \u2115} (m k : \u2115) (hn : n = m + k) (p : MonicDegreeEq R n) :Module.Finite R (UniversalFactorizationRing m k hn p)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/UniversalFactorizationRing.lean#L485-L489"},"Polynomial.natTrailingDegree_ne_zero":{"signature":"theorem Polynomial.natTrailingDegree_ne_zero {R : Type u} [Semiring R] {p : Polynomial R} :p.natTrailingDegree \u2260 0 \u2194 p \u2260 0 \u2227 p.coeff 0 = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L144-L145"},"Polynomial.disjoint_ker_aeval_of_isCoprime":{"signature":"theorem Polynomial.disjoint_ker_aeval_of_isCoprime {R : Type u} {M : Type w} [CommRing R] [AddCommGroup M] [Module R M] (f : M \u2192\u2097[R] M) {p q : Polynomial R} (hpq : IsCoprime p q) :Disjoint ((aeval f) p).ker ((aeval f) q).ker","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Basic.lean#L907-L914"},"Polynomial.contract_one":{"signature":"theorem Polynomial.contract_one {R : Type u} [CommSemiring R] {f : Polynomial R} :contract 1 f = f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Expand.lean#L205-L206"},"Polynomial.expand_contract'":{"signature":"theorem Polynomial.expand_contract' {R : Type u} [CommSemiring R] (p : \u2115) [ExpChar R p] [NoZeroDivisors R] {f : Polynomial R} (hf : derivative f = 0) :(expand R p) (contract p f) = f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Expand.lean#L256-L260"},"Polynomial.map_scaleRoots":{"signature":"theorem Polynomial.map_scaleRoots {R : Type u_1} {S : Type u_2} [Semiring R] [Semiring S] (p : Polynomial R) (x : R) (f : R \u2192+* S) (h : f p.leadingCoeff \u2260 0) :map f (p.scaleRoots x) = (map f p).scaleRoots (f x)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ScaleRoots.lean#L90-L93"},"Polynomial.derivative_pow_eq_zero":{"signature":"theorem Polynomial.derivative_pow_eq_zero {R : Type u} [CommSemiring R] [NoZeroDivisors R] {n : \u2115} (chn : \u2191n \u2260 0) {a : Polynomial R} :derivative (a ^ n) = 0 \u2194 derivative a = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L751-L755"},"Polynomial.opRingEquiv_op_C":{"signature":"theorem Polynomial.opRingEquiv_op_C {R : Type u_1} [Semiring R] (a : R) :(opRingEquiv R) (MulOpposite.op (C a)) = C (MulOpposite.op a)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Opposites.lean#L46-L48"},"Polynomial.wronskianBilin_apply":{"signature":"theorem Polynomial.wronskianBilin_apply {R : Type u_1} [CommRing R] (a b : Polynomial R) :((wronskianBilin R) a) b = a.wronskian b","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Wronskian.lean#L51-L52"},"Polynomial.mod_eq_of_dvd_sub":{"signature":"theorem Polynomial.mod_eq_of_dvd_sub {R : Type u} [Field R] {p\u2081 p\u2082 q : Polynomial R} (h : q \u2223 p\u2081 - p\u2082) :p\u2081 % q = p\u2082 % q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L724-L730"},"Polynomial.toMvPolynomial_C":{"signature":"theorem Polynomial.toMvPolynomial_C {R : Type u_1} {\u03c3 : Type u_3} [CommSemiring R] (i : \u03c3) (r : R) :(toMvPolynomial i) (C r) = MvPolynomial.C r","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/MvPolynomial/Equiv.lean#L779-L781"},"Polynomial.degree_div_lt":{"signature":"theorem Polynomial.degree_div_lt {R : Type u} [Field R] {p q : Polynomial R} (hp : p \u2260 0) (hq : 0 < q.degree) :(p / q).degree < p.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L401-L405"},"Polynomial.isCoveringMapOn_eval":{"signature":"theorem Polynomial.isCoveringMapOn_eval {\ud835\udd5c : Type u_1} [NontriviallyNormedField \ud835\udd5c] [ProperSpace \ud835\udd5c] (p : Polynomial \ud835\udd5c) :IsCoveringMapOn (fun (x : \ud835\udd5c) => eval x p) ((fun (x : \ud835\udd5c) => eval x p) '' {k : \ud835\udd5c | eval k (derivative p) = 0})\u1d9c","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Complex/CoveringMap.lean#L54-L61"},"Polynomial.support_C":{"signature":"theorem Polynomial.support_C {R : Type u} [Semiring R] {a : R} (h : a \u2260 0) :(C a).support = {0}","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L748-L749"},"Polynomial.gaussNorm_C":{"signature":"theorem Polynomial.gaussNorm_C {R : Type u_1} {F : Type u_2} [Semiring R] [FunLike F R \u211d] (v : F) (c : \u211d) [ZeroHomClass F R \u211d] (r : R) :gaussNorm v c (C r) = v r","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/GaussNorm.lean#L61-L63"},"Polynomial.differentiableWithinAt_aeval":{"signature":"theorem Polynomial.differentiableWithinAt_aeval {\ud835\udd5c : Type u} [NontriviallyNormedField \ud835\udd5c] {x : \ud835\udd5c} {s : Set \ud835\udd5c} {R : Type u_1} [CommSemiring R] [Algebra R \ud835\udd5c] (q : Polynomial R) :DifferentiableWithinAt \ud835\udd5c (fun (x : \ud835\udd5c) => (aeval x) q) s x","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Calculus/Deriv/Polynomial.lean#L90-L92"},"Polynomial.derivative_X_sq":{"signature":"theorem Polynomial.derivative_X_sq {R : Type u} [Semiring R] :derivative (X ^ 2) = C 2 * X","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L107-L108"},"Polynomial.scaleRoots_C":{"signature":"theorem Polynomial.scaleRoots_C {R : Type u_1} [Semiring R] (r c : R) :(C c).scaleRoots r = C c","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ScaleRoots.lean#L95-L97"},"Polynomial.X_add_C_scaleRoots":{"signature":"theorem Polynomial.X_add_C_scaleRoots {R : Type u_1} [Semiring R] (r s : R) :(X + C r).scaleRoots s = X + C (r * s)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ScaleRoots.lean#L119-L122"},"Polynomial.splits_neg_iff":{"signature":"theorem Polynomial.splits_neg_iff {R : Type u_1} [Ring R] {f : Polynomial R} :(-f).Splits \u2194 f.Splits","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L236-L238"},"Polynomial.exists_image_comap_of_monic":{"signature":"theorem Polynomial.exists_image_comap_of_monic {R : Type u_1} [CommRing R] (f g : Polynomial R) (hg : g.Monic) :\u2203 (t : Finset R), PrimeSpectrum.comap C '' (PrimeSpectrum.zeroLocus {g} \\ PrimeSpectrum.zeroLocus {f}) = (PrimeSpectrum.zeroLocus \u2191t)\u1d9c","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Spectrum/Prime/Polynomial.lean#L174-L178"},"Polynomial.isOpen_image_comap_of_monic":{"signature":"theorem Polynomial.isOpen_image_comap_of_monic {R : Type u_1} [CommRing R] (f g : Polynomial R) (hg : g.Monic) :IsOpen (PrimeSpectrum.comap C '' (PrimeSpectrum.zeroLocus {g} \\ PrimeSpectrum.zeroLocus {f}))","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Spectrum/Prime/Polynomial.lean#L187-L191"},"Polynomial.coeffs_zero":{"signature":"theorem Polynomial.coeffs_zero {R : Type u} [Semiring R] :coeffs 0 = \u2205","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L1018-L1020"},"Polynomial.ofFn_coeff_eq_zero_of_ge":{"signature":"theorem Polynomial.ofFn_coeff_eq_zero_of_ge {R : Type u_1} [Semiring R] [DecidableEq R] {n i : \u2115} (v : Fin n \u2192 R) (hi : n \u2264 i) :((ofFn n) v).coeff i = 0","doc":"If n \u2264 i the i-th coefficient of ofFn n v is 0.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/OfFn.lean#L73-L76"},"Polynomial.ofFn_natDegree_lt":{"signature":"theorem Polynomial.ofFn_natDegree_lt {R : Type u_1} [Semiring R] [DecidableEq R] {n : \u2115} (h : 1 \u2264 n) (v : Fin n \u2192 R) :((ofFn n) v).natDegree < n","doc":"ofFn n v has natDegree smaller than n.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/OfFn.lean#L78-L81"},"Polynomial.leadingCoeff_add_of_degree_lt'":{"signature":"theorem Polynomial.leadingCoeff_add_of_degree_lt' {R : Type u} [Semiring R] {p q : Polynomial R} (h : q.degree < p.degree) :(p + q).leadingCoeff = p.leadingCoeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L269-L272"},"Polynomial.leadingCoeff_sub_of_degree_lt":{"signature":"theorem Polynomial.leadingCoeff_sub_of_degree_lt {R : Type u} [Ring R] {p q : Polynomial R} (h : q.degree < p.degree) :(p - q).leadingCoeff = p.leadingCoeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L555-L558"},"Polynomial.natDegree_mul_X_pow":{"signature":"theorem Polynomial.natDegree_mul_X_pow {R : Type u} [Semiring R] [Nontrivial R] {p : Polynomial R} (n : \u2115) (hp : p \u2260 0) :(p * X ^ n).natDegree = p.natDegree + n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L520-L521"},"Polynomial.roots_scaleRoots":{"signature":"theorem Polynomial.roots_scaleRoots {R : Type u_1} [CommRing R] [IsDomain R] (p : Polynomial R) {r : R} (hr : IsUnit r) :(p.scaleRoots r).roots = Multiset.map (fun (x : R) => r * x) p.roots","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ScaleRoots.lean#L348-L355"},"Polynomial.resultant_scaleRoots":{"signature":"theorem Polynomial.resultant_scaleRoots {R : Type u_1} [CommRing R] (f g : Polynomial R) (r : R) :(f.scaleRoots r).resultant (g.scaleRoots r) = r ^ (f.natDegree * g.natDegree) * f.resultant g","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L691-L734"},"Polynomial.toContinuousMapOn_apply":{"signature":"theorem Polynomial.toContinuousMapOn_apply {R : Type u_1} [Semiring R] [TopologicalSpace R] [IsTopologicalSemiring R] (p : Polynomial R) (X : Set R) (x : \u2191X) :(p.toContinuousMapOn X) x = p.toContinuousMap \u2191x","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Topology/ContinuousMap/Polynomial.lean#L56-L56"},"Polynomial.natDegree_hasseDeriv":{"signature":"theorem Polynomial.natDegree_hasseDeriv {R : Type u_1} [Semiring R] [IsAddTorsionFree R] (p : Polynomial R) (n : \u2115) :((hasseDeriv n) p).natDegree = p.natDegree - n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/HasseDeriv.lean#L193-L198"},"Polynomial.evalEval_one":{"signature":"theorem Polynomial.evalEval_one {R : Type u_1} [Semiring R] (x y : R) :evalEval x y 1 = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Bivariate.lean#L60-L62"},"Polynomial.signVariations_monomial":{"signature":"theorem Polynomial.signVariations_monomial {R : Type u_1} [Semiring R] [LinearOrder R] (d : \u2115) (c : R) :((monomial d) c).signVariations = 0","doc":"Sign variations of a monomial are always zero.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/RuleOfSigns.lean#L60-L65"},"Polynomial.newtonMap_apply_of_not_isUnit":{"signature":"theorem Polynomial.newtonMap_apply_of_not_isUnit {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [Algebra R S] {P : Polynomial R} {x : S} (h : \u00acIsUnit ((aeval x) (derivative P))) :P.newtonMap x = x","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Dynamics/Newton.lean#L60-L62"},"Polynomial.splits_of_splits_of_dvd":{"signature":"theorem Polynomial.splits_of_splits_of_dvd {R : Type u_1} [CommRing R] {f g : Polynomial R} [IsDomain R] (hg : g.Splits) (hg\u2080 : g \u2260 0) (hfg : f \u2223 g) :f.Splits","doc":"Alias of Polynomial.Splits.of_dvd.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L753-L754"},"Polynomial.natDegree_reflect_le":{"signature":"theorem Polynomial.natDegree_reflect_le {R : Type u_1} [Semiring R] {N : \u2115} {p : Polynomial R} :(reflect N p).natDegree \u2264 max N p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Reverse.lean#L183-L186"},"Polynomial.reflect_reflect":{"signature":"theorem Polynomial.reflect_reflect {R : Type u_1} [Semiring R] {N : \u2115} {p : Polynomial R} :reflect N (reflect N p) = p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Reverse.lean#L105-L105"},"Polynomial.irreducible_comp":{"signature":"theorem Polynomial.irreducible_comp {K : Type u} [Field K] {f g : Polynomial K} (hfm : f.Monic) (hgm : g.Monic) (hf : Irreducible f) (hg : \u2200 (E : Type u) [inst : Field E] [inst_1 : Algebra K E] (x : E), minpoly K x = f \u2192 Irreducible (map (algebraMap K \u21a5K\u27eex\u27ef) g - C (IntermediateField.AdjoinSimple.gen K x))) :Irreducible (f.comp g)","doc":"Let f, g be monic polynomials over K. If f is irreducible, and g(x) - \u03b1 is irreducible in K\u27ee\u03b1\u27ef with \u03b1 a root of f, then f(g(x)) is irreducible.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/IntermediateField/Adjoin/Basic.lean#L604-L652"},"Polynomial.X_add_C_ne_zero":{"signature":"theorem Polynomial.X_add_C_ne_zero {R : Type u} [Nontrivial R] [Semiring R] (r : R) :X + C r \u2260 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L626-L627"},"Polynomial.lt_rootMultiplicity_of_isRoot_iterate_derivative":{"signature":"theorem Polynomial.lt_rootMultiplicity_of_isRoot_iterate_derivative {R : Type u} [CommRing R] [NoZeroDivisors R] [CharZero R] {p : Polynomial R} {t : R} {n : \u2115} (h : p \u2260 0) (hroot : \u2200 m \u2264 n, ((\u21d1derivative)^[m] p).IsRoot t) :n < rootMultiplicity t p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L168-L173"},"Polynomial.lt_rootMultiplicity_iff_isRoot_iterate_derivative":{"signature":"theorem Polynomial.lt_rootMultiplicity_iff_isRoot_iterate_derivative {R : Type u} [CommRing R] [NoZeroDivisors R] [CharZero R] {p : Polynomial R} {t : R} {n : \u2115} (h : p \u2260 0) :n < rootMultiplicity t p \u2194 \u2200 m \u2264 n, ((\u21d1derivative)^[m] p).IsRoot t","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L175-L179"},"Polynomial.ofFinsupp_ofNat":{"signature":"theorem Polynomial.ofFinsupp_ofNat {R : Type u} [Semiring R] (n : \u2115) [n.AtLeastTwo] :{ toFinsupp := OfNat.ofNat n } = OfNat.ofNat n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L261-L262"},"Polynomial.contentIdeal_map_eq_map_contentIdeal":{"signature":"theorem Polynomial.contentIdeal_map_eq_map_contentIdeal {R : Type u_1} {S : Type u_2} [Semiring R] [Semiring S] (p : Polynomial R) (f : R \u2192+* S) :(map f p).contentIdeal = Ideal.map f p.contentIdeal","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ContentIdeal.lean#L90-L98"},"Polynomial.contentIdeal_eq_bot_iff":{"signature":"theorem Polynomial.contentIdeal_eq_bot_iff {R : Type u_1} [Semiring R] (p : Polynomial R) :p.contentIdeal = \u22a5 \u2194 p = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ContentIdeal.lean#L60-L66"},"Polynomial.mul_contentIdeal_le_radical_contentIdeal_mul":{"signature":"theorem Polynomial.mul_contentIdeal_le_radical_contentIdeal_mul {R : Type u_3} [CommRing R] {p q : Polynomial R} :p.contentIdeal * q.contentIdeal \u2264 (p * q).contentIdeal.radical","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ContentIdeal.lean#L172-L179"},"Polynomial.contentIdeal_mul_eq_top_of_contentIdeal_eq_top":{"signature":"theorem Polynomial.contentIdeal_mul_eq_top_of_contentIdeal_eq_top {R : Type u_3} [CommRing R] {p q : Polynomial R} (hp : p.contentIdeal = \u22a4) (hq : q.contentIdeal = \u22a4) :(p * q).contentIdeal = \u22a4","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ContentIdeal.lean#L181-L187"},"Polynomial.eq_mul_mul_of_aroots_quadratic_eq_pair":{"signature":"theorem Polynomial.eq_mul_mul_of_aroots_quadratic_eq_pair {T : Type u_2} {S : Type u_3} [CommRing T] [CommRing S] [IsDomain S] [Algebra T S] {a b c : T} {x1 x2 : S} (haroots : (C a * X ^ 2 + C b * X + C c).aroots S = {x1, x2}) :(algebraMap T S) c = (algebraMap T S) a * x1 * x2","doc":"Vieta's formula for quadratics (aroots version).","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/SmallDegreeVieta.lean#L62-L68"},"Polynomial.trailingDegree_eq_iff_natTrailingDegree_eq_of_pos":{"signature":"theorem Polynomial.trailingDegree_eq_iff_natTrailingDegree_eq_of_pos {R : Type u} [Semiring R] {p : Polynomial R} {n : \u2115} (hn : n \u2260 0) :p.trailingDegree = \u2191n \u2194 p.natTrailingDegree = n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L101-L103"},"Polynomial.bernoulli_eval_one_sub":{"signature":"theorem Polynomial.bernoulli_eval_one_sub (n : \u2115) (x : \u211a) :eval (1 - x) (bernoulli n) = (-1) ^ n * eval x (bernoulli n)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/NumberTheory/BernoulliPolynomials.lean#L257-L259"},"Polynomial.degree_pow_le_of_le":{"signature":"theorem Polynomial.degree_pow_le_of_le {R : Type u} [Semiring R] {p : Polynomial R} {a : WithBot \u2115} (b : \u2115) (hp : p.degree \u2264 a) :(p ^ b).degree \u2264 \u2191b * a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L408-L414"},"Polynomial.coeffList_C_mul":{"signature":"theorem Polynomial.coeffList_C_mul {R : Type u_1} [Semiring R] [NoZeroDivisors R] (P : Polynomial R) {x : R} (hx : x \u2260 0) :(C x * P).coeffList = List.map (fun (x_1 : R) => x * x_1) P.coeffList","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/CoeffList.lean#L178-L181"},"Polynomial.signVariations_C_mul":{"signature":"theorem Polynomial.signVariations_C_mul {R : Type u_1} [Ring R] [LinearOrder R] [IsStrictOrderedRing R] {\u03b7 : R} (P : Polynomial R) (hx : \u03b7 \u2260 0) :(C \u03b7 * P).signVariations = P.signVariations","doc":"The number of sign changes does not change if we multiply by any nonzero scalar.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/RuleOfSigns.lean#L150-L160"},"Polynomial.natTrailingDegree_monomial_le":{"signature":"theorem Polynomial.natTrailingDegree_monomial_le {R : Type u} {a : R} {n : \u2115} [Semiring R] :((monomial n) a).natTrailingDegree \u2264 n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L178-L180"},"Polynomial.natTrailingDegree_C":{"signature":"theorem Polynomial.natTrailingDegree_C {R : Type u} [Semiring R] (a : R) :(C a).natTrailingDegree = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L197-L199"},"Polynomial.natTrailingDegree_intCast":{"signature":"theorem Polynomial.natTrailingDegree_intCast {R : Type u} [Ring R] (n : \u2124) :(\u2191n).natTrailingDegree = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L390-L392"},"Polynomial.derivRootWeight":{"signature":"noncomputable def Polynomial.derivRootWeight (P : Polynomial \u2102) (z w : \u2102) :\u211d","doc":"Given a polynomial P of positive degree and a root z of its derivative, derivRootWeight P z w gives the weight of a root w of P in a convex combination that is equal to z.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Complex/Polynomial/GaussLucas.lean#L23-L28"},"Polynomial.derivRootWeight_nonneg":{"signature":"theorem Polynomial.derivRootWeight_nonneg (P : Polynomial \u2102) (z w : \u2102) :0 \u2264 P.derivRootWeight z w","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Complex/Polynomial/GaussLucas.lean#L30-L32"},"Polynomial.algEquivCMulXAddC_symm_apply":{"signature":"theorem Polynomial.algEquivCMulXAddC_symm_apply {R : Type u_3} [CommRing R] (a b : R) [Invertible a] (a\u271d : Polynomial R) :(algEquivCMulXAddC a b).symm a\u271d = (aeval (C \u215fa * (X - C b))) a\u271d","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L341-L341"},"Polynomial.one_mem_nthRootsFinset":{"signature":"theorem Polynomial.one_mem_nthRootsFinset {R : Type u} {n : \u2115} [CommRing R] [IsDomain R] (hn : 0 < n) :1 \u2208 nthRootsFinset n 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L413-L414"},"Polynomial.eval\u2082_natCast":{"signature":"theorem Polynomial.eval\u2082_natCast {R : Type u} {S : Type v} [Semiring R] [Semiring S] (f : R \u2192+* S) (x : S) (n : \u2115) :eval\u2082 f x \u2191n = \u2191n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L102-L106"},"Polynomial.eval\u2082_ofNat":{"signature":"theorem Polynomial.eval\u2082_ofNat {R : Type u} [Semiring R] {S : Type u_1} [Semiring S] (n : \u2115) [n.AtLeastTwo] (f : R \u2192+* S) (a : S) :eval\u2082 f a (OfNat.ofNat n) = OfNat.ofNat n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L108-L111"},"Polynomial.eval\u2082_one_cyclotomic_prime_pow":{"signature":"theorem Polynomial.eval\u2082_one_cyclotomic_prime_pow {R : Type u_1} {S : Type u_2} [CommRing R] [Semiring S] (f : R \u2192+* S) {p : \u2115} (k : \u2115) [Fact (Nat.Prime p)] :eval\u2082 f 1 (cyclotomic (p ^ (k + 1)) R) = \u2191p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Eval.lean#L45-L46"},"Polynomial.wronskian_self_eq_zero":{"signature":"theorem Polynomial.wronskian_self_eq_zero {R : Type u_1} [CommRing R] (a : Polynomial R) :a.wronskian a = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Wronskian.lean#L73-L74"},"Polynomial.isAlt_wronskianBilin":{"signature":"theorem Polynomial.isAlt_wronskianBilin {R : Type u_1} [CommRing R] :(wronskianBilin R).IsAlt","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Wronskian.lean#L76-L76"},"Polynomial.wronskian_neg_eq":{"signature":"theorem Polynomial.wronskian_neg_eq {R : Type u_1} [CommRing R] (a b : Polynomial R) :-a.wronskian b = b.wronskian a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Wronskian.lean#L78-L79"},"Polynomial.roots_C_mul_X_add_C":{"signature":"theorem Polynomial.roots_C_mul_X_add_C {R : Type u} {a : R} [Field R] (b : R) (ha : a \u2260 0) :(C a * X + C b).roots = {-(a\u207b\u00b9 * b)}","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L525-L526"},"Polynomial.roots_degree_eq_one":{"signature":"theorem Polynomial.roots_degree_eq_one {R : Type u} [Field R] {p : Polynomial R} (h : p.degree = 1) :p.roots = {-((p.coeff 1)\u207b\u00b9 * p.coeff 0)}","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L528-L531"},"Polynomial.wronskian_eq_of_sum_zero":{"signature":"theorem Polynomial.wronskian_eq_of_sum_zero {R : Type u_1} [CommRing R] {a b c : Polynomial R} (hAdd : a + b + c = 0) :a.wronskian b = b.wronskian c","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Wronskian.lean#L81-L82"},"Polynomial.wronskian_zero_left":{"signature":"theorem Polynomial.wronskian_zero_left {R : Type u_1} [CommRing R] (a : Polynomial R) :wronskian 0 a = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Wronskian.lean#L54-L56"},"Polynomial.degree_wronskian_lt_add":{"signature":"theorem Polynomial.degree_wronskian_lt_add {R : Type u_1} [CommRing R] {a b : Polynomial R} (ha : a \u2260 0) (hb : b \u2260 0) :(a.wronskian b).degree < a.degree + b.degree","doc":"Degree of W(a,b) is strictly less than the sum of degrees of a and b (both nonzero).","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Wronskian.lean#L84-L104"},"Polynomial.natDegree_wronskian_lt_add":{"signature":"theorem Polynomial.natDegree_wronskian_lt_add {R : Type u_1} [CommRing R] {a b : Polynomial R} (hw : a.wronskian b \u2260 0) :(a.wronskian b).natDegree < a.natDegree + b.natDegree","doc":"natDegree version of the above theorem. Note this would be false with just (ha : a \u2260 0) and (hb : b \u2260 0), since when a = b = 1 we have (wronskian a b).natDegree = a.natDegree = b.natDegree = 0.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Wronskian.lean#L106-L119"},"Polynomial.abc":{"signature":"theorem Polynomial.abc {k : Type u_1} [Field k] [DecidableEq k] {a b c : Polynomial k} (ha : a \u2260 0) (hb : b \u2260 0) (hc : c \u2260 0) (hab : IsCoprime a b) (hsum : a + b + c = 0) :a.natDegree + 1 \u2264 (UniqueFactorizationMonoid.radical (a * b * c)).natDegree \u2227 b.natDegree + 1 \u2264 (UniqueFactorizationMonoid.radical (a * b * c)).natDegree \u2227 c.natDegree + 1 \u2264 (UniqueFactorizationMonoid.radical (a * b * c)).natDegree \u2228 derivative a = 0 \u2227 derivative b = 0 \u2227 derivative c = 0","doc":"Polynomial ABC theorem.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/NumberTheory/FLT/MasonStothers.lean#L54-L101"},"Polynomial.iterate_derivative_C_mul":{"signature":"theorem Polynomial.iterate_derivative_C_mul {R : Type u} [Semiring R] (a : R) (p : Polynomial R) (k : \u2115) :(\u21d1derivative)^[k] (C a * p) = C a * (\u21d1derivative)^[k] p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L150-L153"},"Polynomial.derivative_C_mul":{"signature":"theorem Polynomial.derivative_C_mul {R : Type u} [Semiring R] (a : R) (p : Polynomial R) :derivative (C a * p) = C a * derivative p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L155-L156"},"Polynomial.flt_catalan":{"signature":"theorem Polynomial.flt_catalan {k : Type u_1} [Field k] {p q r : \u2115} (hp : p \u2260 0) (hq : q \u2260 0) (hr : r \u2260 0) (hineq : q * r + r * p + p * q \u2264 p * q * r) (chp : \u2191p \u2260 0) (chq : \u2191q \u2260 0) (chr : \u2191r \u2260 0) {a b c : Polynomial k} (ha : a \u2260 0) (hb : b \u2260 0) (hc : c \u2260 0) (hab : IsCoprime a b) {u v w : k} (hu : u \u2260 0) (hv : v \u2260 0) (hw : w \u2260 0) (heq : C u * a ^ p + C v * b ^ q + C w * c ^ r = 0) :a.natDegree = 0 \u2227 b.natDegree = 0 \u2227 c.natDegree = 0","doc":"Nonsolvability of the Fermat-Catalan equation.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/NumberTheory/FLT/Polynomial.lean#L196-L219"},"Polynomial.splits_prod":{"signature":"theorem Polynomial.splits_prod {R : Type u_1} [CommSemiring R] {\u03b9 : Type u_2} {f : \u03b9 \u2192 Polynomial R} {s : Finset \u03b9} (h : \u2200 i \u2208 s, (f i).Splits) :(\u220f i \u2208 s, f i).Splits","doc":"Alias of Polynomial.Splits.prod.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L698-L699"},"Polynomial.zero_modByMonic":{"signature":"theorem Polynomial.zero_modByMonic {R : Type u} [Ring R] (p : Polynomial R) :0 %\u2098 p = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L180-L182"},"Polynomial.neg_modByMonic":{"signature":"theorem Polynomial.neg_modByMonic {R : Type u} [CommRing R] (p q : Polynomial R) :-p %\u2098 q = -(p %\u2098 q)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L683-L684"},"Polynomial.sub_modByMonic":{"signature":"theorem Polynomial.sub_modByMonic {R : Type u} [CommRing R] (p\u2081 p\u2082 q : Polynomial R) :(p\u2081 - p\u2082) %\u2098 q = p\u2081 %\u2098 q - p\u2082 %\u2098 q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L686-L687"},"Polynomial.sub_mod":{"signature":"theorem Polynomial.sub_mod {R : Type u} [Field R] (p\u2081 p\u2082 q : Polynomial R) :(p\u2081 - p\u2082) % q = p\u2081 % q - p\u2082 % q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L445-L446"},"Polynomial.sum_derivRootWeight_pos":{"signature":"theorem Polynomial.sum_derivRootWeight_pos {P : Polynomial \u2102} (hP : 0 < P.degree) (z : \u2102) :0 < \u2211 w \u2208 P.roots.toFinset, P.derivRootWeight z w","doc":"The sum of the weights derivRootWeight P z w of all the roots w of P is positive, provided that P is not a constant polynomial.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Complex/Polynomial/GaussLucas.lean#L36-L52"},"Polynomial.toContinuousMap_X_eq_id":{"signature":"theorem Polynomial.toContinuousMap_X_eq_id {R : Type u_1} [Semiring R] [TopologicalSpace R] [IsTopologicalSemiring R] :X.toContinuousMap = ContinuousMap.id R","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Topology/ContinuousMap/Polynomial.lean#L48-L49"},"Polynomial.algebraMap_def":{"signature":"theorem Polynomial.algebraMap_def (R : Type u_1) (A : Type u_2) [CommSemiring R] [Semiring A] [Algebra R A] :algebraMap (Polynomial R) (Polynomial A) = mapRingHom (algebraMap R A)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/PolynomialAlgebra.lean#L211-L212"},"Polynomial.contentIdeal_one":{"signature":"theorem Polynomial.contentIdeal_one {R : Type u_1} [Semiring R] :contentIdeal 1 = \u22a4","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ContentIdeal.lean#L84-L86"},"Polynomial.roots_expand":{"signature":"theorem Polynomial.roots_expand {R : Type u_1} [CommRing R] [IsDomain R] {p : \u2115} [ExpChar R p] {f : Polynomial R} [PerfectRing R p] :((expand R p) f).roots = p \u2022 Multiset.map (\u21d1(frobeniusEquiv R p).symm) f.roots","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Perfect.lean#L350-L352"},"Polynomial.roots_expand_map_frobenius":{"signature":"theorem Polynomial.roots_expand_map_frobenius {R : Type u_1} [CommRing R] [IsDomain R] {p : \u2115} [ExpChar R p] {f : Polynomial R} [PerfectRing R p] :Multiset.map (\u21d1(frobenius R p)) ((expand R p) f).roots = p \u2022 f.roots","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Perfect.lean#L378-L379"},"Polynomial.roots_expand_image_frobenius":{"signature":"theorem Polynomial.roots_expand_image_frobenius {R : Type u_1} [CommRing R] [IsDomain R] {p : \u2115} [ExpChar R p] {f : Polynomial R} [PerfectRing R p] [DecidableEq R] :Finset.image (\u21d1(frobenius R p)) ((expand R p) f).roots.toFinset = f.roots.toFinset","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Perfect.lean#L386-L389"},"Polynomial.rootsExpandEquivRoots":{"signature":"noncomputable def Polynomial.rootsExpandEquivRoots {R : Type u_1} [CommRing R] [IsDomain R] (p : \u2115) [ExpChar R p] (f : Polynomial R) [DecidableEq R] [PerfectRing R p] :\u21a5((expand R p) f).roots.toFinset \u2243 \u21a5f.roots.toFinset","doc":"If f is a polynomial over a perfect integral domain R of characteristic p, then there is a bijection from the set of roots of Polynomial.expand R p f to the set of roots of f. It's given by x \u21a6 x ^ p, see rootsExpandEquivRoots_apply.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Perfect.lean#L420-L427"},"Polynomial.toContinuousMap_apply":{"signature":"theorem Polynomial.toContinuousMap_apply {R : Type u_1} [Semiring R] [TopologicalSpace R] [IsTopologicalSemiring R] (p : Polynomial R) (x : R) :p.toContinuousMap x = eval x p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Topology/ContinuousMap/Polynomial.lean#L43-L43"},"Polynomial.opRingEquiv_symm_C":{"signature":"theorem Polynomial.opRingEquiv_symm_C {R : Type u_1} [Semiring R] (a : R\u1d50\u1d52\u1d56) :(opRingEquiv R).symm (C a) = MulOpposite.op (C (MulOpposite.unop a))","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Opposites.lean#L67-L69"},"Polynomial.continuousWithinAt":{"signature":"theorem Polynomial.continuousWithinAt {R : Type u_1} [Semiring R] [TopologicalSpace R] [IsTopologicalSemiring R] (p : Polynomial R) {s : Set R} {a : R} :ContinuousWithinAt (fun (x : R) => eval x p) s a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Topology/Algebra/Polynomial.lean#L64-L66"},"Polynomial.degree_pos_of_not_isUnit_of_dvd_monic":{"signature":"theorem Polynomial.degree_pos_of_not_isUnit_of_dvd_monic {R : Type u} [CommSemiring R] {a p : Polynomial R} (hp : p.Monic) (ha : \u00acIsUnit a) (hap : a \u2223 p) :0 < a.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Units.lean#L89-L92"},"Polynomial.leadingCoeff_dvd_leadingCoeff":{"signature":"theorem Polynomial.leadingCoeff_dvd_leadingCoeff {R : Type u} [Semiring R] [NoZeroDivisors R] {a p : Polynomial R} (hap : a \u2223 p) :a.leadingCoeff \u2223 p.leadingCoeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L716-L718"},"Polynomial.irreducible_iff_degree_lt":{"signature":"theorem Polynomial.irreducible_iff_degree_lt {R : Type u} [Field R] (p : Polynomial R) (hp0 : p \u2260 0) (hpu : \u00acIsUnit p) :Irreducible p \u2194 \u2200 (q : Polynomial R), q.degree \u2264 \u2191(p.natDegree / 2) \u2192 q \u2223 p \u2192 IsUnit q","doc":"To check a polynomial over a field is irreducible, it suffices to check only for divisors that have smaller degree. See also: Polynomial.Monic.irreducible_iff_natDegree.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L663-L674"},"Polynomial.evalEvalRingHom_apply":{"signature":"theorem Polynomial.evalEvalRingHom_apply {R : Type u_1} [CommSemiring R] (x y : R) (a\u271d : Polynomial (Polynomial R)) :(evalEvalRingHom x y) a\u271d = eval x (eval (C y) a\u271d)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Bivariate.lean#L145-L145"},"Polynomial.degree_modByMonic_le_left":{"signature":"theorem Polynomial.degree_modByMonic_le_left {R : Type u} [Ring R] {p q : Polynomial R} :(p %\u2098 q).degree \u2264 p.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L220-L226"},"Polynomial.natDegree_modByMonic_le_left":{"signature":"theorem Polynomial.natDegree_modByMonic_le_left {R : Type u} [Ring R] {p q : Polynomial R} :(p %\u2098 q).natDegree \u2264 p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L232-L233"},"Polynomial.splits_zero":{"signature":"theorem Polynomial.splits_zero {R : Type u_1} [Semiring R] :Splits 0","doc":"Alias of Polynomial.Splits.zero.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L653-L654"},"Polynomial.aroots_map":{"signature":"theorem Polynomial.aroots_map (R : Type u) (S : Type v) {T : Type w} [CommRing R] [IsDomain R] [CommRing T] (p : Polynomial T) [CommRing S] [Algebra T S] [Algebra S R] [Algebra T R] [IsScalarTower T S R] :(map (algebraMap T S) p).aroots R = p.aroots R","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L526-L530"},"Polynomial.cyclotomic_six":{"signature":"theorem Polynomial.cyclotomic_six (R : Type u_1) [Ring R] :cyclotomic 6 R = X ^ 2 - X + 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Expand.lean#L76-L83"},"Polynomial.algebraMap_pi_self_eq_eval":{"signature":"theorem Polynomial.algebraMap_pi_self_eq_eval (R : Type u_1) [CommSemiring R] :\u21d1(algebraMap (Polynomial R) (R \u2192 R)) = fun (p : Polynomial R) (z : R) => eval z p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Algebraic/Pi.lean#L74-L77"},"Polynomial.valuation_X_eq_neg_one":{"signature":"theorem Polynomial.valuation_X_eq_neg_one (K : Type u_1) [Field K] :(IsDedekindDomain.HeightOneSpectrum.valuation (RatFunc K) (idealX K)) RatFunc.X = WithZero.exp (-1)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/RatFunc/AsPolynomial.lean#L250-L254"},"Polynomial.scaleRoots_one":{"signature":"theorem Polynomial.scaleRoots_one {R : Type u_1} [Semiring R] (p : Polynomial R) :p.scaleRoots 1 = p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ScaleRoots.lean#L99-L101"},"Polynomial.evalEval_zero":{"signature":"theorem Polynomial.evalEval_zero {R : Type u_1} [Semiring R] (x y : R) :evalEval x y 0 = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Bivariate.lean#L56-L58"},"Polynomial.splits_X_pow":{"signature":"theorem Polynomial.splits_X_pow {R : Type u_1} [Semiring R] (n : \u2115) :(X ^ n).Splits","doc":"Alias of Polynomial.Splits.X_pow.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L704-L705"},"Polynomial.isMonicOfDegree_add_add_two":{"signature":"theorem Polynomial.isMonicOfDegree_add_add_two {R : Type u_1} [Semiring R] [Nontrivial R] (a b : R) :(X ^ 2 + C a * X + C b).IsMonicOfDegree 2","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/IsMonicOfDegree.lean#L187-L193"},"Polynomial.ofFn_zero":{"signature":"theorem Polynomial.ofFn_zero {R : Type u_1} [Semiring R] [DecidableEq R] (n : \u2115) :(ofFn n) 0 = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/OfFn.lean#L57-L57"},"Polynomial.card_roots_le_derivative":{"signature":"theorem Polynomial.card_roots_le_derivative (p : Polynomial \u211d) :p.roots.card \u2264 (derivative p).roots.card + 1","doc":"The number of roots of a real polynomial (counted with multiplicities) is at most the number of roots of its derivative (counted with multiplicities) plus one.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Calculus/LocalExtr/Polynomial.lean#L57-L85"},"Polynomial.root_gcd_iff_root_left_right":{"signature":"theorem Polynomial.root_gcd_iff_root_left_right {R : Type u} {k : Type y} [Field R] [CommSemiring k] [DecidableEq R] {\u03d5 : R \u2192+* k} {f g : Polynomial R} {\u03b1 : k} :eval\u2082 \u03d5 \u03b1 (EuclideanDomain.gcd f g) = 0 \u2194 eval\u2082 \u03d5 \u03b1 f = 0 \u2227 eval\u2082 \u03d5 \u03b1 g = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L483-L486"},"Polynomial.isRoot_gcd_iff_isRoot_left_right":{"signature":"theorem Polynomial.isRoot_gcd_iff_isRoot_left_right {R : Type u} [Field R] [DecidableEq R] {f g : Polynomial R} {\u03b1 : R} :(EuclideanDomain.gcd f g).IsRoot \u03b1 \u2194 f.IsRoot \u03b1 \u2227 g.IsRoot \u03b1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L488-L490"},"Polynomial.map_list_prod":{"signature":"theorem Polynomial.map_list_prod {R : Type u} {S : Type v} [Semiring R] [Semiring S] (f : R \u2192+* S) (L : List (Polynomial R)) :map f L.prod = (List.map (map f) L).prod","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L558-L559"},"Polynomial.isMonicOfDegree_sub_add_two":{"signature":"theorem Polynomial.isMonicOfDegree_sub_add_two {R : Type u_1} [Ring R] [Nontrivial R] (a b : R) :(X ^ 2 - C a * X + C b).IsMonicOfDegree 2","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/IsMonicOfDegree.lean#L236-L243"},"Polynomial.support_update_ne_zero":{"signature":"theorem Polynomial.support_update_ne_zero {R : Type u} [Semiring R] (p : Polynomial R) (n : \u2115) {a : R} (ha : a \u2260 0) :(p.update n a).support = insert n p.support","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L1008-L1009"},"Polynomial.degree_natCast_le":{"signature":"theorem Polynomial.degree_natCast_le {R : Type u} [Semiring R] (n : \u2115) :(\u2191n).degree \u2264 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L187-L187"},"Polynomial.rootsExpandEquivRoots_apply":{"signature":"theorem Polynomial.rootsExpandEquivRoots_apply {R : Type u_1} [CommRing R] [IsDomain R] (p : \u2115) [ExpChar R p] (f : Polynomial R) [DecidableEq R] [PerfectRing R p] (x : \u21a5((expand R p) f).roots.toFinset) :\u2191((rootsExpandEquivRoots p f) x) = \u2191x ^ p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Perfect.lean#L429-L430"},"Polynomial.smul_eval":{"signature":"theorem Polynomial.smul_eval (S : Type u_3) [CommSemiring S] (G : Type u_4) [Group G] [MulSemiringAction G S] (g : G) (f : Polynomial S) (x : S) :eval x (g \u2022 f) = g \u2022 eval (g\u207b\u00b9 \u2022 x) f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/GroupRingAction.lean#L68-L70"},"Polynomial.rootSet_one":{"signature":"theorem Polynomial.rootSet_one {T : Type w} [CommRing T] (S : Type u_1) [CommRing S] [IsDomain S] [Algebra T S] :rootSet 1 S = \u2205","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L554-L556"},"Polynomial.ite_le_natDegree_coeff":{"signature":"theorem Polynomial.ite_le_natDegree_coeff {R : Type u} [Semiring R] (p : Polynomial R) (n : \u2115) (I : Decidable (n < 1 + p.natDegree)) :(if n < 1 + p.natDegree then p.coeff n else 0) = p.coeff n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L111-L115"},"Polynomial.smeval_smul":{"signature":"theorem Polynomial.smeval_smul (R : Type u_1) [Semiring R] (p : Polynomial R) {S : Type u_2} [AddCommMonoid S] [Pow S \u2115] [Module R S] (x : S) (r : R) :(r \u2022 p).smeval x = r \u2022 p.smeval x","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Smeval.lean#L120-L124"},"Polynomial.coeff_update_same":{"signature":"theorem Polynomial.coeff_update_same {R : Type u} [Semiring R] (p : Polynomial R) (n : \u2115) (a : R) :(p.update n a).coeff n = a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L987-L989"},"Polynomial.degreeLT_mono":{"signature":"theorem Polynomial.degreeLT_mono {R : Type u} [Semiring R] {m n : \u2115} (H : m \u2264 n) :degreeLT R m \u2264 degreeLT R n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Basic.lean#L92-L94"},"Polynomial.coeff_restriction'":{"signature":"theorem Polynomial.coeff_restriction' {R : Type u} [Ring R] {p : Polynomial R} {n : \u2115} :\u2191(p.restriction.coeff n) = p.coeff n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Basic.lean#L314-L315"},"Polynomial.ne_zero_of_ofFn_ne_zero":{"signature":"theorem Polynomial.ne_zero_of_ofFn_ne_zero {R : Type u_1} [Semiring R] [DecidableEq R] {n : \u2115} {v : Fin n \u2192 R} (h : (ofFn n) v \u2260 0) :n \u2260 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/OfFn.lean#L62-L65"},"Polynomial.cyclotomic'_splits":{"signature":"theorem Polynomial.cyclotomic'_splits {K : Type u_1} [Field K] (n : \u2115) :(cyclotomic' n K).Splits","doc":"cyclotomic' n K splits.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean#L137-L141"},"Polynomial.isIntegrallyClosed_iff'":{"signature":"theorem Polynomial.isIntegrallyClosed_iff' {R : Type u_1} [CommRing R] {K : Type u_2} [Field K] [Algebra R K] [IsFractionRing R K] [IsDomain R] :IsIntegrallyClosed R \u2194 \u2200 (p : Polynomial R), p.Monic \u2192 (Irreducible p \u2194 Irreducible (map (algebraMap R K) p))","doc":"Integrally closed domains are precisely the domains for in which Gauss's lemma holds for monic polynomials","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/GaussLemma.lean#L181-L195"},"Polynomial.bernoulli_eval_one_add":{"signature":"theorem Polynomial.bernoulli_eval_one_add (n : \u2115) (x : \u211a) :eval (1 + x) (bernoulli n) = eval x (bernoulli n) + \u2191n * x ^ (n - 1)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/NumberTheory/BernoulliPolynomials.lean#L220-L223"},"Polynomial.image_rootSet":{"signature":"theorem Polynomial.image_rootSet {R : Type u_1} [CommRing R] {f : Polynomial R} {A : Type u_2} {B : Type u_3} [CommRing A] [CommRing B] [IsDomain A] [IsDomain B] [Algebra R A] [Algebra R B] [IsSimpleRing A] (hf : (map (algebraMap R A) f).Splits) (g : A \u2192\u2090[R] B) :\u21d1g '' f.rootSet A = f.rootSet B","doc":"Alias of Polynomial.Splits.image_rootSet.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L793-L794"},"Polynomial.coeff_add_eq_right_of_lt":{"signature":"theorem Polynomial.coeff_add_eq_right_of_lt {R : Type u} {n : \u2115} [Semiring R] {p q : Polynomial R} (pn : p.natDegree < n) :(p + q).coeff n = q.coeff n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L175-L177"},"Polynomial.coeff_sub_eq_neg_right_of_lt":{"signature":"theorem Polynomial.coeff_sub_eq_neg_right_of_lt {R : Type u} {n : \u2115} [Ring R] {p q : Polynomial R} (df : p.natDegree < n) :(p - q).coeff n = -q.coeff n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L335-L336"},"Polynomial.trailingDegree_C_mul_X_pow":{"signature":"theorem Polynomial.trailingDegree_C_mul_X_pow {R : Type u} {a : R} [Semiring R] (n : \u2115) (ha : a \u2260 0) :(C a * X ^ n).trailingDegree = \u2191n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L209-L211"},"Polynomial.sum_taylor_eq":{"signature":"theorem Polynomial.sum_taylor_eq {R : Type u_1} [CommRing R] (f : Polynomial R) (r : R) :(((taylor r) f).sum fun (i : \u2115) (a : R) => C a * (X - C r) ^ i) = f","doc":"Taylor's formula.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Taylor.lean#L199-L203"},"Polynomial.splits_pow":{"signature":"theorem Polynomial.splits_pow {R : Type u_1} [Semiring R] {f : Polynomial R} (hf : f.Splits) (n : \u2115) :(f ^ n).Splits","doc":"Alias of Polynomial.Splits.pow.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L701-L702"},"Polynomial.derivative_comp_one_sub_X":{"signature":"theorem Polynomial.derivative_comp_one_sub_X {R : Type u} [CommRing R] (p : Polynomial R) :derivative (p.comp (1 - X)) = -(derivative p).comp (1 - X)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L646-L647"},"Polynomial.hermite_zero":{"signature":"theorem Polynomial.hermite_zero :hermite 0 = C 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Hermite/Basic.lean#L62-L64"},"Polynomial.deriv_aeval":{"signature":"theorem Polynomial.deriv_aeval {\ud835\udd5c : Type u} [NontriviallyNormedField \ud835\udd5c] {x : \ud835\udd5c} {R : Type u_1} [CommSemiring R] [Algebra R \ud835\udd5c] (q : Polynomial R) :deriv (fun (x : \ud835\udd5c) => (aeval x) q) x = (aeval x) (derivative q)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Calculus/Deriv/Polynomial.lean#L110-L112"},"Polynomial.deriv_gaussian_eq_hermite_mul_gaussian":{"signature":"theorem Polynomial.deriv_gaussian_eq_hermite_mul_gaussian (n : \u2115) (x : \u211d) :deriv^[n] (fun (y : \u211d) => Real.exp (-(y ^ 2 / 2))) x = (-1) ^ n * (aeval x) (hermite n) * Real.exp (-(x ^ 2 / 2))","doc":"hermite n is (up to sign) the factor appearing in deriv^[n] of a Gaussian.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Hermite/Gaussian.lean#L41-L59"},"Polynomial.hermite_eq_deriv_gaussian":{"signature":"theorem Polynomial.hermite_eq_deriv_gaussian (n : \u2115) (x : \u211d) :(aeval x) (hermite n) = (-1) ^ n * deriv^[n] (fun (y : \u211d) => Real.exp (-(y ^ 2 / 2))) x / Real.exp (-(x ^ 2 / 2))","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Hermite/Gaussian.lean#L61-L66"},"Polynomial.leadingCoeff_map'":{"signature":"theorem Polynomial.leadingCoeff_map' {R : Type u} {S : Type v} [Semiring R] [Semiring S] {f : R \u2192+* S} (hf : Function.Injective \u21d1f) (p : Polynomial R) :(map f p).leadingCoeff = f p.leadingCoeff","doc":"Alias of Polynomial.leadingCoeff_map_of_injective.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Monic.lean#L413-L414"},"Polynomial.ofFinsupp_intCast":{"signature":"theorem Polynomial.ofFinsupp_intCast {R : Type u} [Ring R] (z : \u2124) :{ toFinsupp := \u2191z } = \u2191z","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L1084-L1085"},"Polynomial.smul_mem_rootSet_iff_of_isUnit":{"signature":"theorem Polynomial.smul_mem_rootSet_iff_of_isUnit {R : Type u} {S : Type v} [CommRing R] [IsDomain R] [CommRing S] [Algebra S R] {G : Type u_1} [Monoid G] [MulSemiringAction G R] [SMulCommClass G S R] {f : Polynomial S} {g : G} (hg : IsUnit g) {x : R} :g \u2022 x \u2208 f.rootSet R \u2194 x \u2208 f.rootSet R","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L651-L656"},"Polynomial.ext_iff_degree_le":{"signature":"theorem Polynomial.ext_iff_degree_le {R : Type u} [Semiring R] {p q : Polynomial R} {n : \u2115} (hp : p.degree \u2264 \u2191n) (hq : q.degree \u2264 \u2191n) :p = q \u2194 \u2200 i \u2264 n, p.coeff i = q.coeff i","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L102-L104"},"Polynomial.dvd_comp_neg_X_iff":{"signature":"theorem Polynomial.dvd_comp_neg_X_iff {R : Type u} [CommRing R] (p q : Polynomial R) :p \u2223 q.comp (-X) \u2194 p.comp (-X) \u2223 q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L676-L679"},"Polynomial.evalEval_C":{"signature":"theorem Polynomial.evalEval_C {R : Type u_1} [Semiring R] (x y : R) (p : Polynomial R) :evalEval x y (C p) = eval x p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Bivariate.lean#L49-L50"},"Polynomial.evalEval_CC":{"signature":"theorem Polynomial.evalEval_CC {R : Type u_1} [Semiring R] (x y p : R) :evalEval x y (CC p) = p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Bivariate.lean#L52-L54"},"Polynomial.evalEval_surjective":{"signature":"theorem Polynomial.evalEval_surjective {R : Type u_1} [Semiring R] (x y : R) :Function.Surjective (evalEval x y)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Bivariate.lean#L90-L91"},"Polynomial.coeff_mem_radical_span_coeff_of_dvd":{"signature":"theorem Polynomial.coeff_mem_radical_span_coeff_of_dvd {R : Type u_1} [CommRing R] (p q : Polynomial R) (hp : p.Monic) (hq : q.Monic) (H : q \u2223 p) (i : \u2115) (hi : i \u2260 q.natDegree) :q.coeff i \u2208 (Ideal.span {x : R | \u2203 i < p.natDegree, p.coeff i = x}).radical","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/IntegralClosure/GoingDown.lean#L27-L45"},"Polynomial.homogenize_X":{"signature":"theorem Polynomial.homogenize_X {R : Type u_1} [CommSemiring R] {n : \u2115} (hn : n \u2260 0) :X.homogenize n = MvPolynomial.X 0 * MvPolynomial.X 1 ^ (n - 1)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Homogenize.lean#L98-L101"},"Polynomial.one_le_mahlerMeasure_of_ne_zero":{"signature":"theorem Polynomial.one_le_mahlerMeasure_of_ne_zero {p : Polynomial \u2124} (hp : p \u2260 0) :1 \u2264 (map (Int.castRingHom \u2102) p).mahlerMeasure","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/NumberTheory/MahlerMeasure.lean#L42-L47"},"Polynomial.scaleRoots_mul":{"signature":"theorem Polynomial.scaleRoots_mul {R : Type u_1} [CommSemiring R] (p : Polynomial R) (r s : R) :p.scaleRoots (r * s) = (p.scaleRoots r).scaleRoots s","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ScaleRoots.lean#L182-L185"},"Polynomial.scaleRoots_dvd'":{"signature":"theorem Polynomial.scaleRoots_dvd' {R : Type u_1} [CommSemiring R] (p q : Polynomial R) {r : R} (hr : IsUnit r) (hpq : p \u2223 q) :p.scaleRoots r \u2223 q.scaleRoots r","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ScaleRoots.lean#L256-L261"},"Polynomial.scaleRoots_dvd_iff":{"signature":"theorem Polynomial.scaleRoots_dvd_iff {R : Type u_1} [CommSemiring R] (p q : Polynomial R) {r : R} (hr : IsUnit r) :p.scaleRoots r \u2223 q.scaleRoots r \u2194 p \u2223 q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ScaleRoots.lean#L270-L273"},"Polynomial.aevalTower_ofId":{"signature":"theorem Polynomial.aevalTower_ofId {S : Type v} {A' : Type u_1} [CommSemiring A'] [CommSemiring S] [Algebra S A'] :aevalTower (Algebra.ofId S A') = aeval","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L557-L560"},"Polynomial.wronskian_neg_left":{"signature":"theorem Polynomial.wronskian_neg_left {R : Type u_1} [CommRing R] (a b : Polynomial R) :(-a).wronskian b = -a.wronskian b","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Wronskian.lean#L61-L62"},"Polynomial.divX_C_mul":{"signature":"theorem Polynomial.divX_C_mul {R : Type u} {a : R} [Semiring R] {p : Polynomial R} :(C a * p).divX = C a * p.divX","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Inductions.lean#L72-L75"},"Polynomial.natDegree_divX_eq_natDegree_tsub_one":{"signature":"theorem Polynomial.natDegree_divX_eq_natDegree_tsub_one {R : Type u} [Semiring R] {p : Polynomial R} :p.divX.natDegree = p.natDegree - 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Inductions.lean#L92-L100"},"Polynomial.coeff_ofNat_succ":{"signature":"theorem Polynomial.coeff_ofNat_succ {R : Type u} [Semiring R] (a n : \u2115) [h : a.AtLeastTwo] :(OfNat.ofNat a).coeff (n + 1) = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L654-L658"},"Polynomial.taylorLinearEquiv_symm":{"signature":"theorem Polynomial.taylorLinearEquiv_symm {R : Type u_1} [CommRing R] {n : \u2115} (r : R) :(taylorLinearEquiv r n).symm = taylorLinearEquiv (-r) n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/DegreeLT.lean#L173-L175"},"Polynomial.eval\u2082_congr":{"signature":"theorem Polynomial.eval\u2082_congr {R : Type u_1} {S : Type u_2} [Semiring R] [Semiring S] {f g : R \u2192+* S} {s t : S} {\u03c6 \u03c8 : Polynomial R} :f = g \u2192 s = t \u2192 \u03c6 = \u03c8 \u2192 eval\u2082 f s \u03c6 = eval\u2082 g t \u03c8","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L63-L65"},"Polynomial.leadingCoeff_divByMonic_X_sub_C":{"signature":"theorem Polynomial.leadingCoeff_divByMonic_X_sub_C {R : Type u} [CommRing R] [IsDomain R] (p : Polynomial R) (hp : p.degree \u2260 0) (a : R) :(p /\u2098 (X - C a)).leadingCoeff = p.leadingCoeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L798-L804"},"Polynomial.natTrailingDegree_eq_of_trailingDegree_eq":{"signature":"theorem Polynomial.natTrailingDegree_eq_of_trailingDegree_eq {R : Type u} {S : Type v} [Semiring R] {p : Polynomial R} [Semiring S] {q : Polynomial S} (h : p.trailingDegree = q.trailingDegree) :p.natTrailingDegree = q.natTrailingDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L113-L116"},"Polynomial.degree_multiset_prod":{"signature":"theorem Polynomial.degree_multiset_prod {R : Type u} [CommSemiring R] [NoZeroDivisors R] (t : Multiset (Polynomial R)) [Nontrivial R] :t.prod.degree = (Multiset.map (fun (f : Polynomial R) => f.degree) t).sum","doc":"The degree of a product of polynomials is equal to the sum of the degrees, where the degree of the zero polynomial is \u22a5.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/BigOperators.lean#L341-L345"},"Polynomial.trunc_toLaurent":{"signature":"theorem Polynomial.trunc_toLaurent {R : Type u_1} [Semiring R] (f : Polynomial R) :LaurentPolynomial.trunc (toLaurent f) = f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Laurent.lean#L326-L328"},"Polynomial.natDegree_X_pow_mul":{"signature":"theorem Polynomial.natDegree_X_pow_mul {R : Type u} [Semiring R] [Nontrivial R] {p : Polynomial R} (n : \u2115) (hp : p \u2260 0) :(X ^ n * p).natDegree = p.natDegree + n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L523-L524"},"Polynomial.hasseDeriv_X":{"signature":"theorem Polynomial.hasseDeriv_X {R : Type u_1} [Semiring R] (k : \u2115) (hk : 1 < k) :(hasseDeriv k) X = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/HasseDeriv.lean#L124-L126"},"Polynomial.monic_finprod_of_monic":{"signature":"theorem Polynomial.monic_finprod_of_monic {R : Type u} [CommSemiring R] (\u03b1 : Type u_1) (f : \u03b1 \u2192 Polynomial R) (hf : \u2200 i \u2208 Function.mulSupport f, (f i).Monic) :(finprod f).Monic","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Monic.lean#L292-L299"},"Polynomial.monic_finprod_X_sub_C":{"signature":"theorem Polynomial.monic_finprod_X_sub_C {R : Type u} [CommRing R] {\u03b1 : Type u_1} (b : \u03b1 \u2192 R) :(\u220f\u1da0 (k : \u03b1), (X - C (b k))).Monic","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L756-L757"},"Polynomial.degree_intCast_le":{"signature":"theorem Polynomial.degree_intCast_le {R : Type u} [Ring R] (n : \u2124) :(\u2191n).degree \u2264 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L293-L293"},"Polynomial.separable_C_mul_X_pow_add_C_mul_X_add_C'":{"signature":"theorem Polynomial.separable_C_mul_X_pow_add_C_mul_X_add_C' {R : Type u} [CommRing R] (p n : \u2115) (a b c : R) [CharP R p] (hn : p \u2223 n) (hb : IsUnit b) :(C a * X ^ n + C b * X + C c).Separable","doc":"If R is of characteristic p, p \u2223 n and b is a unit, then a * X ^ n + b * X + c is separable.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Separable.lean#L277-L282"},"Polynomial.cauchyBound_X_add_C":{"signature":"theorem Polynomial.cauchyBound_X_add_C {K : Type u_1} [NormedDivisionRing K] (x : K) :(X + C x).cauchyBound = \u2016x\u2016\u208a + 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/CauchyBound.lean#L55-L57"},"Polynomial.rootsExpandPowEquivRoots_apply":{"signature":"theorem Polynomial.rootsExpandPowEquivRoots_apply {R : Type u_1} [CommRing R] [IsDomain R] (p : \u2115) [ExpChar R p] (f : Polynomial R) [DecidableEq R] [PerfectRing R p] (n : \u2115) (x : \u21a5((expand R (p ^ n)) f).roots.toFinset) :\u2191((rootsExpandPowEquivRoots p f n) x) = \u2191x ^ p ^ n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Perfect.lean#L443-L445"},"Polynomial.splits_of_degree_le_one_of_invertible":{"signature":"theorem Polynomial.splits_of_degree_le_one_of_invertible {R : Type u_1} [Semiring R] {f : Polynomial R} (hf : f.degree \u2264 1) (h : Invertible f.leadingCoeff) :f.Splits","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L111-L113"},"Polynomial.int_monic_iff":{"signature":"theorem Polynomial.int_monic_iff {K : Type u_1} [Field K] (R : Subring K) (P : Polynomial K) (hP : \u2200 (n : \u2115), P.coeff n \u2208 R) :(int R P hP).Monic \u2194 P.Monic","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Ring/Subring/IntPolynomial.lean#L48-L50"},"Polynomial.content_monomial":{"signature":"theorem Polynomial.content_monomial {R : Type u_1} [CommRing R] [NormalizedGCDMonoid R] {r : R} {k : \u2115} :((monomial k) r).content = normalize r","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Content.lean#L150-L152"},"Polynomial.nontrivial_iff":{"signature":"theorem Polynomial.nontrivial_iff {R : Type u} [Semiring R] :Nontrivial (Polynomial R) \u2194 Nontrivial R","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L1187-L1192"},"Polynomial.isLeftCancelMulZero_iff":{"signature":"theorem Polynomial.isLeftCancelMulZero_iff {R : Type u_1} [Semiring R] :IsLeftCancelMulZero (Polynomial R) \u2194 IsLeftCancelMulZero R \u2227 IsCancelAdd R","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Opposites.lean#L101-L113"},"Polynomial.isRightCancelMulZero_iff":{"signature":"theorem Polynomial.isRightCancelMulZero_iff {R : Type u_1} [Semiring R] :IsRightCancelMulZero (Polynomial R) \u2194 IsRightCancelMulZero R \u2227 IsCancelAdd R","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Opposites.lean#L115-L118"},"Polynomial.isCancelMulZero_iff":{"signature":"theorem Polynomial.isCancelMulZero_iff {R : Type u_1} [Semiring R] :IsCancelMulZero (Polynomial R) \u2194 IsCancelMulZero R \u2227 IsCancelAdd R","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Opposites.lean#L120-L123"},"Polynomial.isDomain_iff":{"signature":"theorem Polynomial.isDomain_iff {R : Type u_1} [Semiring R] :IsDomain (Polynomial R) \u2194 IsDomain R \u2227 IsCancelAdd R","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Opposites.lean#L125-L127"},"Polynomial.toAlgHom_taylorEquiv":{"signature":"theorem Polynomial.toAlgHom_taylorEquiv {R : Type u_1} [CommRing R] (r : R) :\u2191(taylorEquiv r) = taylorAlgHom r","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Taylor.lean#L189-L189"},"Polynomial.eval_derivative_eq_eval_mul_sum_of_splits":{"signature":"theorem Polynomial.eval_derivative_eq_eval_mul_sum_of_splits {R : Type u_1} [Field R] {f : Polynomial R} (hf : f.Splits) {x : R} (hx : eval x f \u2260 0) :eval x (derivative f) = eval x f * (Multiset.map (fun (z : R) => 1 / (x - z)) f.roots).sum","doc":"Alias of Polynomial.Splits.eval_derivative_eq_eval_mul_sum.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L877-L878"},"Polynomial.trinomial_monic":{"signature":"theorem Polynomial.trinomial_monic {R : Type u_1} [Semiring R] {k m n : \u2115} {u v : R} (hkm : k < m) (hmn : m < n) :(trinomial k m n u v 1).Monic","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/UnitTrinomial.lean#L100-L102"},"Polynomial.X_pow_sub_X_sub_one_irreducible_rat":{"signature":"theorem Polynomial.X_pow_sub_X_sub_one_irreducible_rat {n : \u2115} (hn1 : n \u2260 1) :Irreducible (X ^ n - X - 1)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Selmer.lean#L69-L80"},"Polynomial.natTrailingDegree_natCast":{"signature":"theorem Polynomial.natTrailingDegree_natCast {R : Type u} [Semiring R] (n : \u2115) :(\u2191n).natTrailingDegree = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L205-L207"},"Polynomial.aroots_smul_nonzero":{"signature":"theorem Polynomial.aroots_smul_nonzero {S : Type v} {T : Type w} [CommRing T] [IsDomain T] [CommRing S] [IsDomain S] [Algebra T S] [Module.IsTorsionFree T S] {a : T} (p : Polynomial T) (ha : a \u2260 0) :(a \u2022 p).aroots S = p.aroots S","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L499-L503"},"Polynomial.isEisensteinAt_iff":{"signature":"theorem Polynomial.isEisensteinAt_iff {R : Type u} [CommSemiring R] (f : Polynomial R) (\ud835\udcdf : Ideal R) :f.IsEisensteinAt \ud835\udcdf \u2194 f.leadingCoeff \u2209 \ud835\udcdf \u2227 (\u2200 {n : \u2115}, n < f.natDegree \u2192 f.coeff n \u2208 \ud835\udcdf) \u2227 f.coeff 0 \u2209 \ud835\udcdf ^ 2","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Eisenstein/Basic.lean#L54-L54"},"Polynomial.fiberEquivQuotient":{"signature":"noncomputable def Polynomial.fiberEquivQuotient {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [Algebra R S] (f : Polynomial R \u2192\u2090[R] S) (hf : Function.Surjective \u21d1f) (p : Ideal R) [p.IsPrime] :p.Fiber S \u2243\u2090[p.ResidueField] Polynomial p.ResidueField \u29f8 Ideal.map (mapRingHom (algebraMap R p.ResidueField)) (RingHom.ker \u2191f)","doc":"\u03ba(p) \u2297[R] (R[X] \u29f8 I) = \u03ba(p)[X] / I","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/LocalRing/ResidueField/Polynomial.lean#L93-L114"},"Polynomial.not_ker_le_map_C_of_surjective_of_quasiFiniteAt":{"signature":"theorem Polynomial.not_ker_le_map_C_of_surjective_of_quasiFiniteAt {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [Algebra R S] (f : Polynomial R \u2192\u2090[R] S) (hf : Function.Surjective \u21d1f) (P : Ideal S) [P.IsPrime] [Algebra.QuasiFiniteAt R P] :\u00acRingHom.ker f \u2264 Ideal.map C (Ideal.under R P)","doc":"If P is a prime of R[X]/I that is quasi finite over R, then I is not contained in (P \u2229 R)[X]. For usability, we replace I by the kernel of a surjective map R[X] \u2192\u2090[R] S.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/QuasiFinite/Basic.lean#L528-L551"},"Polynomial.coeff_opRingEquiv":{"signature":"theorem Polynomial.coeff_opRingEquiv {R : Type u_1} [Semiring R] (p : (Polynomial R)\u1d50\u1d52\u1d56) (n : \u2115) :((opRingEquiv R) p).coeff n = MulOpposite.op ((MulOpposite.unop p).coeff n)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Opposites.lean#L81-L83"},"Polynomial.support_opRingEquiv":{"signature":"theorem Polynomial.support_opRingEquiv {R : Type u_1} [Semiring R] (p : (Polynomial R)\u1d50\u1d52\u1d56) :((opRingEquiv R) p).support = (MulOpposite.unop p).support","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Opposites.lean#L85-L87"},"Polynomial.natDegree_eq_support_max'":{"signature":"theorem Polynomial.natDegree_eq_support_max' {R : Type u} [Semiring R] {p : Polynomial R} (h : p \u2260 0) :p.natDegree = p.support.max' \u22ef","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Support.lean#L128-L131"},"Polynomial.natDegree_opRingEquiv":{"signature":"theorem Polynomial.natDegree_opRingEquiv {R : Type u_1} [Semiring R] (p : (Polynomial R)\u1d50\u1d52\u1d56) :((opRingEquiv R) p).natDegree = (MulOpposite.unop p).natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Opposites.lean#L89-L94"},"Polynomial.monomial_left_inj":{"signature":"theorem Polynomial.monomial_left_inj {R : Type u} [Semiring R] {a : R} (ha : a \u2260 0) {i j : \u2115} :(monomial i) a = (monomial j) a \u2194 i = j","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L811-L813"},"Polynomial.toFinsupp_eq_one":{"signature":"theorem Polynomial.toFinsupp_eq_one {R : Type u} [Semiring R] {a : Polynomial R} :a.toFinsupp = 1 \u2194 a = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L235-L237"},"Polynomial.mkDerivation_one_eq_derivative'":{"signature":"theorem Polynomial.mkDerivation_one_eq_derivative' (R : Type u_1) [CommSemiring R] :(mkDerivation R) 1 = derivative'","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivation.lean#L74-L76"},"Polynomial.cyclotomic_mahlerMeasure_eq_one":{"signature":"theorem Polynomial.cyclotomic_mahlerMeasure_eq_one {R : Type u_1} [CommRing R] [Algebra R \u2102] (n : \u2115) :(map (algebraMap R \u2102) (cyclotomic n R)).mahlerMeasure = 1","doc":"The Mahler measure of a cyclotomic polynomial is 1.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/NumberTheory/MahlerMeasure.lean#L126-L137"},"Polynomial.roots_X_pow_char_sub_C_pow":{"signature":"theorem Polynomial.roots_X_pow_char_sub_C_pow {R : Type u_1} [CommRing R] [IsDomain R] {p : \u2115} [ExpChar R p] [PerfectRing R p] {y : R} {m : \u2115} :((X ^ p - C y) ^ m).roots = (m * p) \u2022 {(frobeniusEquiv R p).symm y}","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Perfect.lean#L368-L371"},"Polynomial.natSepDegree_X":{"signature":"theorem Polynomial.natSepDegree_X {F : Type u} [Field F] :X.natSepDegree = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/SeparableDegree.lean#L312-L314"},"Polynomial.eq_zero_of_eq_zero":{"signature":"theorem Polynomial.eq_zero_of_eq_zero {R : Type u} [Semiring R] (h : 0 = 1) (p : Polynomial R) :p = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L736-L737"},"Polynomial.monomial_mem_lifts":{"signature":"theorem Polynomial.monomial_mem_lifts {R : Type u} [Semiring R] {S : Type v} [Semiring S] {f : R \u2192+* S} {s : S} (n : \u2115) (h : s \u2208 Set.range \u21d1f) :(monomial n) s \u2208 lifts f","doc":"If (s : S) is in the image of f, then monomial n s lifts.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Lifts.lean#L116-L120"},"Polynomial.contentIdeal_FG":{"signature":"theorem Polynomial.contentIdeal_FG {R : Type u_1} [Semiring R] (p : Polynomial R) :p.contentIdeal.FG","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ContentIdeal.lean#L88-L88"},"Polynomial.isPrimitive_iff_contentIdeal_eq_top":{"signature":"theorem Polynomial.isPrimitive_iff_contentIdeal_eq_top {R : Type u_3} [CommSemiring R] [IsBezout R] (p : Polynomial R) :p.IsPrimitive \u2194 p.contentIdeal = \u22a4","doc":"The polynomial p is primitive if and only if the coefficients of p generate the whole ring.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ContentIdeal.lean#L215-L218"},"Polynomial.erase_same":{"signature":"theorem Polynomial.erase_same {R : Type u} [Semiring R] (p : Polynomial R) (n : \u2115) :(erase n p).coeff n = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L963-L964"},"Polynomial.erase_ne":{"signature":"theorem Polynomial.erase_ne {R : Type u} [Semiring R] (p : Polynomial R) (n i : \u2115) (h : i \u2260 n) :(erase n p).coeff i = p.coeff i","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L966-L968"},"Polynomial.erase_mem_lifts":{"signature":"theorem Polynomial.erase_mem_lifts {R : Type u} [Semiring R] {S : Type v} [Semiring S] {f : R \u2192+* S} {p : Polynomial S} (n : \u2115) (h : p \u2208 lifts f) :erase n p \u2208 lifts f","doc":"If p lifts then p.erase n lifts.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Lifts.lean#L122-L131"},"Polynomial.root_mul":{"signature":"theorem Polynomial.root_mul {R : Type u} {a : R} [CommSemiring R] {p q : Polynomial R} [NoZeroDivisors R] :(p * q).IsRoot a \u2194 p.IsRoot a \u2228 q.IsRoot a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L694-L695"},"Polynomial.toFinsupp_intCast":{"signature":"theorem Polynomial.toFinsupp_intCast {R : Type u} [Ring R] (z : \u2124) :(\u2191z).toFinsupp = \u2191z","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L1087-L1088"},"Polynomial.evalEval_smul":{"signature":"theorem Polynomial.evalEval_smul {R : Type u_1} {S : Type u_2} [Semiring R] [DistribSMul S R] [IsScalarTower S R R] (x y : R) (s : S) (p : Polynomial (Polynomial R)) :evalEval x y (s \u2022 p) = s \u2022 evalEval x y p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Bivariate.lean#L85-L88"},"Polynomial.degree_list_sum_le":{"signature":"theorem Polynomial.degree_list_sum_le {S : Type u_1} [Semiring S] (l : List (Polynomial S)) :l.sum.degree \u2264 (List.map natDegree l).maximum","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/BigOperators.lean#L92-L102"},"Polynomial.map_dickson":{"signature":"theorem Polynomial.map_dickson {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] {k : \u2115} {a : R} (f : R \u2192+* S) (n : \u2115) :map f (dickson k a n) = dickson k (f a) n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Dickson.lean#L91-L97"},"Polynomial.splits_one":{"signature":"theorem Polynomial.splits_one {R : Type u_1} [Semiring R] :Splits 1","doc":"Alias of Polynomial.Splits.one.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L689-L690"},"Polynomial.map_intCast":{"signature":"theorem Polynomial.map_intCast {R : Type u} [Ring R] {S : Type u_1} [Ring S] (f : R \u2192+* S) (n : \u2124) :map f \u2191n = \u2191n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L731-L732"},"Polynomial.eval\u2082_multiset_prod":{"signature":"theorem Polynomial.eval\u2082_multiset_prod {R : Type u} {S : Type v} [CommSemiring R] [CommSemiring S] (f : R \u2192+* S) (s : Multiset (Polynomial R)) (x : S) :eval\u2082 f x s.prod = (Multiset.map (eval\u2082 f x) s).prod","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L641-L643"},"Polynomial.dvd_iterate_derivative_pow":{"signature":"theorem Polynomial.dvd_iterate_derivative_pow {R : Type u} [CommSemiring R] (f : Polynomial R) (n : \u2115) {m : \u2115} (c : R) (hm : m \u2260 0) :\u2191n \u2223 eval c ((\u21d1derivative)^[m] (f ^ n))","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L480-L485"},"Polynomial.ne_zero_of_trailingDegree_lt":{"signature":"theorem Polynomial.ne_zero_of_trailingDegree_lt {R : Type u} [Semiring R] {p : Polynomial R} {n : \u2115\u221e} (h : p.trailingDegree < n) :p \u2260 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L427-L428"},"Polynomial.coeffs_one":{"signature":"theorem Polynomial.coeffs_one {R : Type u} [Semiring R] :coeffs 1 \u2286 {1}","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L1025-L1028"},"Polynomial.homogenize_C":{"signature":"theorem Polynomial.homogenize_C {R : Type u_1} [CommSemiring R] (c : R) (n : \u2115) :(C c).homogenize n = MvPolynomial.C c * MvPolynomial.X 1 ^ n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Homogenize.lean#L103-L105"},"Polynomial.homogenize_one":{"signature":"theorem Polynomial.homogenize_one {R : Type u_1} [CommSemiring R] (n : \u2115) :homogenize 1 n = MvPolynomial.X 1 ^ n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Homogenize.lean#L107-L109"},"Polynomial.homogenize_finsetProd":{"signature":"theorem Polynomial.homogenize_finsetProd {R : Type u_1} [CommSemiring R] {\u03b9 : Type u_2} {s : Finset \u03b9} {p : \u03b9 \u2192 Polynomial R} {n : \u03b9 \u2192 \u2115} (h : \u2200 i \u2208 s, (p i).natDegree \u2264 n i) :(\u220f i \u2208 s, p i).homogenize (\u2211 i \u2208 s, n i) = \u220f i \u2208 s, (p i).homogenize (n i)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Homogenize.lean#L172-L180"},"Polynomial.aeval_fn_apply":{"signature":"theorem Polynomial.aeval_fn_apply {R : Type u} [CommSemiring R] {X : Type u_3} (g : Polynomial R) (f : X \u2192 R) (x : X) :(aeval f) g x = (aeval (f x)) g","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L460-L463"},"Polynomial.isCoprime_iff_aeval_ne_zero":{"signature":"theorem Polynomial.isCoprime_iff_aeval_ne_zero {K : Type v} [Field K] (f g : Polynomial K) :IsCoprime f g \u2194 \u2200 {A : Type v} [inst : CommRing A] [IsDomain A] [inst_2 : Algebra K A] (a : A), (aeval a) f \u2260 0 \u2228 (aeval a) g \u2260 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/SplittingField/Construction.lean#L78-L87"},"Polynomial.natDegree_notMem_eraseLead_support":{"signature":"theorem Polynomial.natDegree_notMem_eraseLead_support {R : Type u_1} [Semiring R] {f : Polynomial R} :f.natDegree \u2209 f.eraseLead.support","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/EraseLead.lean#L94-L95"},"Polynomial.isUnit_leadingCoeff_mul_right_eq_zero_iff":{"signature":"theorem Polynomial.isUnit_leadingCoeff_mul_right_eq_zero_iff {R : Type u} [Semiring R] {p : Polynomial R} (h : IsUnit p.leadingCoeff) {q : Polynomial R} :p * q = 0 \u2194 q = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Monic.lean#L561-L574"},"Polynomial.taylor_pow":{"signature":"theorem Polynomial.taylor_pow {R : Type u_1} [CommSemiring R] (r : R) (f : Polynomial R) (n : \u2115) :(taylor r) (f ^ n) = (taylor r) f ^ n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Taylor.lean#L150-L152"},"Polynomial.eval_C_X_comp_eval\u2082_map_C_X":{"signature":"theorem Polynomial.eval_C_X_comp_eval\u2082_map_C_X {R : Type u_1} [CommSemiring R] :(evalRingHom (C X)).comp (eval\u2082RingHom (mapRingHom (algebraMap R (Polynomial (Polynomial R)))) (C X)) = RingHom.id (Polynomial (Polynomial R))","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Bivariate.lean#L192-L195"},"Polynomial.eval_C_X_eval\u2082_map_C_X":{"signature":"theorem Polynomial.eval_C_X_eval\u2082_map_C_X {R : Type u_1} [CommSemiring R] {p : Polynomial (Polynomial R)} :eval (C X) (eval\u2082 (mapRingHom (algebraMap R (Polynomial (Polynomial R)))) (C X) p) = p","doc":"Viewing R[X,Y,X'] as an R[X']-algebra, a polynomial p : R[X',Y'] can be evaluated at Y : R[X,Y,X'] (substitution of Y' by Y), obtaining another polynomial in R[X,Y,X']. When this polynomial is then evaluated at X' = X, the original polynomial p is recovered.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Bivariate.lean#L197-L202"},"Polynomial.hasFDerivAt":{"signature":"theorem Polynomial.hasFDerivAt {\ud835\udd5c : Type u} [NontriviallyNormedField \ud835\udd5c] (p : Polynomial \ud835\udd5c) (x : \ud835\udd5c) :HasFDerivAt (fun (x : \ud835\udd5c) => eval x p) (ContinuousLinearMap.smulRight 1 (eval x (derivative p))) x","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Calculus/Deriv/Polynomial.lean#L124-L126"},"Polynomial.hasFDerivWithinAt":{"signature":"theorem Polynomial.hasFDerivWithinAt {\ud835\udd5c : Type u} [NontriviallyNormedField \ud835\udd5c] {s : Set \ud835\udd5c} (p : Polynomial \ud835\udd5c) (x : \ud835\udd5c) :HasFDerivWithinAt (fun (x : \ud835\udd5c) => eval x p) (ContinuousLinearMap.smulRight 1 (eval x (derivative p))) s x","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Calculus/Deriv/Polynomial.lean#L132-L134"},"Polynomial.fderivWithin":{"signature":"theorem Polynomial.fderivWithin {\ud835\udd5c : Type u} [NontriviallyNormedField \ud835\udd5c] {x : \ud835\udd5c} {s : Set \ud835\udd5c} (p : Polynomial \ud835\udd5c) (hxs : UniqueDiffWithinAt \ud835\udd5c s x) :fderivWithin \ud835\udd5c (fun (x : \ud835\udd5c) => eval x p) s x = ContinuousLinearMap.smulRight 1 (eval x (derivative p))","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Calculus/Deriv/Polynomial.lean#L151-L153"},"Polynomial.toFinsuppIsoLinear_symm_apply_toFinsupp":{"signature":"theorem Polynomial.toFinsuppIsoLinear_symm_apply_toFinsupp (R : Type u) [Semiring R] (a\u271d : AddMonoidAlgebra R \u2115) :((toFinsuppIsoLinear R).symm a\u271d).toFinsupp = a\u271d","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L339-L339"},"Polynomial.reverse_X_mul":{"signature":"theorem Polynomial.reverse_X_mul {R : Type u_1} [Semiring R] (p : Polynomial R) :(X * p).reverse = p.reverse","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Reverse.lean#L309-L310"},"Polynomial.gaussNorm_intAdicAbv_le_one":{"signature":"theorem Polynomial.gaussNorm_intAdicAbv_le_one {R : Type u_1} [CommRing R] [IsDedekindDomain R] (v : IsDedekindDomain.HeightOneSpectrum R) {b : NNReal} (hb : 1 < b) (p : Polynomial R) :gaussNorm (v.intAdicAbv hb) 1 p \u2264 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/DedekindDomain/GaussLemma.lean#L28-L31"},"Polynomial.rootMultiplicity_add":{"signature":"theorem Polynomial.rootMultiplicity_add {R : Type u} [CommRing R] {p q : Polynomial R} (a : R) (hzero : p + q \u2260 0) :min (rootMultiplicity a p) (rootMultiplicity a q) \u2264 rootMultiplicity a (p + q)","doc":"The multiplicity of p + q is at least the minimum of the multiplicities.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L730-L734"},"Polynomial.aevalTower_algebraMap":{"signature":"theorem Polynomial.aevalTower_algebraMap {R : Type u} {S : Type v} {A' : Type u_1} [CommSemiring R] [CommSemiring A'] [CommSemiring S] [Algebra S R] [Algebra S A'] (g : R \u2192\u2090[S] A') (y : A') (x : R) :(aevalTower g y) ((algebraMap R (Polynomial R)) x) = g x","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L539-L540"},"Polynomial.hasSMulPi":{"signature":"def Polynomial.hasSMulPi (R : Type u_1) (S : Type u_2) [Semiring R] [SMul R S] :SMul (Polynomial R) (R \u2192 S)","doc":"This is not an instance as it forms a diamond with Pi.instSMul. See the instance_diamonds test for details.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Algebraic/Pi.lean#L26-L30"},"Polynomial.trailingDegree_ne_zero":{"signature":"theorem Polynomial.trailingDegree_ne_zero {R : Type u} [Semiring R] {p : Polynomial R} :p.trailingDegree \u2260 0 \u2194 p.coeff 0 = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L147-L148"},"Polynomial.monic_X_pow_sub":{"signature":"theorem Polynomial.monic_X_pow_sub {R : Type u} [Ring R] {p : Polynomial R} {n : \u2115} (H : p.degree < \u2191n) :(X ^ n - p).Monic","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Monic.lean#L437-L438"},"Polynomial.neg_cancelLeads":{"signature":"theorem Polynomial.neg_cancelLeads {R : Type u_1} [Ring R] {p q : Polynomial R} :-p.cancelLeads q = q.cancelLeads p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/CancelLeads.lean#L47-L49"},"Polynomial.toFinsupp_X":{"signature":"theorem Polynomial.toFinsupp_X {R : Type u} [Semiring R] :X.toFinsupp = AddMonoidAlgebra.single 1 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L492-L494"},"Polynomial.iterate_derivative_prod_X_sub_C":{"signature":"theorem Polynomial.iterate_derivative_prod_X_sub_C {R : Type u} [CommRing R] {k : \u2115} {S : Finset R} (hk : k \u2264 S.card) :(\u21d1derivative)^[k] (\u220f a \u2208 S, (X - C a)) = \u2191k.factorial * \u2211 T \u2208 Finset.powersetCard (S.card - k) S, \u220f a \u2208 T, (X - C a)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L697-L729"},"Polynomial.logMahlerMeasure_X":{"signature":"theorem Polynomial.logMahlerMeasure_X :X.logMahlerMeasure = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/MahlerMeasure.lean#L62-L64"},"Polynomial.support_scaleRoots_eq":{"signature":"theorem Polynomial.support_scaleRoots_eq {R : Type u_1} [Semiring R] (p : Polynomial R) {s : R} (hs : s \u2208 nonZeroDivisors R) :(p.scaleRoots s).support = p.support","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ScaleRoots.lean#L58-L65"},"Polynomial.ofReal_eval":{"signature":"theorem Polynomial.ofReal_eval {K : Type u_1} [RCLike K] (p : Polynomial \u211d) (x : \u211d) :\u2191(eval x p) = (aeval \u2191x) p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/RCLike/Lemmas.lean#L28-L29"},"Polynomial.aeval_pi_apply\u2082":{"signature":"theorem Polynomial.aeval_pi_apply\u2082 {R : Type u} [CommSemiring R] {I : Type u_3} {A : I \u2192 Type u_4} [(i : I) \u2192 Semiring (A i)] [(i : I) \u2192 Algebra R (A i)] (x : (i : I) \u2192 A i) (p : Polynomial R) (j : I) :(aeval x) p j = (aeval (x j)) p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L440-L441"},"Polynomial.addHom_ext'":{"signature":"theorem Polynomial.addHom_ext' {R : Type u} [Semiring R] {M : Type u_1} [AddZeroClass M] {f g : Polynomial R \u2192+ M} (h : \u2200 (n : \u2115), f.comp (monomial n).toAddMonoidHom = g.comp (monomial n).toAddMonoidHom) :f = g","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L725-L728"},"Polynomial.natSepDegree_one":{"signature":"theorem Polynomial.natSepDegree_one {F : Type u} [Field F] :natSepDegree 1 = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/SeparableDegree.lean#L327-L329"},"Polynomial.lt_rootMultiplicity_of_isRoot_iterate_derivative_of_mem_nonZeroDivisors'":{"signature":"theorem Polynomial.lt_rootMultiplicity_of_isRoot_iterate_derivative_of_mem_nonZeroDivisors' {R : Type u} [CommRing R] {p : Polynomial R} {t : R} {n : \u2115} (h : p \u2260 0) (hroot : \u2200 m \u2264 n, ((\u21d1derivative)^[m] p).IsRoot t) (hnzd : \u2200 m \u2264 n, m \u2260 0 \u2192 \u2191m \u2208 nonZeroDivisors R) :n < rootMultiplicity t p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L98-L111"},"Polynomial.toFinsupp_sum":{"signature":"theorem Polynomial.toFinsupp_sum {R : Type u} [Semiring R] {\u03b9 : Type u_1} (s : Finset \u03b9) (f : \u03b9 \u2192 Polynomial R) :(\u2211 i \u2208 s, f i).toFinsupp = \u2211 i \u2208 s, (f i).toFinsupp","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L350-L352"},"Polynomial.mul_eq_sum_sum":{"signature":"theorem Polynomial.mul_eq_sum_sum {R : Type u} [Semiring R] {p q : Polynomial R} :p * q = \u2211 i \u2208 p.support, q.sum fun (j : \u2115) (a : R) => (monomial (i + j)) (p.coeff i * a)","doc":"Expressing the product of two polynomials as a double sum.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L837-L842"},"Polynomial.lsum_apply":{"signature":"theorem Polynomial.lsum_apply {R : Type u_1} {A : Type u_2} {M : Type u_3} [Semiring R] [Semiring A] [AddCommMonoid M] [Module R A] [Module R M] (f : \u2115 \u2192 A \u2192\u2097[R] M) (p : Polynomial A) :(lsum f) p = p.sum fun (x1 : \u2115) (x2 : A) => (f x1) x2","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L68-L68"},"Polynomial.generalizedEisenstein":{"signature":"theorem Polynomial.generalizedEisenstein {R : Type u_1} [CommRing R] [IsDomain R] {K : Type u_2} [Field K] [Algebra R K] {q f : Polynomial R} {p : \u2115} (hq_irr : Irreducible (map (algebraMap R K) q)) (hq_monic : q.Monic) (hf_prim : f.IsPrimitive) (hfd0 : 0 < f.natDegree) (hfP : (algebraMap R K) f.leadingCoeff \u2260 0) (hfmodP : map (algebraMap R K) f = C ((algebraMap R K) f.leadingCoeff) * map (algebraMap R K) q ^ p) (hfmodP2 : map (Ideal.Quotient.mk (RingHom.ker (algebraMap R K) ^ 2)) (f %\u2098 q) \u2260 0) :Irreducible f","doc":"A generalized Eisenstein criterion Let R be an integral domain and K an R-algebra which is a domain. Let q : R[X] be a monic polynomial which is prime in K[X]. Let f : R[X] be a primitive polynomial of strictly positive degree whose leading coefficient is not zero in K and such that the image f in K[X] is a power of q. Assume moreover that f.modByMonic q is not zero in (R \u29f8 (P ^ 2))[X], where P is the kernel of algebraMap R K. Then f is irreducible.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Eisenstein/Criterion.lean#L112-L169"},"Polynomial.irreducible_of_eisenstein_criterion":{"signature":"theorem Polynomial.irreducible_of_eisenstein_criterion {R : Type u_1} [CommRing R] [IsDomain R] {f : Polynomial R} {P : Ideal R} (hP : P.IsPrime) (hfl : f.leadingCoeff \u2209 P) (hfP : \u2200 (n : \u2115), \u2191n < f.degree \u2192 f.coeff n \u2208 P) (hfd0 : 0 < f.degree) (h0 : f.coeff 0 \u2209 P ^ 2) (hu : f.IsPrimitive) :Irreducible f","doc":"If f is a nonconstant polynomial with coefficients in R, and P is a prime ideal in R, then if every coefficient in R except the leading coefficient is in P, and the trailing coefficient is not in P^2 and no nonunits in R divide f, then f is irreducible.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Eisenstein/Criterion.lean#L171-L208"},"Polynomial.nextCoeff_map_of_leadingCoeff_ne_zero":{"signature":"theorem Polynomial.nextCoeff_map_of_leadingCoeff_ne_zero {R : Type u} {S : Type v} [Semiring R] [Semiring S] {p : Polynomial R} (f : R \u2192+* S) (hf : f p.leadingCoeff \u2260 0) :(map f p).nextCoeff = f p.nextCoeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Degree.lean#L189-L191"},"Polynomial.coeff_hermite_succ_zero":{"signature":"theorem Polynomial.coeff_hermite_succ_zero (n : \u2115) :(hermite (n + 1)).coeff 0 = -(hermite n).coeff 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Hermite/Basic.lean#L75-L76"},"Polynomial.natDegree_map_lt":{"signature":"theorem Polynomial.natDegree_map_lt {R : Type u} {S : Type v} [Semiring R] [Semiring S] {f : R \u2192+* S} {p : Polynomial R} (hp : f p.leadingCoeff = 0) (hp\u2080 : map f p \u2260 0) :(map f p).natDegree < p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Degree.lean#L159-L161"},"Polynomial.lcoeff_comp_mapAlgHom_eq":{"signature":"theorem Polynomial.lcoeff_comp_mapAlgHom_eq {R : Type u} {A : Type z} {B : Type u_2} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A \u2192\u2090[R] B) (n : \u2115) :\u2191R (lcoeff B n) \u2218\u2097 (mapAlgHom f).toLinearMap = f.toLinearMap \u2218\u2097 \u2191R (lcoeff A n)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L197-L200"},"Polynomial.resultant_taylor":{"signature":"theorem Polynomial.resultant_taylor {R : Type u_1} [CommRing R] (f g : Polynomial R) (r : R) :((taylor r) f).resultant ((taylor r) g) = f.resultant g","doc":"Res(f(x + r), g(x + r)) = Res(f, g).","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L765-L798"},"Polynomial.eval_natCast_map":{"signature":"theorem Polynomial.eval_natCast_map {R : Type u} {S : Type v} [Semiring R] [Semiring S] (f : R \u2192+* S) (p : Polynomial R) (n : \u2115) :eval (\u2191n) (map f p) = f (eval (\u2191n) p)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Coeff.lean#L152-L157"},"Polynomial.natDegree_C_mul_eq_of_mul_eq_one":{"signature":"theorem Polynomial.natDegree_C_mul_eq_of_mul_eq_one {R : Type u} {a : R} [Semiring R] {p : Polynomial R} {ai : R} (au : ai * a = 1) :(C a * p).natDegree = p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L106-L112"},"Polynomial.instFintypeGal":{"signature":"instance Polynomial.instFintypeGal {F : Type u_1} [Field F] (p : Polynomial F) :Fintype p.Gal","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/PolynomialGaloisGroup.lean#L57-L57"},"Polynomial.hasseDeriv_C":{"signature":"theorem Polynomial.hasseDeriv_C {R : Type u_1} [Semiring R] (k : \u2115) (r : R) (hk : 0 < k) :(hasseDeriv k) (C r) = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/HasseDeriv.lean#L117-L119"},"Polynomial.hasseDeriv_apply_one":{"signature":"theorem Polynomial.hasseDeriv_apply_one {R : Type u_1} [Semiring R] (k : \u2115) (hk : 0 < k) :(hasseDeriv k) 1 = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/HasseDeriv.lean#L121-L122"},"Polynomial.coeff_hermite_explicit":{"signature":"theorem Polynomial.coeff_hermite_explicit (n k : \u2115) :(hermite (2 * n + k)).coeff k = (-1) ^ n * \u2191(2 * n - 1).doubleFactorial * \u2191((2 * n + k).choose k)","doc":"Because of coeff_hermite_of_odd_add, every nonzero coefficient is described as follows.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Hermite/Basic.lean#L141-L182"},"Polynomial.coeff_hermite_of_even_add":{"signature":"theorem Polynomial.coeff_hermite_of_even_add {n k : \u2115} (hnk : Even (n + k)) :(hermite n).coeff k = (-1) ^ ((n - k) / 2) * \u2191(n - k - 1).doubleFactorial * \u2191(n.choose k)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Hermite/Basic.lean#L184-L191"},"Polynomial.newtonMap_apply_of_isUnit":{"signature":"theorem Polynomial.newtonMap_apply_of_isUnit {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [Algebra R S] {P : Polynomial R} {x : S} (h : IsUnit ((aeval x) (derivative P))) :P.newtonMap x = x - \u2191h.unit\u207b\u00b9 * (aeval x) P","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Dynamics/Newton.lean#L56-L58"},"Polynomial.eq_prod_roots_of_splits_id":{"signature":"theorem Polynomial.eq_prod_roots_of_splits_id {R : Type u_1} [CommRing R] {f : Polynomial R} [IsDomain R] (hf : f.Splits) :f = C f.leadingCoeff * (Multiset.map (fun (x : R) => X - C x) f.roots).prod","doc":"Alias of Polynomial.Splits.eq_prod_roots.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L802-L803"},"Polynomial.comp_C_integral_of_surjective_of_isJacobsonRing":{"signature":"theorem Polynomial.comp_C_integral_of_surjective_of_isJacobsonRing {R : Type u_1} [CommRing R] [IsJacobsonRing R] {S : Type u_2} [Field S] (f : Polynomial R \u2192+* S) (hf : Function.Surjective \u21d1f) :(f.comp C).IsIntegral","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Jacobson/Ring.lean#L549-L559"},"Polynomial.degree_prod":{"signature":"theorem Polynomial.degree_prod {R : Type u} {\u03b9 : Type w} (s : Finset \u03b9) [CommSemiring R] [NoZeroDivisors R] (f : \u03b9 \u2192 Polynomial R) [Nontrivial R] :(\u220f i \u2208 s, f i).degree = \u2211 i \u2208 s, (f i).degree","doc":"The degree of a product of polynomials is equal to the sum of the degrees, where the degree of the zero polynomial is \u22a5.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/BigOperators.lean#L347-L351"},"Polynomial.le_rootMultiplicity_mul":{"signature":"theorem Polynomial.le_rootMultiplicity_mul {R : Type u} [CommRing R] {p q : Polynomial R} (x : R) (hpq : p * q \u2260 0) :rootMultiplicity x p + rootMultiplicity x q \u2264 rootMultiplicity x (p * q)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L736-L739"},"Polynomial.taylorAlgHom_apply":{"signature":"theorem Polynomial.taylorAlgHom_apply {R : Type u_1} [CommSemiring R] (r : R) (a : Polynomial R) :(taylorAlgHom r) a = (taylor r) a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Taylor.lean#L146-L146"},"Polynomial.evalRingHom_zero":{"signature":"theorem Polynomial.evalRingHom_zero :evalRingHom 0 = constantCoeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Coeff.lean#L196-L197"},"Polynomial.dickson_one_one_eq_chebyshev_T":{"signature":"theorem Polynomial.dickson_one_one_eq_chebyshev_T (R : Type u_1) [CommRing R] [Invertible 2] (n : \u2115) :dickson 1 1 n = 2 * (Chebyshev.T R \u2191n).comp (C \u215f2 * X)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Dickson.lean#L153-L155"},"Polynomial.dickson_one_one_mul":{"signature":"theorem Polynomial.dickson_one_one_mul (R : Type u_1) [CommRing R] (m n : \u2115) :dickson 1 1 (m * n) = (dickson 1 1 m).comp (dickson 1 1 n)","doc":"The (m * n)-th Dickson polynomial of the first kind is the composition of the m-th and n-th.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Dickson.lean#L181-L196"},"Polynomial.dickson_one_one_comp_comm":{"signature":"theorem Polynomial.dickson_one_one_comp_comm (R : Type u_1) [CommRing R] (m n : \u2115) :(dickson 1 1 m).comp (dickson 1 1 n) = (dickson 1 1 n).comp (dickson 1 1 m)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Dickson.lean#L198-L200"},"Polynomial.isPrimitive_iff_ne_zero":{"signature":"theorem Polynomial.isPrimitive_iff_ne_zero {F : Type u_2} [Field F] (p : Polynomial F) :p.IsPrimitive \u2194 p \u2260 0","doc":"In a field, the notion of primitive polynomials is degenerate.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Content.lean#L76-L79"},"Polynomial.algEquivOfCompEqX_symm":{"signature":"theorem Polynomial.algEquivOfCompEqX_symm {R : Type u} [CommSemiring R] (p q : Polynomial R) (hpq : p.comp q = X) (hqp : q.comp p = X) :(p.algEquivOfCompEqX q hpq hqp).symm = q.algEquivOfCompEqX p hqp hpq","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L335-L337"},"Polynomial.exists_irreducible_of_degree_pos":{"signature":"theorem Polynomial.exists_irreducible_of_degree_pos {R : Type u_1} [CommSemiring R] [NoZeroDivisors R] [WfDvdMonoid R] {f : Polynomial R} [Nontrivial R] (hf : 0 < f.degree) :\u2203 (g : Polynomial R), Irreducible g \u2227 g \u2223 f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/UniqueFactorization.lean#L70-L72"},"Polynomial.exists_irreducible_of_natDegree_pos":{"signature":"theorem Polynomial.exists_irreducible_of_natDegree_pos {R : Type u_1} [CommSemiring R] [NoZeroDivisors R] [WfDvdMonoid R] {f : Polynomial R} [Nontrivial R] (hf : 0 < f.natDegree) :\u2203 (g : Polynomial R), Irreducible g \u2227 g \u2223 f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/UniqueFactorization.lean#L74-L77"},"Polynomial.aeval_algebraMap_eq_zero_iff_of_injective":{"signature":"theorem Polynomial.aeval_algebraMap_eq_zero_iff_of_injective {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [CommSemiring A] [Semiring B] [Algebra R A] [Algebra A B] [Algebra R B] [IsScalarTower R A B] {x : A} {p : Polynomial R} (h : Function.Injective \u21d1(algebraMap A B)) :(aeval ((algebraMap A B) x)) p = 0 \u2194 (aeval x) p = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Tower.lean#L62-L64"},"Polynomial.flt":{"signature":"theorem Polynomial.flt {k : Type u_1} [Field k] {n : \u2115} (hn : 3 \u2264 n) (chn : \u2191n \u2260 0) {a b c : Polynomial k} (ha : a \u2260 0) (hb : b \u2260 0) (hc : c \u2260 0) (hab : IsCoprime a b) (heq : a ^ n + b ^ n = c ^ n) :a.natDegree = 0 \u2227 b.natDegree = 0 \u2227 c.natDegree = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/NumberTheory/FLT/Polynomial.lean#L221-L236"},"Polynomial.length_coeffList_eq_withBotSucc_degree":{"signature":"theorem Polynomial.length_coeffList_eq_withBotSucc_degree {R : Type u_1} [Semiring R] (P : Polynomial R) :P.coeffList.length = P.degree.succ","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/CoeffList.lean#L83-L84"},"Polynomial.ringHom_ext'_iff":{"signature":"theorem Polynomial.ringHom_ext'_iff {R : Type u} [Semiring R] {S : Type u_1} [Semiring S] {f g : Polynomial R \u2192+* S} :f = g \u2194 f.comp C = g.comp C \u2227 f X = g X","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Monomial.lean#L74-L74"},"Polynomial.natSepDegree_X_sub_C_pow":{"signature":"theorem Polynomial.natSepDegree_X_sub_C_pow {F : Type u} [Field F] {x : F} {n : \u2115} :((X - C x) ^ n).natSepDegree = if n = 0 then 0 else 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/SeparableDegree.lean#L414-L416"},"Polynomial.continuousWithinAt_aeval":{"signature":"theorem Polynomial.continuousWithinAt_aeval {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] [TopologicalSpace A] [IsTopologicalSemiring A] (p : Polynomial R) {s : Set A} {a : A} :ContinuousWithinAt (fun (x : A) => (aeval x) p) s a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Topology/Algebra/Polynomial.lean#L87-L90"},"Polynomial.natDegree_hermite":{"signature":"theorem Polynomial.natDegree_hermite {n : \u2115} :(hermite n).natDegree = n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Hermite/Basic.lean#L108-L110"},"Polynomial.X_mem_nonzeroDivisors":{"signature":"theorem Polynomial.X_mem_nonzeroDivisors {R : Type u} [CommSemiring R] :X \u2208 nonZeroDivisors (Polynomial R)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L766-L767"},"Polynomial.logMahlerMeasure_C_mul":{"signature":"theorem Polynomial.logMahlerMeasure_C_mul {a : \u2102} (ha : a \u2260 0) {p : Polynomial \u2102} (hp : p \u2260 0) :(C a * p).logMahlerMeasure = Real.log \u2016a\u2016 + p.logMahlerMeasure","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/MahlerMeasure.lean#L153-L155"},"Polynomial.logMahlerMeasure_X_add_C":{"signature":"theorem Polynomial.logMahlerMeasure_X_add_C (z : \u2102) :(X + C z).logMahlerMeasure = \u2016z\u2016.posLog","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/MahlerMeasure.lean#L163-L165"},"Polynomial.logMahlerMeasure_C_mul_X_add_C":{"signature":"theorem Polynomial.logMahlerMeasure_C_mul_X_add_C {a : \u2102} (ha : a \u2260 0) (b : \u2102) :(C a * X + C b).logMahlerMeasure = Real.log \u2016a\u2016 + \u2016a\u207b\u00b9 * b\u2016.posLog","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/MahlerMeasure.lean#L167-L170"},"Polynomial.ringKrullDim_le":{"signature":"theorem Polynomial.ringKrullDim_le {R : Type u_1} [CommRing R] :ringKrullDim (Polynomial R) \u2264 2 * ringKrullDim R + 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/KrullDimension/Polynomial.lean#L31-L40"},"Polynomial.disc_C":{"signature":"theorem Polynomial.disc_C {R : Type u_1} [CommRing R] (r : R) :(C r).discr = 1","doc":"Alias of Polynomial.discr_C. The discriminant of a constant polynomial is 1.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L974-L974"},"Polynomial.opRingEquiv_symm_X":{"signature":"theorem Polynomial.opRingEquiv_symm_X {R : Type u_1} [Semiring R] :(opRingEquiv R).symm X = MulOpposite.op X","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Opposites.lean#L71-L73"},"Polynomial.int_natDegree":{"signature":"theorem Polynomial.int_natDegree {K : Type u_1} [Field K] (R : Subring K) (P : Polynomial K) (hP : \u2200 (n : \u2115), P.coeff n \u2208 R) :(int R P hP).natDegree = P.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Ring/Subring/IntPolynomial.lean#L52-L53"},"Polynomial.isIntegral_iff_isIntegral_coeff":{"signature":"theorem Polynomial.isIntegral_iff_isIntegral_coeff {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [Algebra R S] {f : Polynomial S} :IsIntegral (Polynomial R) f \u2194 \u2200 (n : \u2115), IsIntegral R (f.coeff n)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/IsIntegral.lean#L163-L168"},"Polynomial.degree_add_eq_of_leadingCoeff_add_ne_zero":{"signature":"theorem Polynomial.degree_add_eq_of_leadingCoeff_add_ne_zero {R : Type u} [Semiring R] {p q : Polynomial R} (h : p.leadingCoeff + q.leadingCoeff \u2260 0) :(p + q).degree = max p.degree q.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L214-L228"},"Polynomial.hilbertPoly_zero_left":{"signature":"theorem Polynomial.hilbertPoly_zero_left {F : Type u_1} [Field F] (d : \u2115) :hilbertPoly 0 d = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/HilbertPoly.lean#L114-L117"},"Polynomial.aeval_algEquiv":{"signature":"theorem Polynomial.aeval_algEquiv {R : Type u} {A : Type z} {B : Type u_2} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A \u2243\u2090[R] B) (x : A) :aeval (f x) = (\u2191f).comp (aeval x)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L412-L413"},"Polynomial.ofFn_coeff_eq_val_of_lt":{"signature":"theorem Polynomial.ofFn_coeff_eq_val_of_lt {R : Type u_1} [Semiring R] [DecidableEq R] {n i : \u2115} (v : Fin n \u2192 R) (hi : i < n) :((ofFn n) v).coeff i = v \u27e8i, hi\u27e9","doc":"If i < n the i-th coefficient of ofFn n v is v i.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/OfFn.lean#L67-L71"},"Polynomial.ofFn_comp_toFn_eq_id_of_natDegree_lt":{"signature":"theorem Polynomial.ofFn_comp_toFn_eq_id_of_natDegree_lt {R : Type u_1} [Semiring R] [DecidableEq R] {n : \u2115} {p : Polynomial R} (h_deg : p.natDegree < n) :(ofFn n) ((toFn n) p) = p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/OfFn.lean#L110-L116"},"Polynomial.card_eq_of_natDegree_le_of_coeff_le":{"signature":"theorem Polynomial.card_eq_of_natDegree_le_of_coeff_le {n : \u2115} {B\u2081 B\u2082 : Fin (n + 1) \u2192 \u211d} :{p : Polynomial \u2124 | p.natDegree \u2264 n \u2227 \u2200 (i : Fin (n + 1)), B\u2081 i \u2264 \u2191(p.coeff \u2191i) \u2227 \u2191(p.coeff \u2191i) \u2264 B\u2082 i}.ncard = \u220f i : Fin (n + 1), (\u230aB\u2082 i\u230b - \u2308B\u2081 i\u2309 + 1).toNat","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/NumberTheory/MahlerMeasure.lean#L57-L74"},"Polynomial.sup_ker_aeval_le_ker_aeval_mul":{"signature":"theorem Polynomial.sup_ker_aeval_le_ker_aeval_mul {R : Type u} {M : Type w} [CommRing R] [AddCommGroup M] [Module R M] {f : M \u2192\u2097[R] M} {p q : Polynomial R} :((aeval f) p).ker \u2294 ((aeval f) q).ker \u2264 ((aeval f) (p * q)).ker","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Basic.lean#L929-L937"},"Polynomial.splits_of_exists_multiset":{"signature":"theorem Polynomial.splits_of_exists_multiset {R : Type u_1} [CommRing R] {f : Polynomial R} :f.Splits \u2194 \u2203 (m : Multiset R), f = C f.leadingCoeff * (Multiset.map (fun (x : R) => X - C x) m).prod","doc":"Alias of Polynomial.splits_iff_exists_multiset.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L840-L841"},"Polynomial.dickson_two_one_eq_chebyshev_U":{"signature":"theorem Polynomial.dickson_two_one_eq_chebyshev_U (R : Type u_1) [CommRing R] [Invertible 2] (n : \u2115) :dickson 2 1 n = (Chebyshev.U R \u2191n).comp (C \u215f2 * X)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Dickson.lean#L173-L175"},"Polynomial.mul_mem_nthRootsFinset":{"signature":"theorem Polynomial.mul_mem_nthRootsFinset {R : Type u} {n : \u2115} [CommRing R] [IsDomain R] {\u03b7\u2081 \u03b7\u2082 a\u2081 a\u2082 : R} (h\u03b7\u2081 : \u03b7\u2081 \u2208 nthRootsFinset n a\u2081) (h\u03b7\u2082 : \u03b7\u2082 \u2208 nthRootsFinset n a\u2082) :\u03b7\u2081 * \u03b7\u2082 \u2208 nthRootsFinset n (a\u2081 * a\u2082)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L393-L401"},"Polynomial.exists_root_of_degree_eq_one":{"signature":"theorem Polynomial.exists_root_of_degree_eq_one {R : Type u} [Field R] {p : Polynomial R} (h : p.degree = 1) :\u2203 (x : R), p.IsRoot x","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L533-L536"},"Polynomial.toFinsupp_eq_zero":{"signature":"theorem Polynomial.toFinsupp_eq_zero {R : Type u} [Semiring R] {a : Polynomial R} :a.toFinsupp = 0 \u2194 a = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L231-L233"},"Polynomial.splits_of_comp":{"signature":"theorem Polynomial.splits_of_comp {R : Type u_1} [CommRing R] {f : Polynomial R} {S : Type u_4} [CommRing S] [IsDomain S] [IsSimpleRing R] (i : R \u2192+* S) (hf : (map i f).Splits) (hi : \u2200 a \u2208 (map i f).roots, a \u2208 i.range) :f.Splits","doc":"Alias of Polynomial.Splits.of_splits_map.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L848-L849"},"Polynomial.coeff_toSubring'":{"signature":"theorem Polynomial.coeff_toSubring' {R : Type u} [Ring R] (p : Polynomial R) (T : Subring R) (hp : \u2191p.coeffs \u2286 \u2191T) {n : \u2115} :\u2191((p.toSubring T hp).coeff n) = p.coeff n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Basic.lean#L383-L384"},"Polynomial.toSubring_one":{"signature":"theorem Polynomial.toSubring_one {R : Type u} [Ring R] (T : Subring R) :toSubring 1 T \u22ef = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Basic.lean#L409-L416"},"Polynomial.expand_pow":{"signature":"theorem Polynomial.expand_pow {R : Type u} [CommSemiring R] (p q : \u2115) (f : Polynomial R) :(expand R (p ^ q)) f = (\u21d1(expand R p))^[q] f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Expand.lean#L82-L84"},"Polynomial.isFixedPt_newtonMap_of_aeval_eq_zero":{"signature":"theorem Polynomial.isFixedPt_newtonMap_of_aeval_eq_zero {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [Algebra R S] {P : Polynomial R} {x : S} (h : (aeval x) P = 0) :Function.IsFixedPt P.newtonMap x","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Dynamics/Newton.lean#L74-L76"},"Polynomial.ofFinsupp_nsmul":{"signature":"theorem Polynomial.ofFinsupp_nsmul {R : Type u} [Semiring R] (a : \u2115) (b : AddMonoidAlgebra R \u2115) :{ toFinsupp := a \u2022 b } = a \u2022 { toFinsupp := b }","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L164-L166"},"Polynomial.rootMultiplicity_le_rootMultiplicity_of_dvd":{"signature":"theorem Polynomial.rootMultiplicity_le_rootMultiplicity_of_dvd {R : Type u} [CommRing R] {p q : Polynomial R} (hq : q \u2260 0) (hpq : p \u2223 q) (x : R) :rootMultiplicity x p \u2264 rootMultiplicity x q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L741-L744"},"Polynomial.lt_rootMultiplicity_iff_isRoot_iterate_derivative_of_mem_nonZeroDivisors'":{"signature":"theorem Polynomial.lt_rootMultiplicity_iff_isRoot_iterate_derivative_of_mem_nonZeroDivisors' {R : Type u} [CommRing R] {p : Polynomial R} {t : R} {n : \u2115} (h : p \u2260 0) (hnzd : \u2200 m \u2264 n, m \u2260 0 \u2192 \u2191m \u2208 nonZeroDivisors R) :n < rootMultiplicity t p \u2194 \u2200 m \u2264 n, ((\u21d1derivative)^[m] p).IsRoot t","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L120-L125"},"Polynomial.iterate_derivative_neg":{"signature":"theorem Polynomial.iterate_derivative_neg {R : Type u} [Ring R] {f : Polynomial R} {k : \u2115} :(\u21d1derivative)^[k] (-f) = -(\u21d1derivative)^[k] f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L612-L613"},"Polynomial.eq_zero_of_hasseDeriv_eq_zero":{"signature":"theorem Polynomial.eq_zero_of_hasseDeriv_eq_zero {R : Type u_1} [Semiring R] (f : Polynomial R) (r : R) (h : \u2200 (k : \u2115), eval r ((hasseDeriv k) f) = 0) :f = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Taylor.lean#L114-L118"},"Polynomial.hasDerivWithinAt":{"signature":"theorem Polynomial.hasDerivWithinAt {\ud835\udd5c : Type u} [NontriviallyNormedField \ud835\udd5c] (p : Polynomial \ud835\udd5c) (x : \ud835\udd5c) (s : Set \ud835\udd5c) :HasDerivWithinAt (fun (x : \ud835\udd5c) => eval x p) (eval x (derivative p)) s x","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Calculus/Deriv/Polynomial.lean#L73-L75"},"Polynomial.degree_C_mul_of_isUnit":{"signature":"theorem Polynomial.degree_C_mul_of_isUnit {R : Type u} {a : R} [Semiring R] (ha : IsUnit a) (p : Polynomial R) :(C a * p).degree = p.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L382-L388"},"Polynomial.natDegree_C_mul_of_isUnit":{"signature":"theorem Polynomial.natDegree_C_mul_of_isUnit {R : Type u} {a : R} [Semiring R] (ha : IsUnit a) (p : Polynomial R) :(C a * p).natDegree = p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L398-L399"},"Polynomial.leadingCoeff_C_mul_of_isUnit":{"signature":"theorem Polynomial.leadingCoeff_C_mul_of_isUnit {R : Type u} {a : R} [Semiring R] (ha : IsUnit a) (p : Polynomial R) :(C a * p).leadingCoeff = a * p.leadingCoeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L404-L406"},"Polynomial.toFinsupp_natCast":{"signature":"theorem Polynomial.toFinsupp_natCast {R : Type u} [Semiring R] (n : \u2115) :(\u2191n).toFinsupp = \u2191n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L258-L259"},"Polynomial.derivWithin_aeval":{"signature":"theorem Polynomial.derivWithin_aeval {\ud835\udd5c : Type u} [NontriviallyNormedField \ud835\udd5c] {x : \ud835\udd5c} {s : Set \ud835\udd5c} {R : Type u_1} [CommSemiring R] [Algebra R \ud835\udd5c] (q : Polynomial R) (hxs : UniqueDiffWithinAt \ud835\udd5c s x) :derivWithin (fun (x : \ud835\udd5c) => (aeval x) q) s x = (aeval x) (derivative q)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Calculus/Deriv/Polynomial.lean#L119-L122"},"Polynomial.trailingDegree_le_trailingDegree":{"signature":"theorem Polynomial.trailingDegree_le_trailingDegree {R : Type u} [Semiring R] {p q : Polynomial R} (h : q.coeff p.natTrailingDegree \u2260 0) :q.trailingDegree \u2264 p.trailingDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L150-L152"},"Polynomial.natDegree_monomial_le":{"signature":"theorem Polynomial.natDegree_monomial_le {R : Type u} [Semiring R] (a : R) {m : \u2115} :((monomial m) a).natDegree \u2264 m","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L227-L231"},"Polynomial.eval\u2082AddMonoidHom_apply":{"signature":"theorem Polynomial.eval\u2082AddMonoidHom_apply {R : Type u} {S : Type v} [Semiring R] [Semiring S] (f : R \u2192+* S) (x : S) (p : Polynomial R) :(eval\u2082AddMonoidHom f x) p = eval\u2082 f x p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L96-L96"},"Polynomial.mapAlgEquiv_id":{"signature":"theorem Polynomial.mapAlgEquiv_id {R : Type u} {A : Type z} [CommSemiring R] [Semiring A] [Algebra R A] :mapAlgEquiv AlgEquiv.refl = AlgEquiv.refl","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L213-L215"},"Polynomial.X_pow_sub_one_dvd_prod_cyclotomic":{"signature":"theorem Polynomial.X_pow_sub_one_dvd_prod_cyclotomic (R : Type u_1) [CommRing R] {n m : \u2115} (hpos : 0 < n) (hm : m \u2223 n) (hdiff : m \u2260 n) :X ^ m - 1 \u2223 \u220f i \u2208 n.properDivisors, cyclotomic i R","doc":"If m is a proper divisor of n, then X ^ m - 1 divides \u220f i \u2208 Nat.properDivisors n, cyclotomic i R.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean#L461-L471"},"Polynomial.instSMulCommClassElemRootSet":{"signature":"instance Polynomial.instSMulCommClassElemRootSet {R : Type u} {S : Type v} [CommRing R] [IsDomain R] [CommRing S] [Algebra S R] (G : Type u_1) (H : Type u_2) [Monoid G] [MulSemiringAction G R] [SMulCommClass G S R] [Monoid H] [MulSemiringAction H R] [SMulCommClass H S R] [SMulCommClass G H R] (f : Polynomial S) :SMulCommClass G H \u2191(f.rootSet R)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L676-L680"},"Polynomial.evalEval_dvd":{"signature":"theorem Polynomial.evalEval_dvd {R : Type u_1} [CommSemiring R] (x y : R) {p q : Polynomial (Polynomial R)} :p \u2223 q \u2192 evalEval x y p \u2223 evalEval x y q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Bivariate.lean#L139-L140"},"Polynomial.algebraMap_hahnSeries_apply":{"signature":"theorem Polynomial.algebraMap_hahnSeries_apply (\u0393 : Type u_1) {R : Type u_2} [CommSemiring R] [Semiring \u0393] [PartialOrder \u0393] [IsStrictOrderedRing \u0393] (f : Polynomial R) :(algebraMap (Polynomial R) (HahnSeries \u0393 R)) f = (HahnSeries.ofPowerSeries \u0393 R) \u2191f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/HahnSeries/PowerSeries.lean#L227-L230"},"Polynomial.aeval_eq_prod_aroots_sub_of_monic_of_splits":{"signature":"theorem Polynomial.aeval_eq_prod_aroots_sub_of_monic_of_splits {R : Type u_1} [CommRing R] {f : Polynomial R} {A : Type u_2} [CommRing A] [IsDomain A] [Algebra R A] (hf : (map (algebraMap R A) f).Splits) (hm : f.Monic) (x : A) :(aeval x) f = (Multiset.map (fun (x_1 : A) => x - x_1) (f.aroots A)).prod","doc":"Alias of Polynomial.Splits.aeval_eq_prod_aroots_of_monic.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L814-L815"},"Polynomial.mapAlgHom_monomial":{"signature":"theorem Polynomial.mapAlgHom_monomial {R : Type u} {A : Type z} {B : Type u_2} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A \u2192\u2090[R] B) (n : \u2115) (a : A) :(mapAlgHom f) ((monomial n) a) = (monomial n) (f a)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L202-L203"},"Polynomial.isRoot_of_aeval_algebraMap_eq_zero":{"signature":"theorem Polynomial.isRoot_of_aeval_algebraMap_eq_zero {R : Type u} {S : Type v} [CommSemiring R] [Semiring S] [Algebra R S] [FaithfulSMul R S] {p : Polynomial R} {r : R} (hr : (aeval ((algebraMap R S) r)) p = 0) :p.IsRoot r","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L508-L510"},"Polynomial.discr_of_degree_eq_three":{"signature":"theorem Polynomial.discr_of_degree_eq_three {R : Type u_1} [CommRing R] {f : Polynomial R} (hf : f.degree = 3) :f.discr = f.coeff 2 ^ 2 * f.coeff 1 ^ 2 - 4 * f.coeff 3 * f.coeff 1 ^ 3 - 4 * f.coeff 2 ^ 3 * f.coeff 0 - 27 * f.coeff 3 ^ 2 * f.coeff 0 ^ 2 + 18 * f.coeff 3 * f.coeff 2 * f.coeff 1 * f.coeff 0","doc":"Standard formula for the discriminant of a cubic polynomial.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L1018-L1030"},"Polynomial.disc_of_degree_eq_three":{"signature":"theorem Polynomial.disc_of_degree_eq_three {R : Type u_1} [CommRing R] {f : Polynomial R} (hf : f.degree = 3) :f.discr = f.coeff 2 ^ 2 * f.coeff 1 ^ 2 - 4 * f.coeff 3 * f.coeff 1 ^ 3 - 4 * f.coeff 2 ^ 3 * f.coeff 0 - 27 * f.coeff 3 ^ 2 * f.coeff 0 ^ 2 + 18 * f.coeff 3 * f.coeff 2 * f.coeff 1 * f.coeff 0","doc":"Alias of Polynomial.discr_of_degree_eq_three. Standard formula for the discriminant of a cubic polynomial.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L1032-L1032"},"Polynomial.natTrailingDegree_one":{"signature":"theorem Polynomial.natTrailingDegree_one {R : Type u} [Semiring R] :natTrailingDegree 1 = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L201-L203"},"Polynomial.dvd_comp_X_sub_C_iff":{"signature":"theorem Polynomial.dvd_comp_X_sub_C_iff {R : Type u} [CommRing R] (p q : Polynomial R) (a : R) :p \u2223 q.comp (X - C a) \u2194 p.comp (X + C a) \u2223 q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L667-L670"},"Polynomial.height_eq_height_add_one":{"signature":"theorem Polynomial.height_eq_height_add_one {R : Type u_1} [CommRing R] [IsNoetherianRing R] (p : Ideal R) (P : Ideal (Polynomial R)) [P.IsMaximal] [P.LiesOver p] :P.height = p.height + 1","doc":"Let p be a prime ideal of R. If P is a maximal ideal of R[X] lying over p, ht(P) = ht(p) + 1.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/KrullDimension/Polynomial.lean#L75-L100"},"Polynomial.ringKrullDim_of_isNoetherianRing":{"signature":"theorem Polynomial.ringKrullDim_of_isNoetherianRing {R : Type u_1} [CommRing R] [IsNoetherianRing R] :ringKrullDim (Polynomial R) = ringKrullDim R + 1","doc":"If R is Noetherian, dim R[X] = dim R + 1.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/KrullDimension/Polynomial.lean#L102-L111"},"Polynomial.coeff_mul_of_natDegree_le":{"signature":"theorem Polynomial.coeff_mul_of_natDegree_le {R : Type u} [Semiring R] {df dg : \u2115} {f g : Polynomial R} (hdf : f.natDegree \u2264 df) (hdg : g.natDegree \u2264 dg) :(f * g).coeff (df + dg) = f.coeff df * g.coeff dg","doc":"Alias of Polynomial.coeff_mul_add_eq_of_natDegree_le.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L151-L152"},"Polynomial.splits_of_degree_eq_one":{"signature":"theorem Polynomial.splits_of_degree_eq_one {R : Type u_1} [DivisionSemiring R] {f : Polynomial R} (hf : f.degree = 1) :f.Splits","doc":"Alias of Polynomial.Splits.of_degree_eq_one.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L669-L670"},"Polynomial.isRoot_cyclotomic_iff_charZero":{"signature":"theorem Polynomial.isRoot_cyclotomic_iff_charZero {n : \u2115} {R : Type u_2} [CommRing R] [IsDomain R] [CharZero R] {\u03bc : R} (hn : 0 < n) :(cyclotomic n R).IsRoot \u03bc \u2194 IsPrimitiveRoot \u03bc n","doc":"If R is of characteristic zero, then \u03b6 is a root of cyclotomic n R if and only if it is a primitive n-th root of unity.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Roots.lean#L123-L128"},"Polynomial.splits_of_natDegree_le_one_of_monic":{"signature":"theorem Polynomial.splits_of_natDegree_le_one_of_monic {R : Type u_1} [Semiring R] {f : Polynomial R} (hf : f.natDegree \u2264 1) (h : f.Monic) :f.Splits","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L115-L117"},"Polynomial.natDegree_cyclotomic_le":{"signature":"theorem Polynomial.natDegree_cyclotomic_le {R : Type u_1} [Ring R] {n : \u2115} :(cyclotomic n R).natDegree \u2264 n.totient","doc":"The natural degree of cyclotomic n is at most totient n. If the base ring is nontrivial, then the degree is exactly \u03c6 n, otherwise it's zero.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean#L321-L328"},"Polynomial.map_div":{"signature":"theorem Polynomial.map_div {R : Type u} {k : Type y} [Field R] {p q : Polynomial R} [Field k] (f : R \u2192+* k) :map f (p / q) = map f p / map f q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L410-L414"},"Polynomial.natSepDegree_C_mul":{"signature":"theorem Polynomial.natSepDegree_C_mul {F : Type u} [Field F] (f : Polynomial F) {x : F} (hx : x \u2260 0) :(C x * f).natSepDegree = f.natSepDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/SeparableDegree.lean#L388-L392"},"Polynomial.natSepDegree_C_mul_X_sub_C_pow":{"signature":"theorem Polynomial.natSepDegree_C_mul_X_sub_C_pow {F : Type u} [Field F] {x y : F} {n : \u2115} (hx : x \u2260 0) :(C x * (X - C y) ^ n).natSepDegree = if n = 0 then 0 else 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/SeparableDegree.lean#L418-L420"},"Polynomial.roots_ne_zero_of_splits'":{"signature":"theorem Polynomial.roots_ne_zero_of_splits' {R : Type u_1} [CommRing R] {f : Polynomial R} [IsDomain R] (hf : f.Splits) (hf0 : f.natDegree \u2260 0) :f.roots \u2260 0","doc":"Alias of Polynomial.Splits.roots_ne_zero.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L722-L723"},"Polynomial.coe_smul":{"signature":"theorem Polynomial.coe_smul {R : Type u_1} [Semiring R] (\u03c6 : Polynomial R) (r : R) :\u2191(r \u2022 \u03c6) = r \u2022 \u2191\u03c6","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/PowerSeries/Basic.lean#L837-L839"},"Polynomial.separable_X_pow_sub_C'":{"signature":"theorem Polynomial.separable_X_pow_sub_C' {F : Type u} [Field F] (p n : \u2115) (a : F) [CharP F p] (hn : \u00acp \u2223 n) (ha : a \u2260 0) :(X ^ n - C a).Separable","doc":"If F is of characteristic p and p \u2224 n, then X ^ n - a is separable for any a \u2260 0.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Separable.lean#L433-L436"},"Polynomial.isUnit_of_isUnit_leadingCoeff_of_isUnit_map":{"signature":"theorem Polynomial.isUnit_of_isUnit_leadingCoeff_of_isUnit_map {R : Type u} {S : Type v} [Semiring R] [CommRing S] [IsDomain S] (\u03c6 : R \u2192+* S) {f : Polynomial R} (hf : IsUnit f.leadingCoeff) (H : IsUnit (map \u03c6 f)) :IsUnit f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Degree.lean#L235-L250"},"Polynomial.associated_of_dvd_of_degree_eq":{"signature":"theorem Polynomial.associated_of_dvd_of_degree_eq {K : Type u_1} [Field K] {p q : Polynomial K} (hpq : p \u2223 q) (h\u2081 : p.degree = q.degree) :Associated p q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L832-L835"},"Polynomial.resultant_X_pow_left":{"signature":"theorem Polynomial.resultant_X_pow_left {R : Type u_1} [CommRing R] (g : Polynomial R) (m n : \u2115) (hn : g.natDegree \u2264 n) :(X ^ m).resultant g m n = g.coeff 0 ^ m","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L683-L685"},"Polynomial.eq_prod_roots_of_splits":{"signature":"theorem Polynomial.eq_prod_roots_of_splits {R : Type u_1} [CommRing R] {f : Polynomial R} [IsDomain R] (hf : f.Splits) :f = C f.leadingCoeff * (Multiset.map (fun (x : R) => X - C x) f.roots).prod","doc":"Alias of Polynomial.Splits.eq_prod_roots.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L799-L800"},"Polynomial.natCast_coeff_zero":{"signature":"theorem Polynomial.natCast_coeff_zero {n : \u2115} {R : Type u_1} [Semiring R] :(\u2191n).coeff 0 = \u2191n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L342-L343"},"Polynomial.map_restriction":{"signature":"theorem Polynomial.map_restriction {R : Type u} [CommRing R] (p : Polynomial R) :map (algebraMap (\u21a5(Subring.closure \u2191p.coeffs)) R) p.restriction = p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Basic.lean#L324-L327"},"Polynomial.toFinsuppIso_apply":{"signature":"theorem Polynomial.toFinsuppIso_apply (R : Type u) [Semiring R] (self : Polynomial R) :(toFinsuppIso R) self = self.toFinsupp","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L327-L327"},"Polynomial.evalEval_sum":{"signature":"theorem Polynomial.evalEval_sum {R : Type u_1} [Semiring R] (x y : R) (p : Polynomial R) (f : \u2115 \u2192 R \u2192 Polynomial (Polynomial R)) :evalEval x y (p.sum f) = p.sum fun (n : \u2115) (a : R) => evalEval x y (f n a)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Bivariate.lean#L77-L79"},"Polynomial.eq_centerMass_of_eval_derivative_eq_zero":{"signature":"theorem Polynomial.eq_centerMass_of_eval_derivative_eq_zero {P : Polynomial \u2102} {z : \u2102} (hP : 0 < P.degree) (hz : eval z (derivative P) = 0) :z = P.roots.toFinset.centerMass (P.derivRootWeight z) id","doc":"Gauss-Lucas Theorem: if $P$ is a nonconstant polynomial with complex coefficients, then all zeros of $P'$ belong to the convex hull of the set of zeros of $P$. This version provides explicit formulas for the coefficients of the convex combination. See also rootSet_derivative_subset_convexHull_rootSet below.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Complex/Polynomial/GaussLucas.lean#L54-L89"},"Polynomial.rootSet_derivative_subset_convexHull_rootSet":{"signature":"theorem Polynomial.rootSet_derivative_subset_convexHull_rootSet {P : Polynomial \u2102} (h\u2080 : 0 < P.degree) :(derivative P).rootSet \u2102 \u2286 (convexHull \u211d) (P.rootSet \u2102)","doc":"Gauss-Lucas Theorem: if $P$ is a nonconstant polynomial with complex coefficients, then all zeros of $P'$ belong to the convex hull of the set of zeros of $P$. See also eq_centerMass_of_eval_derivative_eq_zero for a version that provides explicit coefficients of the convex combination.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Complex/Polynomial/GaussLucas.lean#L91-L105"},"Polynomial.splits_of_splits_id":{"signature":"theorem Polynomial.splits_of_splits_id {R : Type u_1} [Semiring R] {f : Polynomial R} (hf : f.Splits) {S : Type u_2} [Semiring S] (i : R \u2192+* S) :(map i f).Splits","doc":"Alias of Polynomial.Splits.map.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L843-L844"},"Polynomial.ker_constantCoeff":{"signature":"theorem Polynomial.ker_constantCoeff {R : Type u_1} [CommRing R] :RingHom.ker constantCoeff = Ideal.span {X}","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Ideal.lean#L49-L53"},"Polynomial.homogenize_map":{"signature":"theorem Polynomial.homogenize_map {R : Type u_1} [CommSemiring R] {S : Type u_2} [CommSemiring S] (f : R \u2192+* S) (p : Polynomial R) (n : \u2115) :(map f p).homogenize n = (MvPolynomial.map f) (p.homogenize n)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Homogenize.lean#L69-L71"},"Polynomial.factorial_mul_shiftedLegendre_eq":{"signature":"theorem Polynomial.factorial_mul_shiftedLegendre_eq (n : \u2115) :\u2191n.factorial * shiftedLegendre n = (\u21d1derivative)^[n] (X ^ n * (1 - X) ^ n)","doc":"The shifted Legendre polynomial multiplied by a factorial equals the higher-order derivative of the combinatorial function X ^ n * (1 - X) ^ n. This is the analogue of Rodrigues' formula for the shifted Legendre polynomials.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ShiftedLegendre.lean#L43-L80"},"Polynomial.neg_one_pow_mul_shiftedLegendre_comp_one_sub_X_eq":{"signature":"theorem Polynomial.neg_one_pow_mul_shiftedLegendre_comp_one_sub_X_eq (n : \u2115) :(-1) ^ n * (shiftedLegendre n).comp (1 - X) = shiftedLegendre n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ShiftedLegendre.lean#L102-L107"},"Polynomial.shiftedLegendre_eval_symm":{"signature":"theorem Polynomial.shiftedLegendre_eval_symm (n : \u2115) {R : Type u_1} [Ring R] (x : R) :(aeval x) (shiftedLegendre n) = (-1) ^ n * (aeval (1 - x)) (shiftedLegendre n)","doc":"The values \u200b\u200bof the Legendre polynomial at x and 1 - x differ by a factor (-1)\u207f.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ShiftedLegendre.lean#L109-L113"},"Polynomial.nodup_of_separable_prod":{"signature":"theorem Polynomial.nodup_of_separable_prod {R : Type u} [CommRing R] [Nontrivial R] {s : Multiset R} (hs : (Multiset.map (fun (a : R) => X - C a) s).prod.Separable) :s.Nodup","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Separable.lean#L239-L244"},"Polynomial.exists_finset_of_splits":{"signature":"theorem Polynomial.exists_finset_of_splits {F : Type u} [Field F] {K : Type v} [Field K] (i : F \u2192+* K) {f : Polynomial F} (sep : f.Separable) (sp : (map i f).Splits) :\u2203 (s : Finset K), map i f = C (i f.leadingCoeff) * \u220f a \u2208 s, (X - C a)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Separable.lean#L511-L521"},"Polynomial.roots_quadratic_eq_pair_iff_of_ne_zero'":{"signature":"theorem Polynomial.roots_quadratic_eq_pair_iff_of_ne_zero' {R : Type u_1} [Field R] {a b c x1 x2 : R} (ha : a \u2260 0) :(C a * X ^ 2 + C b * X + C c).roots = {x1, x2} \u2194 x1 + x2 = -b / a \u2227 x1 * x2 = c / a","doc":"Vieta's formula for quadratics as an iff (Field version).","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/SmallDegreeVieta.lean#L95-L102"},"Polynomial.aroots_quadratic_eq_pair_iff_of_ne_zero'":{"signature":"theorem Polynomial.aroots_quadratic_eq_pair_iff_of_ne_zero' {T : Type u_2} {S : Type u_3} [CommRing T] [Field S] [Algebra T S] {a b c : T} {x1 x2 : S} (ha : (algebraMap T S) a \u2260 0) :(C a * X ^ 2 + C b * X + C c).aroots S = {x1, x2} \u2194 x1 + x2 = -(algebraMap T S) b / (algebraMap T S) a \u2227 x1 * x2 = (algebraMap T S) c / (algebraMap T S) a","doc":"Vieta's formula for quadratics as an iff (aroots, Field version).","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/SmallDegreeVieta.lean#L104-L112"},"Polynomial.abs_tendsto_atTop":{"signature":"theorem Polynomial.abs_tendsto_atTop {\ud835\udd5c : Type u_1} [NormedField \ud835\udd5c] [LinearOrder \ud835\udd5c] [IsStrictOrderedRing \ud835\udd5c] (P : Polynomial \ud835\udd5c) [OrderTopology \ud835\udd5c] (hdeg : 0 < P.degree) :Filter.Tendsto (fun (x : \ud835\udd5c) => |eval x P|) Filter.atTop Filter.atTop","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/Basic.lean#L80-L84"},"Polynomial.abs_isBoundedUnder_iff":{"signature":"theorem Polynomial.abs_isBoundedUnder_iff {\ud835\udd5c : Type u_1} [NormedField \ud835\udd5c] [LinearOrder \ud835\udd5c] [IsStrictOrderedRing \ud835\udd5c] (P : Polynomial \ud835\udd5c) [OrderTopology \ud835\udd5c] :(Filter.IsBoundedUnder (fun (x1 x2 : \ud835\udd5c) => x1 \u2264 x2) Filter.atTop fun (x : \ud835\udd5c) => |eval x P|) \u2194 P.degree \u2264 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/Basic.lean#L86-L92"},"Polynomial.trailingDegree_one_le":{"signature":"theorem Polynomial.trailingDegree_one_le {R : Type u} [Semiring R] :0 \u2264 trailingDegree 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L193-L195"},"Polynomial.le_natTrailingDegree_mul":{"signature":"theorem Polynomial.le_natTrailingDegree_mul {R : Type u} [Semiring R] {p q : Polynomial R} (h : p * q \u2260 0) :p.natTrailingDegree + q.natTrailingDegree \u2264 (p * q).natTrailingDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L298-L306"},"Polynomial.mirror_C":{"signature":"theorem Polynomial.mirror_C {R : Type u_1} [Semiring R] (a : R) :(C a).mirror = C a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Mirror.lean#L53-L54"},"Polynomial.sum_range_pow_eq_bernoulli_sub":{"signature":"theorem Polynomial.sum_range_pow_eq_bernoulli_sub (n p : \u2115) :(\u2191p + 1) * \u2211 k \u2208 Finset.range n, \u2191k ^ p = eval (\u2191n) (bernoulli p.succ) - _root_.bernoulli p.succ","doc":"Another version of sum_range_pow.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/NumberTheory/BernoulliPolynomials.lean#L168-L183"},"Polynomial.bernoulli_succ_eval":{"signature":"theorem Polynomial.bernoulli_succ_eval (n p : \u2115) :eval (\u2191n) (bernoulli p.succ) = _root_.bernoulli p.succ + (\u2191p + 1) * \u2211 k \u2208 Finset.range n, \u2191k ^ p","doc":"Rearrangement of Polynomial.sum_range_pow_eq_bernoulli_sub.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/NumberTheory/BernoulliPolynomials.lean#L185-L189"},"Polynomial.natDegree_sub":{"signature":"theorem Polynomial.natDegree_sub {R : Type u} [Ring R] {p q : Polynomial R} :(p - q).natDegree = (q - p).natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L321-L321"},"Polynomial.natDegree_add_le_iff_left":{"signature":"theorem Polynomial.natDegree_add_le_iff_left {R : Type u} [Semiring R] {n : \u2115} (p q : Polynomial R) (qn : q.natDegree \u2264 n) :(p + q).natDegree \u2264 n \u2194 p.natDegree \u2264 n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L82-L87"},"Polynomial.natDegree_sub_le_iff_left":{"signature":"theorem Polynomial.natDegree_sub_le_iff_left {R : Type u} {n : \u2115} [Ring R] {p q : Polynomial R} (qn : q.natDegree \u2264 n) :(p - q).natDegree \u2264 n \u2194 p.natDegree \u2264 n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L323-L326"},"Polynomial.natDegree_sub_le_iff_right":{"signature":"theorem Polynomial.natDegree_sub_le_iff_right {R : Type u} {n : \u2115} [Ring R] {p q : Polynomial R} (pn : p.natDegree \u2264 n) :(p - q).natDegree \u2264 n \u2194 q.natDegree \u2264 n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L328-L329"},"Polynomial.sub_one_pow_totient_le_cyclotomic_eval":{"signature":"theorem Polynomial.sub_one_pow_totient_le_cyclotomic_eval {q : \u211d} (hq' : 1 < q) (n : \u2115) :(q - 1) ^ n.totient \u2264 eval q (cyclotomic n \u211d)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Eval.lean#L219-L223"},"Polynomial.toLaurent_apply":{"signature":"theorem Polynomial.toLaurent_apply {R : Type u_1} [Semiring R] (p : Polynomial R) :toLaurent p = AddMonoidAlgebra.mapDomain Nat.cast p.toFinsupp","doc":"This is not a simp lemma, as it is usually preferable to use the lemmas about C and X instead.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Laurent.lean#L101-L105"},"Polynomial.surjective_toFn":{"signature":"theorem Polynomial.surjective_toFn {R : Type u_1} [Semiring R] (n : \u2115) :Function.Surjective \u21d1(toFn n)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/OfFn.lean#L106-L108"},"Polynomial.nextCoeff_ne_zero":{"signature":"theorem Polynomial.nextCoeff_ne_zero {R : Type u} [Semiring R] {p : Polynomial R} :p.nextCoeff \u2260 0 \u2194 p.natDegree \u2260 0 \u2227 p.coeff (p.natDegree - 1) \u2260 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L313-L314"},"Polynomial.ofFn_degree_lt":{"signature":"theorem Polynomial.ofFn_degree_lt {R : Type u_1} [Semiring R] [DecidableEq R] {n : \u2115} (v : Fin n \u2192 R) :((ofFn n) v).degree < \u2191n","doc":"ofFn n v has degree smaller than n.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/OfFn.lean#L83-L89"},"Polynomial.div_C":{"signature":"theorem Polynomial.div_C {R : Type u} {a : R} [Field R] {p : Polynomial R} :p / C a = p * C a\u207b\u00b9","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L572-L573"},"Polynomial.trailingDegree_ne_of_natTrailingDegree_ne":{"signature":"theorem Polynomial.trailingDegree_ne_of_natTrailingDegree_ne {R : Type u} [Semiring R] {p : Polynomial R} {n : \u2115} :p.natTrailingDegree \u2260 n \u2192 p.trailingDegree \u2260 \u2191n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L157-L159"},"Polynomial.toContinuousMapAlgHom_apply":{"signature":"theorem Polynomial.toContinuousMapAlgHom_apply {R : Type u_1} [CommSemiring R] [TopologicalSpace R] [IsTopologicalSemiring R] (p : Polynomial R) :toContinuousMapAlgHom p = p.toContinuousMap","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Topology/ContinuousMap/Polynomial.lean#L91-L91"},"Polynomial.residueFieldMapCAlgEquiv_algebraMap":{"signature":"theorem Polynomial.residueFieldMapCAlgEquiv_algebraMap {R : Type u_1} [CommRing R] (I : Ideal R) [I.IsPrime] (J : Ideal (Polynomial R)) [J.IsPrime] [J.LiesOver I] (hJ : J = Ideal.map C I) (p : Polynomial R) :(residueFieldMapCAlgEquiv I J hJ) ((algebraMap (Polynomial R) J.ResidueField) p) = (algebraMap (Polynomial I.ResidueField) (RatFunc I.ResidueField)) (map (algebraMap R I.ResidueField) p)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/LocalRing/ResidueField/Polynomial.lean#L77-L81"},"Polynomial.hasseDeriv_mul":{"signature":"theorem Polynomial.hasseDeriv_mul {R : Type u_1} [Semiring R] (k : \u2115) (f g : Polynomial R) :(hasseDeriv k) (f * g) = \u2211 ij \u2208 Finset.antidiagonal k, (hasseDeriv ij.1) f * (hasseDeriv ij.2) g","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/HasseDeriv.lean#L206-L239"},"Polynomial.contentIdeal_zero":{"signature":"theorem Polynomial.contentIdeal_zero {R : Type u_1} [Semiring R] :contentIdeal 0 = \u22a5","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ContentIdeal.lean#L56-L58"},"Polynomial.coeff_ofNat_mul":{"signature":"theorem Polynomial.coeff_ofNat_mul {R : Type u} [Semiring R] {p : Polynomial R} {a k : \u2115} [a.AtLeastTwo] :(OfNat.ofNat a * p).coeff k = OfNat.ofNat a * p.coeff k","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L176-L177"},"Polynomial.homogenize_sub":{"signature":"theorem Polynomial.homogenize_sub {R : Type u_1} [CommRing R] (p q : Polynomial R) (n : \u2115) :(p - q).homogenize n = p.homogenize n - q.homogenize n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Homogenize.lean#L201-L204"},"Polynomial.natDegree_integralNormalization":{"signature":"theorem Polynomial.natDegree_integralNormalization {R : Type u} [Semiring R] {p : Polynomial R} :p.integralNormalization.natDegree = p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/IntegralNormalization.lean#L89-L93"},"Polynomial.integralNormalization_mul_C_leadingCoeff":{"signature":"theorem Polynomial.integralNormalization_mul_C_leadingCoeff {R : Type u} [Semiring R] (p : Polynomial R) :p.integralNormalization * C p.leadingCoeff = p.scaleRoots p.leadingCoeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/IntegralNormalization.lean#L115-L126"},"Polynomial.resultant_integralNormalization":{"signature":"theorem Polynomial.resultant_integralNormalization {R : Type u_1} [CommRing R] (f g : Polynomial R) (hg : g.natDegree \u2260 0) :(f.scaleRoots g.leadingCoeff).resultant g.integralNormalization = g.leadingCoeff ^ (f.natDegree * (g.natDegree - 1)) * f.resultant g","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L736-L763"},"Polynomial.modByMonicHom_apply":{"signature":"theorem Polynomial.modByMonicHom_apply {R : Type u} [CommRing R] (q p : Polynomial R) :q.modByMonicHom p = p %\u2098 q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/RingDivision.lean#L65-L65"},"Polynomial.algHom_eval\u2082_algebraMap":{"signature":"theorem Polynomial.algHom_eval\u2082_algebraMap {R : Type u_3} {A : Type u_4} {B : Type u_5} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (p : Polynomial R) (f : A \u2192\u2090[R] B) (a : A) :f (eval\u2082 (algebraMap R A) a p) = eval\u2082 (algebraMap R B) (f a) p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L128-L133"},"Polynomial.subsingleton_isRoot_of_natDegree_eq_one":{"signature":"theorem Polynomial.subsingleton_isRoot_of_natDegree_eq_one {R : Type u} [Semiring R] {p : Polynomial R} [IsLeftCancelMulZero R] (h : p.natDegree = 1) :{x : R | p.IsRoot x}.Subsingleton","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L362-L367"},"Polynomial.aevalTower_toAlgHom":{"signature":"theorem Polynomial.aevalTower_toAlgHom {R : Type u} {S : Type v} {A' : Type u_1} [CommSemiring R] [CommSemiring A'] [CommSemiring S] [Algebra S R] [Algebra S A'] (g : R \u2192\u2090[S] A') (y : A') (x : R) :(aevalTower g y) ((IsScalarTower.toAlgHom S R (Polynomial R)) x) = g x","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L545-L546"},"Polynomial.eval\u2082_derivative_of_splits":{"signature":"theorem Polynomial.eval\u2082_derivative_of_splits {R : Type u_1} [CommRing R] {f : Polynomial R} [IsDomain R] [DecidableEq R] (hf : f.Splits) (x : R) :eval x (derivative f) = f.leadingCoeff * (Multiset.map (fun (a : R) => (Multiset.map (fun (x_1 : R) => x - x_1) (f.roots.erase a)).prod) f.roots).sum","doc":"Alias of Polynomial.Splits.eval_derivative.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L865-L866"},"Polynomial.smul_mem_rootSet_iff":{"signature":"theorem Polynomial.smul_mem_rootSet_iff {R : Type u} {S : Type v} [CommRing R] [IsDomain R] [CommRing S] [Algebra S R] {G : Type u_1} [Group G] [MulSemiringAction G R] [SMulCommClass G S R] {f : Polynomial S} {g : G} {x : R} :g \u2022 x \u2208 f.rootSet R \u2194 x \u2208 f.rootSet R","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L658-L661"},"Polynomial.hermite_one":{"signature":"theorem Polynomial.hermite_one :hermite 1 = X","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Hermite/Basic.lean#L66-L68"},"Polynomial.C_div":{"signature":"theorem Polynomial.C_div {R : Type u} {a b : R} [Field R] :C (a / b) = C a / C b","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L575-L576"},"Polynomial.eval\u2082_C_X_eq_coe":{"signature":"theorem Polynomial.eval\u2082_C_X_eq_coe {R : Type u_1} [Semiring R] {\u03c6 : Polynomial R} :eval\u2082 PowerSeries.C PowerSeries.X \u03c6 = \u2191\u03c6","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/PowerSeries/Basic.lean#L896-L902"},"Polynomial.ofMultiset_injective":{"signature":"theorem Polynomial.ofMultiset_injective (R : Type u) [CommRing R] [IsDomain R] :Function.Injective \u21d1ofMultiset","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L290-L291"},"Polynomial.hilbertPoly_X_pow_succ":{"signature":"theorem Polynomial.hilbertPoly_X_pow_succ {F : Type u_1} [Field F] (d k : \u2115) :(X ^ k).hilbertPoly (d + 1) = preHilbertPoly F d k","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/HilbertPoly.lean#L124-L126"},"Polynomial.repr":{"signature":"instance Polynomial.repr {R : Type u} [Semiring R] [Repr R] [DecidableEq R] :Repr (Polynomial R)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L1204-L1224"},"Polynomial.roots_X_pow":{"signature":"theorem Polynomial.roots_X_pow {R : Type u} [CommRing R] [IsDomain R] (n : \u2115) :(X ^ n).roots = n \u2022 {0}","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L256-L257"},"Polynomial.roots_C_mul_X_pow":{"signature":"theorem Polynomial.roots_C_mul_X_pow {R : Type u} {a : R} [CommRing R] [IsDomain R] (ha : a \u2260 0) (n : \u2115) :(C a * X ^ n).roots = n \u2022 {0}","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L259-L261"},"Polynomial.roots_monomial":{"signature":"theorem Polynomial.roots_monomial {R : Type u} {a : R} [CommRing R] [IsDomain R] (ha : a \u2260 0) (n : \u2115) :((monomial n) a).roots = n \u2022 {0}","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L263-L265"},"Polynomial.eval\u2082_intCastRingHom_X":{"signature":"theorem Polynomial.eval\u2082_intCastRingHom_X {R : Type u_3} [Ring R] (p : Polynomial \u2124) (f : Polynomial \u2124 \u2192+* R) :eval\u2082 (Int.castRingHom R) (f X) p = f p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L149-L152"},"Polynomial.mul_modByMonic":{"signature":"theorem Polynomial.mul_modByMonic {R : Type u} [CommRing R] (p\u2081 p\u2082 q : Polynomial R) :p\u2081 * p\u2082 %\u2098 q = p\u2081 %\u2098 q * (p\u2082 %\u2098 q) %\u2098 q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L689-L698"},"Polynomial.support_C_mul_X":{"signature":"theorem Polynomial.support_C_mul_X {R : Type u} [Semiring R] {c : R} (h : c \u2260 0) :(C c * X).support = {1}","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L754-L755"},"Polynomial.natSepDegree_smul_nonzero":{"signature":"theorem Polynomial.natSepDegree_smul_nonzero {F : Type u} [Field F] (f : Polynomial F) {x : F} (hx : x \u2260 0) :(x \u2022 f).natSepDegree = f.natSepDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/SeparableDegree.lean#L394-L398"},"Polynomial.degreeLT_eq_span_X_pow":{"signature":"theorem Polynomial.degreeLT_eq_span_X_pow {R : Type u} [Semiring R] [DecidableEq R] {n : \u2115} :degreeLT R n = Submodule.span R \u2191(Finset.image (fun (n : \u2115) => X ^ n) (Finset.range n))","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Basic.lean#L96-L109"},"Polynomial.not_isField":{"signature":"theorem Polynomial.not_isField (R : Type u) [Ring R] :\u00acIsField (Polynomial R)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L472-L476"},"Polynomial.add_mod":{"signature":"theorem Polynomial.add_mod {R : Type u} [Field R] (p\u2081 p\u2082 q : Polynomial R) :(p\u2081 + p\u2082) % q = p\u2081 % q + p\u2082 % q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L442-L443"},"Polynomial.le_trailingDegree_X":{"signature":"theorem Polynomial.le_trailingDegree_X {R : Type u} [Semiring R] :1 \u2264 X.trailingDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L239-L240"},"Polynomial.nthRoots_two_eq_zero_iff":{"signature":"theorem Polynomial.nthRoots_two_eq_zero_iff {R : Type u} [CommRing R] [IsDomain R] {r : R} :nthRoots 2 r = 0 \u2194 \u00acIsSquare r","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L353-L356"},"Polynomial.hasDerivWithinAt_aeval":{"signature":"theorem Polynomial.hasDerivWithinAt_aeval {\ud835\udd5c : Type u} [NontriviallyNormedField \ud835\udd5c] {R : Type u_1} [CommSemiring R] [Algebra R \ud835\udd5c] (q : Polynomial R) (x : \ud835\udd5c) (s : Set \ud835\udd5c) :HasDerivWithinAt (fun (x : \ud835\udd5c) => (aeval x) q) ((aeval x) (derivative q)) s x","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Calculus/Deriv/Polynomial.lean#L77-L79"},"Polynomial.natTrailingDegree_X":{"signature":"theorem Polynomial.natTrailingDegree_X {R : Type u} [Semiring R] [Nontrivial R] :X.natTrailingDegree = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L361-L363"},"Polynomial.coeff_mapAlgHom_apply":{"signature":"theorem Polynomial.coeff_mapAlgHom_apply {R : Type u} {A : Type z} {B : Type u_2} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A \u2192\u2090[R] B) (p : Polynomial A) (n : \u2115) :((mapAlgHom f) p).coeff n = f (p.coeff n)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L193-L195"},"Polynomial.hasFDerivAt_aeval":{"signature":"theorem Polynomial.hasFDerivAt_aeval {\ud835\udd5c : Type u} [NontriviallyNormedField \ud835\udd5c] {R : Type u_1} [CommSemiring R] [Algebra R \ud835\udd5c] (q : Polynomial R) (x : \ud835\udd5c) :HasFDerivAt (fun (x : \ud835\udd5c) => (aeval x) q) (ContinuousLinearMap.smulRight 1 ((aeval x) (derivative q))) x","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Calculus/Deriv/Polynomial.lean#L128-L130"},"Polynomial.hasFDerivWithinAt_aeval":{"signature":"theorem Polynomial.hasFDerivWithinAt_aeval {\ud835\udd5c : Type u} [NontriviallyNormedField \ud835\udd5c] {s : Set \ud835\udd5c} {R : Type u_1} [CommSemiring R] [Algebra R \ud835\udd5c] (q : Polynomial R) (x : \ud835\udd5c) :HasFDerivWithinAt (fun (x : \ud835\udd5c) => (aeval x) q) (ContinuousLinearMap.smulRight 1 ((aeval x) (derivative q))) s x","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Calculus/Deriv/Polynomial.lean#L136-L139"},"Polynomial.valuation_le_one_of_valuation_X_le_one":{"signature":"theorem Polynomial.valuation_le_one_of_valuation_X_le_one (K : Type u_1) [Field K] {\u0393 : Type u_2} [LinearOrderedCommGroupWithZero \u0393] {v : Valuation (RatFunc K) \u0393} (hv : \u2200 (a : K), a \u2260 0 \u2192 v \u2191(C a) = 1) (hle : v RatFunc.X \u2264 1) (p : Polynomial K) :v \u2191p \u2264 1","doc":"If a valuation v is trivial on constants and v RatFunc.X \u2264 1 then for every polynomial p, v p \u2264 1.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/RatFunc/AsPolynomial.lean#L320-L329"},"Polynomial.opRingEquiv_op_X":{"signature":"theorem Polynomial.opRingEquiv_op_X {R : Type u_1} [Semiring R] :(opRingEquiv R) (MulOpposite.op X) = X","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Opposites.lean#L50-L52"},"Polynomial.opRingEquiv_op_C_mul_X_pow":{"signature":"theorem Polynomial.opRingEquiv_op_C_mul_X_pow {R : Type u_1} [Semiring R] (r : R) (n : \u2115) :(opRingEquiv R) (MulOpposite.op (C r * X ^ n)) = C (MulOpposite.op r) * X ^ n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Opposites.lean#L54-L56"},"Polynomial.derivative_intCast_mul":{"signature":"theorem Polynomial.derivative_intCast_mul {R : Type u} [Ring R] {n : \u2124} {f : Polynomial R} :derivative (\u2191n * f) = \u2191n * derivative f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L631-L633"},"Polynomial.monomial_mul_X_pow":{"signature":"theorem Polynomial.monomial_mul_X_pow {R : Type u} [Semiring R] (n : \u2115) (r : R) (k : \u2115) :(monomial n) r * X ^ k = (monomial (n + k)) r","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L548-L553"},"Polynomial.X_pow_mul_monomial":{"signature":"theorem Polynomial.X_pow_mul_monomial {R : Type u} [Semiring R] (k n : \u2115) (r : R) :X ^ k * (monomial n) r = (monomial (n + k)) r","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L559-L561"},"Polynomial.splits_comp_of_splits":{"signature":"theorem Polynomial.splits_comp_of_splits {R : Type u_1} [Semiring R] {f : Polynomial R} (hf : f.Splits) {S : Type u_2} [Semiring S] (i : R \u2192+* S) :(map i f).Splits","doc":"Alias of Polynomial.Splits.map.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L854-L855"},"Polynomial.natDegree_eq_card_roots":{"signature":"theorem Polynomial.natDegree_eq_card_roots {R : Type u_1} [CommRing R] {f : Polynomial R} [IsDomain R] (hf : f.Splits) :f.natDegree = f.roots.card","doc":"Alias of Polynomial.Splits.natDegree_eq_card_roots.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L784-L785"},"Polynomial.splits_map_iff":{"signature":"theorem Polynomial.splits_map_iff {F : Type u} {K : Type v} [CommRing K] [Field F] {L : Type u_2} [CommRing L] (i : K \u2192+* L) (j : L \u2192+* F) {f : Polynomial K} :(map j (map i f)).Splits \u2194 (map (j.comp i) f).Splits","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L684-L687"},"Polynomial.eval\u2082_at_ofNat":{"signature":"theorem Polynomial.eval\u2082_at_ofNat {R : Type u} [Semiring R] {p : Polynomial R} {S : Type u_1} [Semiring S] (f : R \u2192+* S) (n : \u2115) [n.AtLeastTwo] :eval\u2082 f (OfNat.ofNat n) p = f (eval (OfNat.ofNat n) p)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L270-L273"},"Polynomial.fderivWithin_aeval":{"signature":"theorem Polynomial.fderivWithin_aeval {\ud835\udd5c : Type u} [NontriviallyNormedField \ud835\udd5c] {x : \ud835\udd5c} {s : Set \ud835\udd5c} {R : Type u_1} [CommSemiring R] [Algebra R \ud835\udd5c] (q : Polynomial R) (hxs : UniqueDiffWithinAt \ud835\udd5c s x) :fderivWithin \ud835\udd5c (fun (x : \ud835\udd5c) => (aeval x) q) s x = ContinuousLinearMap.smulRight 1 ((aeval x) (derivative q))","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Calculus/Deriv/Polynomial.lean#L155-L157"},"Polynomial.natDegree_map_lt'":{"signature":"theorem Polynomial.natDegree_map_lt' {R : Type u} {S : Type v} [Semiring R] [Semiring S] {f : R \u2192+* S} {p : Polynomial R} (hp : f p.leadingCoeff = 0) (hp\u2080 : 0 < p.natDegree) :(map f p).natDegree < p.natDegree","doc":"Variant of natDegree_map_lt that assumes 0 < natDegree p instead of map f p \u2260 0.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Degree.lean#L163-L168"},"Polynomial.residueFieldMapCAlgEquiv_symm_X":{"signature":"theorem Polynomial.residueFieldMapCAlgEquiv_symm_X {R : Type u_1} [CommRing R] (I : Ideal R) [I.IsPrime] (J : Ideal (Polynomial R)) [J.IsPrime] [J.LiesOver I] (hJ : J = Ideal.map C I) :(residueFieldMapCAlgEquiv I J hJ).symm RatFunc.X = (algebraMap (Polynomial R) J.ResidueField) X","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/LocalRing/ResidueField/Polynomial.lean#L88-L91"},"Polynomial.head?_coeffList":{"signature":"theorem Polynomial.head?_coeffList {R : Type u_1} [Semiring R] {P : Polynomial R} (h : P \u2260 0) :P.coeffList.head? = some P.leadingCoeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/CoeffList.lean#L72-L75"},"Polynomial.splits_X_sub_C":{"signature":"theorem Polynomial.splits_X_sub_C {R : Type u_1} [Ring R] (a : R) :(X - C a).Splits","doc":"Alias of Polynomial.Splits.X_sub_C.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L692-L693"},"Polynomial.ofFinsupp_erase":{"signature":"theorem Polynomial.ofFinsupp_erase {R : Type u} [Semiring R] (p : AddMonoidAlgebra R \u2115) (n : \u2115) :{ toFinsupp := Finsupp.erase n p } = erase n { toFinsupp := p }","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L935-L938"},"Polynomial.eval\u2082_list_sum":{"signature":"theorem Polynomial.eval\u2082_list_sum {R : Type u} {S : Type v} [Semiring R] [Semiring S] (f : R \u2192+* S) (l : List (Polynomial R)) (x : S) :eval\u2082 f x l.sum = (List.map (eval\u2082 f x) l).sum","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L125-L126"},"Polynomial.eval_listSum":{"signature":"theorem Polynomial.eval_listSum {R : Type u} [Semiring R] (l : List (Polynomial R)) (x : R) :eval x l.sum = (List.map (eval x) l).sum","doc":"Polynomial evaluation commutes with List.sum.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L326-L328"},"Polynomial.sup_ker_aeval_eq_ker_aeval_mul_of_coprime":{"signature":"theorem Polynomial.sup_ker_aeval_eq_ker_aeval_mul_of_coprime {R : Type u} {M : Type w} [CommRing R] [AddCommGroup M] [Module R M] (f : M \u2192\u2097[R] M) {p q : Polynomial R} (hpq : IsCoprime p q) :((aeval f) p).ker \u2294 ((aeval f) q).ker = ((aeval f) (p * q)).ker","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Basic.lean#L939-L960"},"Polynomial.chebyshev_U_eq_dickson_two_one":{"signature":"theorem Polynomial.chebyshev_U_eq_dickson_two_one (R : Type u_1) [CommRing R] (n : \u2115) :Chebyshev.U R \u2191n = (dickson 2 1 n).comp (2 * X)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Dickson.lean#L177-L179"},"Polynomial.aevalTower_comp_algebraMap":{"signature":"theorem Polynomial.aevalTower_comp_algebraMap {R : Type u} {S : Type v} {A' : Type u_1} [CommSemiring R] [CommSemiring A'] [CommSemiring S] [Algebra S R] [Algebra S A'] (g : R \u2192\u2090[S] A') (y : A') :(\u2191(aevalTower g y)).comp (algebraMap R (Polynomial R)) = \u2191g","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L542-L543"},"Polynomial.aevalTower_comp_toAlgHom":{"signature":"theorem Polynomial.aevalTower_comp_toAlgHom {R : Type u} {S : Type v} {A' : Type u_1} [CommSemiring R] [CommSemiring A'] [CommSemiring S] [Algebra S R] [Algebra S A'] (g : R \u2192\u2090[S] A') (y : A') :(aevalTower g y).comp (IsScalarTower.toAlgHom S R (Polynomial R)) = g","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L548-L550"},"Polynomial.comap_C_surjective":{"signature":"theorem Polynomial.comap_C_surjective {R : Type u_1} [CommRing R] :Function.Surjective (PrimeSpectrum.comap C)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Spectrum/Prime/Polynomial.lean#L168-L172"},"Polynomial.natDegree_pos_of_eraseLead_ne_zero":{"signature":"theorem Polynomial.natDegree_pos_of_eraseLead_ne_zero {R : Type u_1} [Semiring R] {f : Polynomial R} (h : f.eraseLead \u2260 0) :0 < f.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/EraseLead.lean#L199-L202"},"Polynomial.degree_eq_one_of_irreducible_of_splits":{"signature":"theorem Polynomial.degree_eq_one_of_irreducible_of_splits {R : Type u_1} [Field R] {f : Polynomial R} (hf : f.Splits) (h : Irreducible f) :f.degree = 1","doc":"Alias of Polynomial.Splits.degree_eq_one_of_irreducible.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L766-L767"},"Polynomial.isCompact_image_comap_of_monic":{"signature":"theorem Polynomial.isCompact_image_comap_of_monic {R : Type u_1} [CommRing R] (f g : Polynomial R) (hg : g.Monic) :IsCompact (PrimeSpectrum.comap C '' (PrimeSpectrum.zeroLocus {g} \\ PrimeSpectrum.zeroLocus {f}))","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Spectrum/Prime/Polynomial.lean#L180-L185"},"Polynomial.aeval_pi_apply":{"signature":"theorem Polynomial.aeval_pi_apply {R : Type u} [CommSemiring R] {I : Type u_3} {A : I \u2192 Type u_4} [(i : I) \u2192 Semiring (A i)] [(i : I) \u2192 Algebra R (A i)] (x : (i : I) \u2192 A i) (p : Polynomial R) :(aeval x) p = fun (j : I) => (aeval (x j)) p","doc":"Polynomial evaluation on an indexed tuple is the indexed tuple of the evaluations on the components. Generalizes Polynomial.aeval_prod_apply to indexed products.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L443-L447"},"Polynomial.derivativeFinsupp_derivative":{"signature":"theorem Polynomial.derivativeFinsupp_derivative {R : Type u} [Semiring R] (p : Polynomial R) :derivativeFinsupp (derivative p) = Finsupp.comapDomain Nat.succ (derivativeFinsupp p) \u22ef","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L437-L441"},"Polynomial.natDegree_modByMonic_le":{"signature":"theorem Polynomial.natDegree_modByMonic_le {R : Type u} [Ring R] (p : Polynomial R) {g : Polynomial R} (hg : g.Monic) :(p %\u2098 g).natDegree \u2264 g.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L228-L230"},"Polynomial.disc":{"signature":"def Polynomial.disc {R : Type u_1} [CommRing R] (f : Polynomial R) :R","doc":"Alias of Polynomial.discr. The discriminant of a polynomial, defined as the determinant of f.sylvesterDeriv modified by a sign. The sign is chosen so polynomials over \u211d with all roots real have non-negative discriminant.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L936-L936"},"Polynomial.resultant_prod_right":{"signature":"theorem Polynomial.resultant_prod_right {R : Type u_1} [CommRing R] {\u03b9 : Type u_3} (s : Finset \u03b9) (f : Polynomial R) (g : \u03b9 \u2192 Polynomial R) (m : \u2115) (hm : f.natDegree \u2264 m) (hg : \u220f i \u2208 s, (g i).leadingCoeff \u2260 0) :f.resultant (\u220f i \u2208 s, g i) m = \u220f i \u2208 s, f.resultant (g i) m","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L656-L662"},"Polynomial.resultant_pow_right":{"signature":"theorem Polynomial.resultant_pow_right {R : Type u_1} [CommRing R] (f g : Polynomial R) (m n : \u2115) (hm : f.natDegree \u2264 m) (hg : g.leadingCoeff ^ n \u2260 0) :f.resultant (g ^ n) m = f.resultant g m ^ n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L669-L672"},"Polynomial.degree_sub_C":{"signature":"theorem Polynomial.degree_sub_C {R : Type u} {a : R} [Ring R] {p : Polynomial R} (hp : 0 < p.degree) :(p - C a).degree = p.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L548-L549"},"Polynomial.C_eq_or_isOpenQuotientMap_eval":{"signature":"theorem Polynomial.C_eq_or_isOpenQuotientMap_eval (p : Polynomial \u2102) :(\u2203 (x : \u2102), C x = p) \u2228 IsOpenQuotientMap fun (x : \u2102) => eval x p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Complex/OpenMapping.lean#L179-L184"},"Polynomial.isOpenQuotientMap_eval":{"signature":"theorem Polynomial.isOpenQuotientMap_eval (p : Polynomial \u2102) (hp : p.natDegree \u2260 0) :IsOpenQuotientMap fun (x : \u2102) => eval x p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Complex/OpenMapping.lean#L186-L188"},"Polynomial.natDegree_pos_of_not_isUnit_of_dvd_monic":{"signature":"theorem Polynomial.natDegree_pos_of_not_isUnit_of_dvd_monic {R : Type u} [CommSemiring R] {a p : Polynomial R} (hp : p.Monic) (ha : \u00acIsUnit a) (hap : a \u2223 p) :0 < a.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Units.lean#L94-L95"},"Polynomial.subalgebraNontrivial":{"signature":"instance Polynomial.subalgebraNontrivial {R : Type u} {A : Type z} [CommSemiring R] [Semiring A] [Algebra R A] [Nontrivial A] :Nontrivial (Subalgebra R (Polynomial A))","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L117-L126"},"Polynomial.coeff_natTrailingDegree_ne_zero":{"signature":"theorem Polynomial.coeff_natTrailingDegree_ne_zero {R : Type u} [Semiring R] {p : Polynomial R} :p.coeff p.natTrailingDegree \u2260 0 \u2194 p \u2260 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L134-L135"},"Polynomial.fderiv_aeval":{"signature":"theorem Polynomial.fderiv_aeval {\ud835\udd5c : Type u} [NontriviallyNormedField \ud835\udd5c] {x : \ud835\udd5c} {R : Type u_1} [CommSemiring R] [Algebra R \ud835\udd5c] (q : Polynomial R) :fderiv \ud835\udd5c (fun (x : \ud835\udd5c) => (aeval x) q) x = ContinuousLinearMap.smulRight 1 ((aeval x) (derivative q))","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Calculus/Deriv/Polynomial.lean#L146-L149"},"Polynomial.finite_range_coeff":{"signature":"theorem Polynomial.finite_range_coeff {R : Type u} [Semiring R] (f : Polynomial R) :(Set.range f.coeff).Finite","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L580-L581"},"Polynomial.mapAlgHom_coe_ringHom":{"signature":"theorem Polynomial.mapAlgHom_coe_ringHom {R : Type u} {A : Type z} {B : Type u_2} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A \u2192\u2090[R] B) :\u2191(mapAlgHom f) = mapRingHom \u2191f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L179-L182"},"Polynomial.coeff_divByMonic_X_sub_C_rec":{"signature":"theorem Polynomial.coeff_divByMonic_X_sub_C_rec {R : Type u} [Ring R] (p : Polynomial R) (a : R) (n : \u2115) :(p /\u2098 (X - C a)).coeff n = p.coeff (n + 1) + a * (p /\u2098 (X - C a)).coeff (n + 1)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L440-L448"},"Polynomial.coeff_divByMonic_X_sub_C":{"signature":"theorem Polynomial.coeff_divByMonic_X_sub_C {R : Type u} [Ring R] (p : Polynomial R) (a : R) (n : \u2115) :(p /\u2098 (X - C a)).coeff n = \u2211 i \u2208 Finset.Icc (n + 1) p.natDegree, a ^ (i - (n + 1)) * p.coeff i","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L450-L469"},"Polynomial.contraction_degree_eq_or_insep":{"signature":"theorem Polynomial.contraction_degree_eq_or_insep {F : Type u_1} [Field F] (q : \u2115) [hq : NeZero q] [CharP F q] (g g' : Polynomial F) (m m' : \u2115) (h_expand : (expand F (q ^ m)) g = (expand F (q ^ m')) g') (hg : g.Separable) (hg' : g'.Separable) :g.natDegree = g'.natDegree","doc":"If two expansions (along the positive characteristic) of two separable polynomials g and g' agree, then they have the same degree.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/SeparableDegree.lean#L110-L122"},"Polynomial.coe_taylorEquiv":{"signature":"theorem Polynomial.coe_taylorEquiv {R : Type u_1} [CommRing R] (r : R) :\u2191\u2191(taylorEquiv r) = taylor r","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Taylor.lean#L191-L191"},"Polynomial.zero_notMem_multiset_map_X_sub_C":{"signature":"theorem Polynomial.zero_notMem_multiset_map_X_sub_C {R : Type u} [Ring R] [Nontrivial R] {\u03b1 : Type u_1} (m : Multiset \u03b1) (f : \u03b1 \u2192 R) :0 \u2209 Multiset.map (fun (a : \u03b1) => X - C (f a)) m","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L777-L780"},"Polynomial.evalEval_X":{"signature":"theorem Polynomial.evalEval_X {R : Type u_1} [Semiring R] (x y : R) :evalEval x y X = y","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Bivariate.lean#L68-L70"},"Polynomial.le_rootMultiplicity_map":{"signature":"theorem Polynomial.le_rootMultiplicity_map {A : Type u_1} {B : Type u_2} [CommRing A] [CommRing B] {p : Polynomial A} {f : A \u2192+* B} (hmap : map f p \u2260 0) (a : A) :rootMultiplicity a p \u2264 rootMultiplicity (f a) (map f p)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L867-L871"},"Polynomial.eq_rootMultiplicity_map":{"signature":"theorem Polynomial.eq_rootMultiplicity_map {A : Type u_1} {B : Type u_2} [CommRing A] [CommRing B] {p : Polynomial A} {f : A \u2192+* B} (hf : Function.Injective \u21d1f) (a : A) :rootMultiplicity a p = rootMultiplicity (f a) (map f p)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L873-L879"},"Polynomial.filter_roots_map_range_eq_map_roots":{"signature":"theorem Polynomial.filter_roots_map_range_eq_map_roots {A : Type u_1} {B : Type u_2} [CommRing A] [CommRing B] [IsDomain A] [IsDomain B] {f : A \u2192+* B} [DecidablePred fun (x : B) => x \u2208 f.range] (hf : Function.Injective \u21d1f) (p : Polynomial A) :Multiset.filter (fun (x : B) => x \u2208 f.range) (map f p).roots = Multiset.map (\u21d1f) p.roots","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L930-L940"},"Polynomial.aeval_eq_smeval":{"signature":"theorem Polynomial.aeval_eq_smeval {R : Type u_1} [CommSemiring R] {S : Type u_2} [Semiring S] [Algebra R S] (x : S) (p : Polynomial R) :(aeval x) p = p.smeval x","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Smeval.lean#L324-L328"},"Polynomial.aeval_coe_eq_smeval":{"signature":"theorem Polynomial.aeval_coe_eq_smeval {R : Type u_1} [CommSemiring R] {S : Type u_2} [Semiring S] [Algebra R S] (x : S) :\u21d1(aeval x) = fun (p : Polynomial R) => p.smeval x","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Smeval.lean#L330-L331"},"Polynomial.splits_of_natDegree_le_one":{"signature":"theorem Polynomial.splits_of_natDegree_le_one {R : Type u_1} [DivisionSemiring R] {f : Polynomial R} (hf : f.natDegree \u2264 1) :f.Splits","doc":"Alias of Polynomial.Splits.of_natDegree_le_one.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L672-L673"},"Polynomial.iterate_derivative_one":{"signature":"theorem Polynomial.iterate_derivative_one {R : Type u} [Semiring R] {k : \u2115} (h : 0 < k) :(\u21d1derivative)^[k] 1 = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L218-L220"},"Polynomial.mkDerivation_one_eq_derivative":{"signature":"theorem Polynomial.mkDerivation_one_eq_derivative (R : Type u_1) [CommSemiring R] (f : Polynomial R) :((mkDerivation R) 1) f = derivative f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivation.lean#L78-L80"},"Polynomial.pUnitAlgEquiv_symm_toPowerSeries":{"signature":"theorem Polynomial.pUnitAlgEquiv_symm_toPowerSeries {R : Type u_1} [CommSemiring R] {f : Polynomial R} :\u2191f = \u2191((MvPolynomial.pUnitAlgEquiv R).symm f)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/PowerSeries/Basic.lean#L921-L926"},"Polynomial.hilbertPoly_eq_zero_of_le_rootMultiplicity_one":{"signature":"theorem Polynomial.hilbertPoly_eq_zero_of_le_rootMultiplicity_one {F : Type u_1} [Field F] [CharZero F] {p : Polynomial F} {d : \u2115} (hdp : d \u2264 rootMultiplicity 1 p) :p.hilbertPoly d = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/HilbertPoly.lean#L234-L244"},"Polynomial.natDegree_hilbertPoly_of_ne_zero":{"signature":"theorem Polynomial.natDegree_hilbertPoly_of_ne_zero {F : Type u_1} [Field F] [CharZero F] {p : Polynomial F} {d : \u2115} (hh : p.hilbertPoly d \u2260 0) :(p.hilbertPoly d).natDegree = d - rootMultiplicity 1 p - 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/HilbertPoly.lean#L267-L277"},"Polynomial.resultant_self_eq_zero":{"signature":"theorem Polynomial.resultant_self_eq_zero {R : Type u_1} [CommRing R] (f : Polynomial R) (h : f.natDegree \u2260 0) :f.resultant f = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L606-L608"},"Polynomial.nextCoeff_mul_C":{"signature":"theorem Polynomial.nextCoeff_mul_C {R : Type u} [Semiring R] {p : Polynomial R} {a : R} [NoZeroDivisors R] :(p * C a).nextCoeff = p.nextCoeff * a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L407-L409"},"Polynomial.eq_prod_roots_of_monic_of_splits_id":{"signature":"theorem Polynomial.eq_prod_roots_of_monic_of_splits_id {R : Type u_1} [CommRing R] {f : Polynomial R} [IsDomain R] (hf : f.Splits) (hm : f.Monic) :f = (Multiset.map (fun (x : R) => X - C x) f.roots).prod","doc":"Alias of Polynomial.Splits.eq_prod_roots_of_monic.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L811-L812"},"Polynomial.mul_divByMonic_cancel_left":{"signature":"theorem Polynomial.mul_divByMonic_cancel_left {R : Type u} [Ring R] (p : Polynomial R) {q : Polynomial R} (hmo : q.Monic) :q * p /\u2098 q = p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L433-L438"},"Polynomial.card_roots_le_map":{"signature":"theorem Polynomial.card_roots_le_map {A : Type u_1} {B : Type u_2} [CommRing A] [CommRing B] [IsDomain A] [IsDomain B] {p : Polynomial A} {f : A \u2192+* B} (h : map f p \u2260 0) :p.roots.card \u2264 (map f p).roots.card","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L942-L945"},"Polynomial.card_roots_le_map_of_injective":{"signature":"theorem Polynomial.card_roots_le_map_of_injective {A : Type u_1} {B : Type u_2} [CommRing A] [CommRing B] [IsDomain A] [IsDomain B] {p : Polynomial A} {f : A \u2192+* B} (hf : Function.Injective \u21d1f) :p.roots.card \u2264 (map f p).roots.card","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L947-L951"},"Polynomial.derivativeFinsupp_one":{"signature":"theorem Polynomial.derivativeFinsupp_one {R : Type u} [Semiring R] :derivativeFinsupp 1 = Finsupp.single 0 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L420-L422"},"Polynomial.eval_homogenize":{"signature":"theorem Polynomial.eval_homogenize {K : Type u_1} [Semifield K] {p : Polynomial K} {n : \u2115} (hn : p.natDegree \u2264 n) (x : Fin 2 \u2192 K) (hx : x 1 \u2260 0) :(MvPolynomial.eval x) (p.homogenize n) = eval (x 0 / x 1) p * x 1 ^ n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Homogenize.lean#L210-L218"},"Polynomial.leadingCoeff_hermite":{"signature":"theorem Polynomial.leadingCoeff_hermite (n : \u2115) :(hermite n).leadingCoeff = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Hermite/Basic.lean#L112-L114"},"Polynomial.hermite_monic":{"signature":"theorem Polynomial.hermite_monic (n : \u2115) :(hermite n).Monic","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Hermite/Basic.lean#L116-L117"},"Polynomial.ofFinsupp_natCast":{"signature":"theorem Polynomial.ofFinsupp_natCast {R : Type u} [Semiring R] (n : \u2115) :{ toFinsupp := \u2191n } = \u2191n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L255-L256"},"Polynomial.resultant_X_sub_C_right":{"signature":"theorem Polynomial.resultant_X_sub_C_right {R : Type u_1} [CommRing R] (f : Polynomial R) (m : \u2115) (r : R) (hf : f.natDegree \u2264 m) :f.resultant (X - C r) m 1 = (-1) ^ m * eval r f","doc":"Res(f, X - r) = (-1)^{deg f} * f(r)","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L396-L400"},"Polynomial.leadingCoeff_smul_integralNormalization":{"signature":"theorem Polynomial.leadingCoeff_smul_integralNormalization {S : Type v} [CommSemiring S] (p : Polynomial S) :p.leadingCoeff \u2022 p.integralNormalization = p.scaleRoots p.leadingCoeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/IntegralNormalization.lean#L133-L135"},"Polynomial.nextCoeff_map_eq_of_isUnit_leadingCoeff":{"signature":"theorem Polynomial.nextCoeff_map_eq_of_isUnit_leadingCoeff {R : Type u} {S : Type v} [Semiring R] {p : Polynomial R} [Semiring S] [Nontrivial S] (f : R \u2192+* S) (hp : IsUnit p.leadingCoeff) :(map f p).nextCoeff = f p.nextCoeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L289-L291"},"Polynomial.leadingCoeff_one":{"signature":"theorem Polynomial.leadingCoeff_one {R : Type u} [Semiring R] :leadingCoeff 1 = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L448-L449"},"Polynomial.height_map_C":{"signature":"theorem Polynomial.height_map_C {R : Type u_1} [CommRing R] [IsNoetherianRing R] (p : Ideal R) [p.IsMaximal] :(Ideal.map C p).height = p.height","doc":"Let p be a maximal ideal of R. Then the height of p[X] equals the height of p.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/KrullDimension/Polynomial.lean#L69-L72"},"Polynomial.logMahlerMeasure_mul_eq_add_logMahelerMeasure":{"signature":"theorem Polynomial.logMahlerMeasure_mul_eq_add_logMahelerMeasure {p q : Polynomial \u2102} (hpq : p * q \u2260 0) :(p * q).logMahlerMeasure = p.logMahlerMeasure + q.logMahlerMeasure","doc":"Alias of Polynomial.logMahlerMeasure_mul_eq_add_logMahlerMeasure.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/MahlerMeasure.lean#L150-L151"},"Polynomial.support_map_of_injective":{"signature":"theorem Polynomial.support_map_of_injective {R : Type u} {S : Type v} [Semiring R] [Semiring S] (p : Polynomial R) {f : R \u2192+* S} (hf : Function.Injective \u21d1f) :(map f p).support = p.support","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Coeff.lean#L211-L214"},"Polynomial.cauchyBound_zero":{"signature":"theorem Polynomial.cauchyBound_zero {K : Type u_1} [NormedDivisionRing K] :cauchyBound 0 = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/CauchyBound.lean#L40-L42"},"Polynomial.coe_aeval_eq_evalRingHom":{"signature":"theorem Polynomial.coe_aeval_eq_evalRingHom {R : Type u} [CommSemiring R] (x : R) :\u2191(aeval x) = evalRingHom x","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L455-L458"},"Polynomial.coeff_zero_eq_zero_of_zero_isRoot":{"signature":"theorem Polynomial.coeff_zero_eq_zero_of_zero_isRoot {R : Type u} [Semiring R] {p : Polynomial R} :p.IsRoot 0 \u2192 p.coeff 0 = 0","doc":"Alias of the forward direction of Polynomial.zero_isRoot_iff_coeff_zero_eq_zero.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Coeff.lean#L68-L69"},"Polynomial.hermite_eq_deriv_gaussian'":{"signature":"theorem Polynomial.hermite_eq_deriv_gaussian' (n : \u2115) (x : \u211d) :(aeval x) (hermite n) = (-1) ^ n * deriv^[n] (fun (y : \u211d) => Real.exp (-(y ^ 2 / 2))) x * Real.exp (x ^ 2 / 2)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Hermite/Gaussian.lean#L68-L71"},"Polynomial.le_natDegree_of_coe_le_degree":{"signature":"theorem Polynomial.le_natDegree_of_coe_le_degree {R : Type u} {n : \u2115} [Semiring R] {p : Polynomial R} (hdeg : \u2191n \u2264 p.degree) :n \u2264 p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/SmallDegree.lean#L65-L67"},"Polynomial.toFinsupp_C_mul_X_pow":{"signature":"theorem Polynomial.toFinsupp_C_mul_X_pow {R : Type u} [Semiring R] (a : R) (n : \u2115) :(C a * X ^ n).toFinsupp = AddMonoidAlgebra.single n a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L665-L667"},"Polynomial.algEquivAevalNegX_apply":{"signature":"theorem Polynomial.algEquivAevalNegX_apply {R : Type u_3} [CommRing R] (a : Polynomial R) :algEquivAevalNegX a = (aeval (-X)) a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L370-L370"},"Polynomial.zero_notMem_multiset_map_X_add_C":{"signature":"theorem Polynomial.zero_notMem_multiset_map_X_add_C {R : Type u} [Nontrivial R] [Semiring R] {\u03b1 : Type u_1} (m : Multiset \u03b1) (f : \u03b1 \u2192 R) :0 \u2209 Multiset.map (fun (a : \u03b1) => X + C (f a)) m","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L629-L632"},"Polynomial.coeff_inj":{"signature":"theorem Polynomial.coeff_inj {R : Type u} [Semiring R] {p q : Polynomial R} :p.coeff = q.coeff \u2194 p = q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L574-L576"},"Polynomial.cauchyBound_one":{"signature":"theorem Polynomial.cauchyBound_one {K : Type u_1} [NormedDivisionRing K] :cauchyBound 1 = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/CauchyBound.lean#L48-L49"},"Polynomial.isFixedPt_newtonMap_of_isUnit_iff":{"signature":"theorem Polynomial.isFixedPt_newtonMap_of_isUnit_iff {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [Algebra R S] {P : Polynomial R} {x : S} (h : IsUnit ((aeval x) (derivative P))) :Function.IsFixedPt P.newtonMap x \u2194 (aeval x) P = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Dynamics/Newton.lean#L78-L80"},"Polynomial.natDegree_eq_card_roots'":{"signature":"theorem Polynomial.natDegree_eq_card_roots' {R : Type u_1} [CommRing R] {f : Polynomial R} [IsDomain R] (hf : f.Splits) :f.natDegree = f.roots.card","doc":"Alias of Polynomial.Splits.natDegree_eq_card_roots.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L731-L732"},"Polynomial.aroots_one":{"signature":"theorem Polynomial.aroots_one {S : Type v} {T : Type w} [CommRing T] [CommRing S] [IsDomain S] [Algebra T S] :aroots 1 S = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L481-L484"},"Polynomial.degree_zero_le":{"signature":"theorem Polynomial.degree_zero_le {R : Type u} [Semiring R] :degree 0 \u2264 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L494-L494"},"Polynomial.trailingDegree_X_pow":{"signature":"theorem Polynomial.trailingDegree_X_pow {R : Type u} [Semiring R] [Nontrivial R] (n : \u2115) :(X ^ n).trailingDegree = \u2191n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L365-L368"},"Polynomial.root_mul_right_of_isRoot":{"signature":"theorem Polynomial.root_mul_right_of_isRoot {R : Type u} {a : R} [CommSemiring R] {p : Polynomial R} (q : Polynomial R) :p.IsRoot a \u2192 (p * q).IsRoot a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L638-L639"},"Polynomial.head_coeffList":{"signature":"theorem Polynomial.head_coeffList {R : Type u_1} [Semiring R] (P : Polynomial R) (hP : P.coeffList \u2260 []) :P.coeffList.head hP = P.leadingCoeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/CoeffList.lean#L77-L81"},"Polynomial.quotientSpanXSubCAlgEquiv_mk":{"signature":"theorem Polynomial.quotientSpanXSubCAlgEquiv_mk {R : Type u_1} [CommRing R] (x : R) (p : Polynomial R) :(quotientSpanXSubCAlgEquiv x) ((Ideal.Quotient.mk (Ideal.span {X - C x})) p) = eval x p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Quotient.lean#L39-L42"},"Polynomial.normUnit_content":{"signature":"theorem Polynomial.normUnit_content {R : Type u_1} [CommRing R] [NormalizedGCDMonoid R] {p : Polynomial R} :normUnit p.content = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Content.lean#L170-L176"},"Polynomial.degree_cubic_lt":{"signature":"theorem Polynomial.degree_cubic_lt {R : Type u} {a b c d : R} [Semiring R] :(C a * X ^ 3 + C b * X ^ 2 + C c * X + C d).degree < 4","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/SmallDegree.lean#L124-L125"},"Polynomial.divX_C_mul_X_pow":{"signature":"theorem Polynomial.divX_C_mul_X_pow {R : Type u} {a : R} {n : \u2115} [Semiring R] :(C a * X ^ n).divX = if n = 0 then 0 else C a * X ^ (n - 1)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Inductions.lean#L105-L106"},"Polynomial.contract_expand":{"signature":"theorem Polynomial.contract_expand {R : Type u} [CommSemiring R] (p : \u2115) {f : Polynomial R} (hp : p \u2260 0) :contract p ((expand R p) f) = f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Expand.lean#L201-L203"},"Polynomial.C_mem_lifts":{"signature":"theorem Polynomial.C_mem_lifts {R : Type u} [Semiring R] {S : Type v} [Semiring S] (f : R \u2192+* S) (r : R) :C (f r) \u2208 lifts f","doc":"If (r : R), then C (f r) lifts.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Lifts.lean#L88-L91"},"Polynomial.mapAlgHom_eq_eval\u2082AlgHom'_CAlgHom":{"signature":"theorem Polynomial.mapAlgHom_eq_eval\u2082AlgHom'_CAlgHom {R : Type u} {A : Type z} {B : Type u_2} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A \u2192\u2090[R] B) :mapAlgHom f = eval\u2082AlgHom' (CAlgHom.comp f) X \u22ef","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L189-L191"},"Polynomial.addHom_ext'_iff":{"signature":"theorem Polynomial.addHom_ext'_iff {R : Type u} [Semiring R] {M : Type u_1} [AddZeroClass M] {f g : Polynomial R \u2192+ M} :f = g \u2194 \u2200 (n : \u2115), f.comp (monomial n).toAddMonoidHom = g.comp (monomial n).toAddMonoidHom","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L725-L725"},"Polynomial.monomial_mem_lifts_and_degree_eq":{"signature":"theorem Polynomial.monomial_mem_lifts_and_degree_eq {R : Type u} [Semiring R] {S : Type v} [Semiring S] {f : R \u2192+* S} {s : S} {n : \u2115} (hl : (monomial n) s \u2208 lifts f) :\u2203 (q : Polynomial R), map f q = (monomial n) s \u2227 q.degree = ((monomial n) s).degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Lifts.lean#L135-L142"},"Polynomial.gaussNorm_monomial":{"signature":"theorem Polynomial.gaussNorm_monomial {R : Type u_1} {F : Type u_2} [Semiring R] [FunLike F R \u211d] (v : F) (c : \u211d) [ZeroHomClass F R \u211d] (n : \u2115) (r : R) :gaussNorm v c ((monomial n) r) = v r * c ^ n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/GaussNorm.lean#L65-L68"},"Polynomial.toLaurent_ne_zero":{"signature":"theorem Polynomial.toLaurent_ne_zero {R : Type u_1} [Semiring R] {f : Polynomial R} :toLaurent f \u2260 0 \u2194 f \u2260 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Laurent.lean#L338-L339"},"Polynomial.disc_of_degree_eq_one":{"signature":"theorem Polynomial.disc_of_degree_eq_one {R : Type u_1} [CommRing R] {f : Polynomial R} (hf : f.degree = 1) :f.discr = 1","doc":"Alias of Polynomial.discr_of_degree_eq_one. The discriminant of a linear polynomial is 1.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L975-L975"},"Polynomial.coeff_zero_eq_leadingCoeff_mul_prod_roots_of_splits":{"signature":"theorem Polynomial.coeff_zero_eq_leadingCoeff_mul_prod_roots_of_splits {R : Type u_1} [CommRing R] {f : Polynomial R} [IsDomain R] (hf : f.Splits) :f.coeff 0 = (-1) ^ f.natDegree * f.leadingCoeff * f.roots.prod","doc":"Alias of Polynomial.Splits.coeff_zero_eq_leadingCoeff_mul_prod_roots.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L883-L885"},"Polynomial.wronskian_neg_right":{"signature":"theorem Polynomial.wronskian_neg_right {R : Type u_1} [CommRing R] (a b : Polynomial R) :a.wronskian (-b) = -a.wronskian b","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Wronskian.lean#L64-L65"},"Polynomial.irreducible_iff_roots_eq_zero_of_degree_le_three":{"signature":"theorem Polynomial.irreducible_iff_roots_eq_zero_of_degree_le_three {K : Type u_1} [Field K] {p : Polynomial K} (hp2 : 2 \u2264 p.natDegree) (hp3 : p.natDegree \u2264 3) :Irreducible p \u2194 p.roots = 0","doc":"A polynomial of degree 2 or 3 is irreducible iff it doesn't have roots.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/SpecificDegree.lean#L46-L56"},"Polynomial.irreducible_of_degree_le_three_of_not_isRoot":{"signature":"theorem Polynomial.irreducible_of_degree_le_three_of_not_isRoot {K : Type u_1} [Field K] {p : Polynomial K} (hdeg : p.natDegree \u2208 Finset.Icc 1 3) (hnot : \u2200 (x : K), \u00acp.IsRoot x) :Irreducible p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/SpecificDegree.lean#L58-L68"},"Polynomial.nextCoeffUp_C_eq_zero":{"signature":"theorem Polynomial.nextCoeffUp_C_eq_zero {R : Type u} [Semiring R] (c : R) :(C c).nextCoeffUp = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L406-L409"},"Polynomial.reverse_C_add":{"signature":"theorem Polynomial.reverse_C_add {R : Type u_1} [Semiring R] (p : Polynomial R) (t : R) :(C t + p).reverse = C t * X ^ p.natDegree + p.reverse","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Reverse.lean#L324-L326"},"Polynomial.splits_of_degree_le_zero":{"signature":"theorem Polynomial.splits_of_degree_le_zero {R : Type u_1} [Semiring R] {f : Polynomial R} (hf : f.degree \u2264 0) :f.Splits","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L92-L94"},"Polynomial.algEquivAevalXAddC_symm_apply":{"signature":"theorem Polynomial.algEquivAevalXAddC_symm_apply {R : Type u_3} [CommRing R] (t : R) (a : Polynomial R) :(algEquivAevalXAddC t).symm a = (aeval (X - C t)) a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L355-L355"},"Polynomial.constantCoeff_coe":{"signature":"theorem Polynomial.constantCoeff_coe {R : Type u_1} [Semiring R] (\u03c6 : Polynomial R) :PowerSeries.constantCoeff \u2191\u03c6 = \u03c6.coeff 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/PowerSeries/Basic.lean#L856-L858"},"Polynomial.toMvPolynomial_injective":{"signature":"theorem Polynomial.toMvPolynomial_injective {R : Type u_1} {\u03c3 : Type u_3} [CommSemiring R] (i : \u03c3) :Function.Injective \u21d1(toMvPolynomial i)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/MvPolynomial/Equiv.lean#L793-L797"},"Polynomial.le_trailingDegree_C_mul_X_pow":{"signature":"theorem Polynomial.le_trailingDegree_C_mul_X_pow {R : Type u} [Semiring R] (n : \u2115) (a : R) :\u2191n \u2264 (C a * X ^ n).trailingDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L213-L216"},"Polynomial.iterate_derivative_mul_X":{"signature":"theorem Polynomial.iterate_derivative_mul_X {R : Type u} [CommSemiring R] {n : \u2115} (p : Polynomial R) :(\u21d1derivative)^[n] (p * X) = (\u21d1derivative)^[n] p * X + n \u2022 (\u21d1derivative)^[n - 1] p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L540-L543"},"Polynomial.integralNormalization_C":{"signature":"theorem Polynomial.integralNormalization_C {R : Type u} [Semiring R] {x : R} (hx : x \u2260 0) :(C x).integralNormalization = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/IntegralNormalization.lean#L47-L49"},"Polynomial.toFinsuppIsoAlg_apply":{"signature":"theorem Polynomial.toFinsuppIsoAlg_apply (R : Type u) [CommSemiring R] (self : Polynomial R) :(toFinsuppIsoAlg R) self = self.toFinsupp","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L111-L111"},"Polynomial.coeff_mul_monomial_zero":{"signature":"theorem Polynomial.coeff_mul_monomial_zero {R : Type u} [Semiring R] (p : Polynomial R) (d : \u2115) (r : R) :(p * (monomial 0) r).coeff d = p.coeff d * r","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L267-L269"},"Polynomial.resultant_zero_zero":{"signature":"theorem Polynomial.resultant_zero_zero {R : Type u_1} [CommRing R] (m n : \u2115) :resultant 0 0 m n = 0 ^ (m + n)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L189-L189"},"Polynomial.dickson_one_one_eval_add_inv":{"signature":"theorem Polynomial.dickson_one_one_eval_add_inv {R : Type u_1} [CommRing R] (x y : R) (h : x * y = 1) (n : \u2115) :eval (x + y) (dickson 1 1 n) = x ^ n + y ^ n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Dickson.lean#L122-L131"},"Polynomial.subsingleton_iff_subsingleton":{"signature":"theorem Polynomial.subsingleton_iff_subsingleton {R : Type u} [Semiring R] :Subsingleton (Polynomial R) \u2194 Subsingleton R","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L689-L692"},"Polynomial.forall_eq_iff_forall_eq":{"signature":"theorem Polynomial.forall_eq_iff_forall_eq {R : Type u} [Semiring R] :(\u2200 (f g : Polynomial R), f = g) \u2194 \u2200 (a b : R), a = b","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L697-L698"},"Polynomial.natDegree_X_mul":{"signature":"theorem Polynomial.natDegree_X_mul {R : Type u} [Semiring R] [Nontrivial R] {p : Polynomial R} (hp : p \u2260 0) :(X * p).natDegree = p.natDegree + 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L517-L518"},"Polynomial.coe_compRingHom_apply":{"signature":"theorem Polynomial.coe_compRingHom_apply {R : Type u} [CommSemiring R] (p q : Polynomial R) :q.compRingHom p = p.comp q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L632-L633"},"Polynomial.exists_dvd_map_of_isAlgebraic":{"signature":"theorem Polynomial.exists_dvd_map_of_isAlgebraic {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [Algebra R S] [alg : Algebra.IsAlgebraic R S] [NoZeroDivisors S] {f : Polynomial S} (hf : f \u2260 0) :\u2203 (g : Polynomial R), g \u2260 0 \u2227 f \u2223 map (algebraMap R S) g","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Algebraic/Integral.lean#L593-L595"},"Polynomial.constantCoeff_surjective":{"signature":"theorem Polynomial.constantCoeff_surjective {R : Type u} [Semiring R] :Function.Surjective \u21d1constantCoeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L133-L134"},"Polynomial.dickson_one_one_zmod_p":{"signature":"theorem Polynomial.dickson_one_one_zmod_p (p : \u2115) [Fact (Nat.Prime p)] :dickson 1 1 p = X ^ p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Dickson.lean#L202-L267"},"Polynomial.dickson_one_one_charP":{"signature":"theorem Polynomial.dickson_one_one_charP (R : Type u_1) [CommRing R] (p : \u2115) [Fact (Nat.Prime p)] [CharP R p] :dickson 1 1 p = X ^ p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Dickson.lean#L269-L273"},"Polynomial.instIsLocalHomRingHomAlgebraMap":{"signature":"instance Polynomial.instIsLocalHomRingHomAlgebraMap {R : Type u_1} [CommRing R] :IsLocalHom (algebraMap R (Polynomial R))","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Nilpotent.lean#L194-L195"},"Polynomial.not_isUnit_of_degree_pos_of_isReduced":{"signature":"theorem Polynomial.not_isUnit_of_degree_pos_of_isReduced {R : Type u_1} [CommRing R] [IsReduced R] (p : Polynomial R) (hpl : 0 < p.degree) :\u00acIsUnit p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Nilpotent.lean#L187-L189"},"Polynomial.monomial_eq_monomial_iff":{"signature":"theorem Polynomial.monomial_eq_monomial_iff {R : Type u} [Semiring R] {m n : \u2115} {a b : R} :(monomial m) a = (monomial n) b \u2194 m = n \u2227 a = b \u2228 a = 0 \u2227 b = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L424-L426"},"Polynomial.X_ne_C":{"signature":"theorem Polynomial.X_ne_C {R : Type u} [Semiring R] [Nontrivial R] (a : R) :X \u2260 C a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L496-L498"},"Polynomial.eval_derivative_of_splits":{"signature":"theorem Polynomial.eval_derivative_of_splits {R : Type u_1} [CommRing R] {f : Polynomial R} [IsDomain R] [DecidableEq R] (hf : f.Splits) (x : R) :eval x (derivative f) = f.leadingCoeff * (Multiset.map (fun (a : R) => (Multiset.map (fun (x_1 : R) => x - x_1) (f.roots.erase a)).prod) f.roots).sum","doc":"Alias of Polynomial.Splits.eval_derivative.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L871-L872"},"Polynomial.splits_of_natDegree_eq_two":{"signature":"theorem Polynomial.splits_of_natDegree_eq_two {R : Type u_1} [Field R] {f : Polynomial R} {x : R} (h\u2081 : f.natDegree = 2) (h\u2082 : eval x f = 0) :f.Splits","doc":"Alias of Polynomial.Splits.of_natDegree_eq_two. A polynomial of degree 2 with a root splits.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L826-L827"},"Polynomial.eval\u2082RingHom'_apply":{"signature":"theorem Polynomial.eval\u2082RingHom'_apply {R : Type u} {S : Type v} [Semiring R] [Semiring S] (f : R \u2192+* S) (x : S) (hf : \u2200 (a : R), Commute (f a) x) (p : Polynomial R) :(eval\u2082RingHom' f x hf) p = eval\u2082 f x p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L175-L175"},"Polynomial.eval_derivative_div_eval_of_ne_zero_of_splits":{"signature":"theorem Polynomial.eval_derivative_div_eval_of_ne_zero_of_splits {R : Type u_1} [Field R] {f : Polynomial R} (hf : f.Splits) {x : R} (hx : eval x f \u2260 0) :eval x (derivative f) / eval x f = (Multiset.map (fun (z : R) => 1 / (x - z)) f.roots).sum","doc":"Alias of Polynomial.Splits.eval_derivative_div_eval_of_ne_zero.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L880-L881"},"Polynomial.dickson_two":{"signature":"theorem Polynomial.dickson_two {R : Type u_1} [CommRing R] (k : \u2115) (a : R) :dickson k a 2 = X ^ 2 - C a * (3 - \u2191k)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Dickson.lean#L76-L77"},"Polynomial.sup_aeval_range_eq_top_of_isCoprime":{"signature":"theorem Polynomial.sup_aeval_range_eq_top_of_isCoprime {R : Type u} {M : Type w} [CommRing R] [AddCommGroup M] [Module R M] (f : M \u2192\u2097[R] M) {p q : Polynomial R} (hpq : IsCoprime p q) :((aeval f) p).range \u2294 ((aeval f) q).range = \u22a4","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Basic.lean#L916-L927"},"Polynomial.eq_of_natDegree_lt_card_of_eval_eq'":{"signature":"theorem Polynomial.eq_of_natDegree_lt_card_of_eval_eq' {R : Type u_1} [CommRing R] [IsDomain R] (p q : Polynomial R) (s : Finset R) (heval : \u2200 i \u2208 s, eval i p = eval i q) (hcard : max p.natDegree q.natDegree < s.card) :p = q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L724-L728"},"Polynomial.degree_eq_card_roots":{"signature":"theorem Polynomial.degree_eq_card_roots {R : Type u_1} [CommRing R] {f : Polynomial R} [IsDomain R] (hf : f.Splits) (hf0 : f \u2260 0) :f.degree = \u2191f.roots.card","doc":"Alias of Polynomial.Splits.degree_eq_card_roots.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L787-L788"},"Polynomial.roots_expand_map_frobenius_le":{"signature":"theorem Polynomial.roots_expand_map_frobenius_le {R : Type u_1} [CommRing R] [IsDomain R] (p : \u2115) [ExpChar R p] (f : Polynomial R) :Multiset.map (\u21d1(frobenius R p)) ((expand R p) f).roots \u2264 p \u2022 f.roots","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Perfect.lean#L321-L324"},"Polynomial.mem_iff_eq_smul_annIdealGenerator":{"signature":"theorem Polynomial.mem_iff_eq_smul_annIdealGenerator (\ud835\udd5c : Type u_1) {A : Type u_2} [Field \ud835\udd5c] [Ring A] [Algebra \ud835\udd5c A] {p : Polynomial \ud835\udd5c} (a : A) :p \u2208 annIdeal \ud835\udd5c a \u2194 \u2203 (s : Polynomial \ud835\udd5c), p = s \u2022 annIdealGenerator \ud835\udd5c a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/LinearAlgebra/AnnihilatingPolynomial.lean#L110-L112"},"Polynomial.eq_of_degree_le_of_eval_finset_eq":{"signature":"theorem Polynomial.eq_of_degree_le_of_eval_finset_eq {R : Type u_1} [CommRing R] [IsDomain R] {f g : Polynomial R} (s : Finset R) (h_deg_le : f.degree \u2264 \u2191s.card) (h_deg_eq : f.degree = g.degree) (hlc : f.leadingCoeff = g.leadingCoeff) (h_eval : \u2200 x \u2208 s, eval x f = eval x g) :f = g","doc":"Two polynomials, with the same degree and leading coefficient, which have the same evaluation on a set of distinct values with cardinality equal to the degree, are equal.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/LinearAlgebra/Lagrange.lean#L71-L84"},"Polynomial.resultant_X_add_C_right":{"signature":"theorem Polynomial.resultant_X_add_C_right {R : Type u_1} [CommRing R] (f : Polynomial R) (m : \u2115) (r : R) (hf : f.natDegree \u2264 m) :f.resultant (X + C r) m 1 = (-1) ^ m * eval (-r) f","doc":"Res(f, X + r) = (-1)^{deg f} * f(-r)","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L402-L405"},"Polynomial.eval\u2082_modByMonic_eq_self_of_root":{"signature":"theorem Polynomial.eval\u2082_modByMonic_eq_self_of_root {R : Type u} {S : Type v} [CommRing R] [CommRing S] {f : R \u2192+* S} {p q : Polynomial R} (hq : q.Monic) {x : S} (hx : eval\u2082 f x q = 0) :eval\u2082 f x (p %\u2098 q) = eval\u2082 f x p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L611-L613"},"Polynomial.isMonicOfDegree_two_iff":{"signature":"theorem Polynomial.isMonicOfDegree_two_iff {R : Type u_1} [Semiring R] [Nontrivial R] {f : Polynomial R} :f.IsMonicOfDegree 2 \u2194 \u2203 (a : R) (b : R), f = X ^ 2 + C a * X + C b","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/IsMonicOfDegree.lean#L195-L203"},"Polynomial.isMonicOfDegree_two_iff'":{"signature":"theorem Polynomial.isMonicOfDegree_two_iff' {R : Type u_1} [Ring R] [Nontrivial R] {f : Polynomial R} :f.IsMonicOfDegree 2 \u2194 \u2203 (a : R) (b : R), f = X ^ 2 - C a * X + C b","doc":"A version of Polynomial.isMonicOfDegree_two_iff with negated middle coefficient.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/IsMonicOfDegree.lean#L245-L251"},"Polynomial.aeval_endomorphism":{"signature":"theorem Polynomial.aeval_endomorphism {R : Type u} [CommRing R] {M : Type u_3} [AddCommGroup M] [Module R M] (f : M \u2192\u2097[R] M) (v : M) (p : Polynomial R) :((aeval f) p) v = p.sum fun (n : \u2115) (b : R) => b \u2022 (f ^ n) v","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L648-L651"},"Polynomial.cyclotomic_eval_le_add_one_pow_totient":{"signature":"theorem Polynomial.cyclotomic_eval_le_add_one_pow_totient {q : \u211d} (hq' : 1 < q) (n : \u2115) :eval q (cyclotomic n \u211d) \u2264 (q + 1) ^ n.totient","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Eval.lean#L290-L295"},"Polynomial.eval\u2082_list_prod_noncomm":{"signature":"theorem Polynomial.eval\u2082_list_prod_noncomm {R : Type u} {S : Type v} [Semiring R] [Semiring S] (f : R \u2192+* S) (x : S) (ps : List (Polynomial R)) (hf : \u2200 p \u2208 ps, \u2200 (k : \u2115), Commute (f (p.coeff k)) x) :eval\u2082 f x ps.prod = (List.map (eval\u2082 f x) ps).prod","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L165-L172"},"Polynomial.leadingCoeff_sub_of_degree_lt'":{"signature":"theorem Polynomial.leadingCoeff_sub_of_degree_lt' {R : Type u} [Ring R] {p q : Polynomial R} (h : p.degree < q.degree) :(p - q).leadingCoeff = -q.leadingCoeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L560-L563"},"Polynomial.natDegree_sum_eq_of_disjoint":{"signature":"theorem Polynomial.natDegree_sum_eq_of_disjoint {R : Type u} {S : Type v} [Semiring R] (f : S \u2192 Polynomial R) (s : Finset S) (h : {i : S | i \u2208 s \u2227 f i \u2260 0}.Pairwise (Function.onFun Ne (natDegree \u2218 f))) :(s.sum f).natDegree = s.sup fun (i : S) => (f i).natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L207-L236"},"Polynomial.eval_monomial_one_add_sub":{"signature":"theorem Polynomial.eval_monomial_one_add_sub {S : Type v} [CommRing S] (d : \u2115) (y : S) :eval (1 + y) ((monomial d) (\u2191d + 1)) - eval y ((monomial d) (\u2191d + 1)) = \u2211 x_1 \u2208 Finset.range (d + 1), \u2191((d + 1).choose x_1) * (\u2191x_1 * y ^ (x_1 - 1))","doc":"A reformulation of the expansion of (1 + y)^d: $$(d + 1) (1 + y)^d - (d + 1)y^d = \\sum_{i = 0}^d {d + 1 \\choose i} \\cdot i \\cdot y^{i - 1}.$$","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Degree.lean#L71-L84"},"Polynomial.isRoot_of_isRoot_of_dvd_derivative_mul":{"signature":"theorem Polynomial.isRoot_of_isRoot_of_dvd_derivative_mul {R : Type u} [CommRing R] [NoZeroDivisors R] [CharZero R] {f g : Polynomial R} (hf0 : f \u2260 0) (hfd : f \u2223 derivative f * g) {a : R} (haf : f.IsRoot a) :g.IsRoot a","doc":"A sufficient condition for the set of roots of a nonzero polynomial f to be a subset of the set of roots of g is that f divides f.derivative * g. Over an algebraically closed field of characteristic zero, this is also a necessary condition. See isRoot_of_isRoot_iff_dvd_derivative_mul","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L181-L199"},"Polynomial.valuation_aeval_monomial_eq_valuation_pow":{"signature":"theorem Polynomial.valuation_aeval_monomial_eq_valuation_pow (K : Type u_1) [Field K] {\u0393 : Type u_2} [LinearOrderedCommGroupWithZero \u0393] (L : Type u_3) [Field L] [Algebra K L] {v : Valuation L \u0393} (hv : \u2200 (a : K), a \u2260 0 \u2192 v ((algebraMap K L) a) = 1) (w : L) (n : \u2115) {a : K} (ha : a \u2260 0) :v ((aeval w) ((monomial n) a)) = v w ^ n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/RatFunc/AsPolynomial.lean#L274-L276"},"Polynomial.valuation_aeval_eq_valuation_X_pow_natDegree_of_one_lt_valuation_X":{"signature":"theorem Polynomial.valuation_aeval_eq_valuation_X_pow_natDegree_of_one_lt_valuation_X (K : Type u_1) [Field K] {\u0393 : Type u_2} [LinearOrderedCommGroupWithZero \u0393] (L : Type u_3) [Field L] [Algebra K L] {v : Valuation L \u0393} (hv : \u2200 (a : K), a \u2260 0 \u2192 v ((algebraMap K L) a) = 1) (w : L) (hpos : 1 < v w) {p : Polynomial K} (hp : p \u2260 0) :v ((aeval w) p) = v w ^ p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/RatFunc/AsPolynomial.lean#L278-L291"},"Polynomial.resultant_C_zero_right":{"signature":"theorem Polynomial.resultant_C_zero_right {R : Type u_1} [CommRing R] (f : Polynomial R) (m : \u2115) (r : R) :f.resultant (C r) m 0 = r ^ m","doc":"Res(a, g) = a ^ deg g","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L174-L175"},"Polynomial.prod_X_sub_C_nextCoeff":{"signature":"theorem Polynomial.prod_X_sub_C_nextCoeff {R : Type u} {\u03b9 : Type w} [CommRing R] {s : Finset \u03b9} (f : \u03b9 \u2192 R) :(\u220f i \u2208 s, (X - C (f i))).nextCoeff = -\u2211 i \u2208 s, f i","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/BigOperators.lean#L260-L262"},"Polynomial.gaussNorm_lt_one_iff_contentIdeal_le":{"signature":"theorem Polynomial.gaussNorm_lt_one_iff_contentIdeal_le {R : Type u_1} [CommRing R] [IsDedekindDomain R] (v : IsDedekindDomain.HeightOneSpectrum R) {b : NNReal} (hb : 1 < b) (p : Polynomial R) :gaussNorm (v.intAdicAbv hb) 1 p < 1 \u2194 p.contentIdeal \u2264 v.asIdeal","doc":"Given a polynomial p in R[X], the v-adic Gauss norm of p is smaller than 1 if and only if the content ideal of p is contained in the prime ideal corresponding to v.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/DedekindDomain/GaussLemma.lean#L33-L52"},"Polynomial.contentIdeal_eq_top_iff_forall_gaussNorm_eq_one":{"signature":"theorem Polynomial.contentIdeal_eq_top_iff_forall_gaussNorm_eq_one {R : Type u_1} [CommRing R] [IsDedekindDomain R] {b : NNReal} (hb : 1 < b) (p : Polynomial R) (hR : \u00acIsField R) :p.contentIdeal = \u22a4 \u2194 \u2200 (v : IsDedekindDomain.HeightOneSpectrum R), gaussNorm (v.intAdicAbv hb) 1 p = 1","doc":"Gauss's Lemma: given a polynomial p in R[X], the content ideal of p is the whole ring if and only if the v-adic Gauss norms of p are equal to 1 for all v.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/DedekindDomain/GaussLemma.lean#L54-L60"},"Polynomial.tendsto_nhds_iff":{"signature":"theorem Polynomial.tendsto_nhds_iff {\ud835\udd5c : Type u_1} [NormedField \ud835\udd5c] [LinearOrder \ud835\udd5c] [IsStrictOrderedRing \ud835\udd5c] (P : Polynomial \ud835\udd5c) [OrderTopology \ud835\udd5c] {c : \ud835\udd5c} :Filter.Tendsto (fun (x : \ud835\udd5c) => eval x P) Filter.atTop (nhds c) \u2194 P.leadingCoeff = c \u2227 P.degree \u2264 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/Basic.lean#L98-L110"},"Polynomial.expand_char":{"signature":"theorem Polynomial.expand_char {R : Type u} [CommSemiring R] (p : \u2115) [ExpChar R p] (f : Polynomial R) :map (frobenius R p) ((expand R p) f) = f ^ p","doc":"Alias of Polynomial.map_frobenius_expand.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Expand.lean#L269-L270"},"Polynomial.rootOfSplits'_eq_rootOfSplits":{"signature":"theorem Polynomial.rootOfSplits'_eq_rootOfSplits {K : Type v} {L : Type w} [Field K] [Field L] (i : K \u2192+* L) {f : Polynomial K} (hf : (map i f).Splits) (hfd : (map i f).degree \u2260 0) :rootOfSplits hf hfd = rootOfSplits hf \u22ef","doc":"rootOfSplits' is definitionally equal to rootOfSplits.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L778-L782"},"Polynomial.intCast_comp":{"signature":"theorem Polynomial.intCast_comp {R : Type u} [Ring R] {p : Polynomial R} (i : \u2124) :(\u2191i).comp p = \u2191i","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L766-L767"},"Polynomial.degree_restriction":{"signature":"theorem Polynomial.degree_restriction {R : Type u} [Ring R] {p : Polynomial R} :p.restriction.degree = p.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Basic.lean#L329-L330"},"Polynomial.natDegree_restriction":{"signature":"theorem Polynomial.natDegree_restriction {R : Type u} [Ring R] {p : Polynomial R} :p.restriction.natDegree = p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Basic.lean#L332-L334"},"Polynomial.monic_restriction":{"signature":"theorem Polynomial.monic_restriction {R : Type u} [Ring R] {p : Polynomial R} :p.restriction.Monic \u2194 p.Monic","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Basic.lean#L336-L340"},"Polynomial.evalEval_natCast":{"signature":"theorem Polynomial.evalEval_natCast {R : Type u_1} [Semiring R] (x y : R) (n : \u2115) :evalEval x y \u2191n = \u2191n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Bivariate.lean#L64-L66"},"Polynomial.add_scaleRoots_of_natDegree_eq":{"signature":"theorem Polynomial.add_scaleRoots_of_natDegree_eq {R : Type u_1} [CommSemiring R] (p q : Polynomial R) (r : R) (h : p.natDegree = q.natDegree) :r ^ (p.natDegree - (p + q).natDegree) \u2022 (p + q).scaleRoots r = p.scaleRoots r + q.scaleRoots r","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ScaleRoots.lean#L242-L254"},"Polynomial.not_isUnit_X_pow_sub_one":{"signature":"theorem Polynomial.not_isUnit_X_pow_sub_one (R : Type u_1) [Ring R] [Nontrivial R] (n : \u2115) :\u00acIsUnit (X ^ n - 1)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Monic.lean#L445-L451"},"Polynomial.units_coeff_zero_smul":{"signature":"theorem Polynomial.units_coeff_zero_smul {R : Type u} [CommRing R] [IsDomain R] (c : (Polynomial R)\u02e3) (p : Polynomial R) :(\u2191c).coeff 0 \u2022 p = \u2191c * p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L683-L684"},"Polynomial.rootsExpandToRoots_apply":{"signature":"theorem Polynomial.rootsExpandToRoots_apply {R : Type u_1} [CommRing R] [IsDomain R] (p : \u2115) [ExpChar R p] (f : Polynomial R) [DecidableEq R] (x : \u21a5((expand R p) f).roots.toFinset) :\u2191((rootsExpandToRoots p f) x) = \u2191x ^ p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Perfect.lean#L402-L403"},"Polynomial.mul_X_pow_comp":{"signature":"theorem Polynomial.mul_X_pow_comp {R : Type u} [Semiring R] {p r : Polynomial R} {k : \u2115} :(p * X ^ k).comp r = p.comp r * r ^ k","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L429-L433"},"Polynomial.mkDerivationEquiv_apply":{"signature":"theorem Polynomial.mkDerivationEquiv_apply (R : Type u_1) {A : Type u_2} [CommSemiring R] [AddCommMonoid A] [Module R A] [Module (Polynomial R) A] [IsScalarTower R (Polynomial R) A] (a : A) :(mkDerivationEquiv R) a = (mkDerivation R) a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivation.lean#L92-L94"},"Polynomial.monic_generator_eq_minpoly":{"signature":"theorem Polynomial.monic_generator_eq_minpoly (\ud835\udd5c : Type u_1) {A : Type u_2} [Field \ud835\udd5c] [Ring A] [Algebra \ud835\udd5c A] (a : A) (p : Polynomial \ud835\udd5c) (p_monic : p.Monic) (p_gen : Ideal.span {p} = annIdeal \ud835\udd5c a) :annIdealGenerator \ud835\udd5c a = p","doc":"If a monic generates the annihilating ideal, it must match our choice of the annihilating ideal generator.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/LinearAlgebra/AnnihilatingPolynomial.lean#L155-L164"},"Polynomial.algEquivCMulXAddC_apply":{"signature":"theorem Polynomial.algEquivCMulXAddC_apply {R : Type u_3} [CommRing R] (a b : R) [Invertible a] (a\u271d : Polynomial R) :(algEquivCMulXAddC a b) a\u271d = (aeval (C a * X + C b)) a\u271d","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L341-L341"},"Polynomial.eval_eq_prod_roots_sub_of_monic_of_splits_id":{"signature":"theorem Polynomial.eval_eq_prod_roots_sub_of_monic_of_splits_id {R : Type u_1} [CommRing R] {f : Polynomial R} [IsDomain R] (hf : f.Splits) (hm : f.Monic) (x : R) :eval x f = (Multiset.map (fun (x_1 : R) => x - x_1) f.roots).prod","doc":"Alias of Polynomial.Splits.eval_eq_prod_roots_of_monic.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L817-L818"},"Polynomial.degreeLE_mono":{"signature":"theorem Polynomial.degreeLE_mono {R : Type u} [Semiring R] {m n : WithBot \u2115} (H : m \u2264 n) :degreeLE R m \u2264 degreeLE R n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Basic.lean#L65-L67"},"Polynomial.degree_neg_le_of_le":{"signature":"theorem Polynomial.degree_neg_le_of_le {R : Type u} [Ring R] {a : WithBot \u2115} {p : Polynomial R} (hp : p.degree \u2264 a) :(-p).degree \u2264 a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L280-L281"},"Polynomial.algEquivAevalXAddC_apply":{"signature":"theorem Polynomial.algEquivAevalXAddC_apply {R : Type u_3} [CommRing R] (t : R) (a : Polynomial R) :(algEquivAevalXAddC t) a = (aeval (X + C t)) a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L355-L355"},"Polynomial.ofFn_zero'":{"signature":"theorem Polynomial.ofFn_zero' {R : Type u_1} [Semiring R] [DecidableEq R] (v : Fin 0 \u2192 R) :(ofFn 0) v = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/OfFn.lean#L59-L60"},"Polynomial.nextCoeffUp_of_constantCoeff_eq_zero":{"signature":"theorem Polynomial.nextCoeffUp_of_constantCoeff_eq_zero {R : Type u} [Semiring R] (p : Polynomial R) (hp : p.coeff 0 = 0) :p.nextCoeffUp = p.coeff (p.natTrailingDegree + 1)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L411-L415"},"Polynomial.divByMonic_one":{"signature":"theorem Polynomial.divByMonic_one {R : Type u} [Ring R] (p : Polynomial R) :p /\u2098 1 = p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L422-L424"},"Polynomial.evalEval_sub":{"signature":"theorem Polynomial.evalEval_sub {R : Type u_1} [Ring R] (x y : R) (p q : Polynomial (Polynomial R)) :evalEval x y (p - q) = evalEval x y p - evalEval x y q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Bivariate.lean#L103-L106"},"Polynomial.monic_geom_sum_X":{"signature":"theorem Polynomial.monic_geom_sum_X {R : Type u} [Semiring R] {n : \u2115} (hn : n \u2260 0) :(\u2211 i \u2208 Finset.range n, X ^ i).Monic","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Basic.lean#L282-L285"},"Polynomial.evalEval_add":{"signature":"theorem Polynomial.evalEval_add {R : Type u_1} [Semiring R] (x y : R) (p q : Polynomial (Polynomial R)) :evalEval x y (p + q) = evalEval x y p + evalEval x y q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Bivariate.lean#L72-L75"},"Polynomial.nthRoots_two_one":{"signature":"theorem Polynomial.nthRoots_two_one {R : Type u} [CommRing R] [IsDomain R] :nthRoots 2 1 = {-1, 1}","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L416-L419"},"Polynomial.homogenize_monomial_of_lt":{"signature":"theorem Polynomial.homogenize_monomial_of_lt {R : Type u_1} [CommSemiring R] {m n : \u2115} (h : n < m) (r : R) :((monomial m) r).homogenize n = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Homogenize.lean#L86-L90"},"Polynomial.coeff_homogenize":{"signature":"theorem Polynomial.coeff_homogenize {R : Type u_1} [CommSemiring R] (p : Polynomial R) (n : \u2115) (m : Fin 2 \u2192\u2080 \u2115) :MvPolynomial.coeff m (p.homogenize n) = if m 0 + m 1 = n then p.coeff (m 0) else 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Homogenize.lean#L111-L121"},"Polynomial.ofMultiset_apply":{"signature":"theorem Polynomial.ofMultiset_apply {R : Type u} [CommRing R] (s : Multiset R) :ofMultiset s = (Multiset.map (fun (a : R) => X - C a) s).prod","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L1195-L1195"},"Polynomial.not_dvd_of_degree_lt":{"signature":"theorem Polynomial.not_dvd_of_degree_lt {R : Type u} [Semiring R] [NoZeroDivisors R] {p q : Polynomial R} (h0 : q \u2260 0) (hl : q.degree < p.degree) :\u00acp \u2223 q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Domain.lean#L79-L81"},"Polynomial.fwdDiff_iter_eq_zero_of_degree_lt":{"signature":"theorem Polynomial.fwdDiff_iter_eq_zero_of_degree_lt {R : Type u_3} [CommRing R] {P : Polynomial R} {n : \u2115} (hP : P.natDegree < n) :((fwdDiff 1)^[n] fun (x : R) => eval x P) = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Group/ForwardDiff.lean#L270-L275"},"Polynomial.fwdDiff_iter_degree_add_one_eq_zero":{"signature":"theorem Polynomial.fwdDiff_iter_degree_add_one_eq_zero {R : Type u_3} [CommRing R] (P : Polynomial R) :((fwdDiff 1)^[P.natDegree + 1] fun (x : R) => eval x P) = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Group/ForwardDiff.lean#L277-L280"},"Polynomial.rootsExpandPowToRoots_apply":{"signature":"theorem Polynomial.rootsExpandPowToRoots_apply {R : Type u_1} [CommRing R] [IsDomain R] (p n : \u2115) [ExpChar R p] (f : Polynomial R) [DecidableEq R] (x : \u21a5((expand R (p ^ n)) f).roots.toFinset) :\u2191((rootsExpandPowToRoots p n f) x) = \u2191x ^ p ^ n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Perfect.lean#L415-L416"},"Polynomial.cyclotomic_eq_prod_X_pow_sub_one_pow_moebius":{"signature":"theorem Polynomial.cyclotomic_eq_prod_X_pow_sub_one_pow_moebius {n : \u2115} (R : Type u_1) [CommRing R] [IsDomain R] :(algebraMap (Polynomial R) (RatFunc R)) (cyclotomic n R) = \u220f i \u2208 n.divisorsAntidiagonal, (algebraMap (Polynomial R) (RatFunc R)) (X ^ i.2 - 1) ^ ArithmeticFunction.moebius i.1","doc":"cyclotomic n R can be expressed as a product in a fraction field of R[X] using M\u00f6bius inversion.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean#L422-L438"},"Polynomial.support_X_empty":{"signature":"theorem Polynomial.support_X_empty {R : Type u} [Semiring R] (H : 1 = 0) :X.support = \u2205","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L805-L806"},"Polynomial.support_update_zero":{"signature":"theorem Polynomial.support_update_zero {R : Type u} [Semiring R] (p : Polynomial R) (n : \u2115) :(p.update n 0).support = p.support.erase n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L1005-L1006"},"Polynomial.eval_map_apply":{"signature":"theorem Polynomial.eval_map_apply {R : Type u} {S : Type v} [Semiring R] {p : Polynomial R} [Semiring S] (f : R \u2192+* S) (x : R) :eval (f x) (map f p) = f (eval x p)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L568-L569"},"Polynomial.algEquivOfCompEqX_symm_apply":{"signature":"theorem Polynomial.algEquivOfCompEqX_symm_apply {R : Type u} [CommSemiring R] (p q : Polynomial R) (hpq : p.comp q = X) (hqp : q.comp p = X) (a : Polynomial R) :(p.algEquivOfCompEqX q hpq hqp).symm a = (aeval q) a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L324-L324"},"Polynomial.derivative'_apply":{"signature":"theorem Polynomial.derivative'_apply {R : Type u_1} [CommSemiring R] (a : Polynomial R) :derivative' a = derivative a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivation.lean#L33-L33"},"Polynomial.card_roots_map_le_degree":{"signature":"theorem Polynomial.card_roots_map_le_degree {A : Type u_3} {B : Type u_4} [Semiring A] [CommRing B] [IsDomain B] {f : A \u2192+* B} (p : Polynomial A) (hp0 : p \u2260 0) :\u2191(map f p).roots.card \u2264 p.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L914-L918"},"Polynomial.splits_X":{"signature":"theorem Polynomial.splits_X {R : Type u_1} [Semiring R] :X.Splits","doc":"Alias of Polynomial.Splits.X.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L695-L696"},"Polynomial.derivative_C_mul_X":{"signature":"theorem Polynomial.derivative_C_mul_X {R : Type u} [Semiring R] (a : R) :derivative (C a * X) = C a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L89-L90"},"Polynomial.eq_of_degree_le_of_eval_index_eq":{"signature":"theorem Polynomial.eq_of_degree_le_of_eval_index_eq {R : Type u_1} [CommRing R] [IsDomain R] {f g : Polynomial R} {\u03b9 : Type u_2} {v : \u03b9 \u2192 R} (s : Finset \u03b9) (hvs : Set.InjOn v \u2191s) (h_deg_le : f.degree \u2264 \u2191s.card) (h_deg_eq : f.degree = g.degree) (hlc : f.leadingCoeff = g.leadingCoeff) (h_eval : \u2200 i \u2208 s, eval (v i) f = eval (v i) g) :f = g","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/LinearAlgebra/Lagrange.lean#L117-L126"},"Polynomial.leadingCoeff_X_pow_add_one":{"signature":"theorem Polynomial.leadingCoeff_X_pow_add_one {R : Type u} [Semiring R] {n : \u2115} (hn : 0 < n) :(X ^ n + 1).leadingCoeff = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L662-L664"},"Polynomial.resultant_C_left":{"signature":"theorem Polynomial.resultant_C_left {R : Type u_1} [CommRing R] (g : Polynomial R) (m n : \u2115) (r : R) :(C r).resultant g m n = (-1) ^ (m * n) * g.coeff n ^ m * r ^ n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L362-L366"},"Polynomial.mod_eq_self_iff":{"signature":"theorem Polynomial.mod_eq_self_iff {R : Type u} [Field R] {p q : Polynomial R} (hq0 : q \u2260 0) :p % q = p \u2194 p.degree < q.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L367-L375"},"Polynomial.div_eq_zero_iff":{"signature":"theorem Polynomial.div_eq_zero_iff {R : Type u} [Field R] {p q : Polynomial R} (hq0 : q \u2260 0) :p / q = 0 \u2194 p.degree < q.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L377-L385"},"Polynomial.isCoprime_scaleRoots":{"signature":"theorem Polynomial.isCoprime_scaleRoots {R : Type u_1} [CommSemiring R] (p q : Polynomial R) (r : R) (hr : IsUnit r) (h : IsCoprime p q) :IsCoprime (p.scaleRoots r) (q.scaleRoots r)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ScaleRoots.lean#L276-L287"},"Polynomial.signVariations_eraseLead":{"signature":"theorem Polynomial.signVariations_eraseLead {R : Type u_1} [Semiring R] [LinearOrder R] (P : Polynomial R) (h : SignType.sign P.leadingCoeff = SignType.sign P.nextCoeff) :P.eraseLead.signVariations = P.signVariations","doc":"If the first two signs are the same, then sign_variations is unchanged by eraseLead","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/RuleOfSigns.lean#L67-L75"},"Polynomial.support_X":{"signature":"theorem Polynomial.support_X {R : Type u} [Semiring R] (H : \u00ac1 = 0) :X.support = {1}","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L808-L809"},"Polynomial.natDegree_cubic_le":{"signature":"theorem Polynomial.natDegree_cubic_le {R : Type u} {a b c d : R} [Semiring R] :(C a * X ^ 3 + C b * X ^ 2 + C c * X + C d).natDegree \u2264 3","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/SmallDegree.lean#L138-L139"},"Polynomial.degree_pos_of_aeval_root":{"signature":"theorem Polynomial.degree_pos_of_aeval_root {R : Type u} {S : Type v} [CommRing R] [Semiring S] [Algebra R S] {p : Polynomial R} (hp : p \u2260 0) {z : S} (hz : (aeval z) p = 0) (inj : \u2200 (x : R), (algebraMap R S) x = 0 \u2192 x = 0) :0 < p.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/RingDivision.lean#L48-L50"},"Polynomial.coeff_sub_eq_left_of_lt":{"signature":"theorem Polynomial.coeff_sub_eq_left_of_lt {R : Type u} {n : \u2115} [Ring R] {p q : Polynomial R} (dg : q.natDegree < n) :(p - q).coeff n = p.coeff n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L331-L333"},"Polynomial.leadingCoeff_opRingEquiv":{"signature":"theorem Polynomial.leadingCoeff_opRingEquiv {R : Type u_1} [Semiring R] (p : (Polynomial R)\u1d50\u1d52\u1d56) :((opRingEquiv R) p).leadingCoeff = MulOpposite.op (MulOpposite.unop p).leadingCoeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Opposites.lean#L96-L99"},"Polynomial.trailingCoeff_mul":{"signature":"theorem Polynomial.trailingCoeff_mul {R : Type u_2} [Semiring R] [NoZeroDivisors R] (p q : Polynomial R) :(p * q).trailingCoeff = p.trailingCoeff * q.trailingCoeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Reverse.lean#L285-L288"},"Polynomial.coe_eq_one_iff":{"signature":"theorem Polynomial.coe_eq_one_iff {R : Type u_1} [Semiring R] {\u03c6 : Polynomial R} :\u2191\u03c6 = 1 \u2194 \u03c6 = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/PowerSeries/Basic.lean#L875-L876"},"Polynomial.card_roots_le_one_of_irreducible":{"signature":"theorem Polynomial.card_roots_le_one_of_irreducible {R : Type u} [CommRing R] [IsDomain R] {p : Polynomial R} (hirr : Irreducible p) :p.roots.card \u2264 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L826-L831"},"Polynomial.coe_neg":{"signature":"theorem Polynomial.coe_neg {R : Type u_1} [CommRing R] (p : Polynomial R) :\u2191(-p) = -\u2191p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/PowerSeries/Basic.lean#L947-L949"},"Polynomial.dvd_mul_leadingCoeff_inv":{"signature":"theorem Polynomial.dvd_mul_leadingCoeff_inv {K : Type u_1} [DivisionRing K] {p q : Polynomial K} (hp0 : p \u2260 0) :q \u2223 p * C p.leadingCoeff\u207b\u00b9 \u2194 q \u2223 p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L445-L447"},"Polynomial.splits_of_splits_gcd_left":{"signature":"theorem Polynomial.splits_of_splits_gcd_left {K : Type v} [Field K] [DecidableEq K] {f g : Polynomial K} (hf0 : f \u2260 0) (hf : f.Splits) :(EuclideanDomain.gcd f g).Splits","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L756-L759"},"Polynomial.leadingCoeff_sum_of_degree_eq":{"signature":"theorem Polynomial.leadingCoeff_sum_of_degree_eq {\u03b9 : Type w} {S : Type u_1} [Semiring S] {f : \u03b9 \u2192 Polynomial S} {s : Finset \u03b9} {d : WithBot \u2115} (hd : \u2200 k \u2208 s, (f k).degree = d) (hf : \u2211 k \u2208 s, (f k).leadingCoeff \u2260 0) :(\u2211 k \u2208 s, f k).leadingCoeff = \u2211 k \u2208 s, (f k).leadingCoeff","doc":"The leading coefficient of a sum of polynomials with the same degree is the sum of the leading coefficients, provided that this sum is nonzero.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/BigOperators.lean#L67-L79"},"Polynomial.support_C_subset":{"signature":"theorem Polynomial.support_C_subset {R : Type u} [Semiring R] (a : R) :(C a).support \u2286 {0}","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L751-L752"},"Polynomial.eval_surjective":{"signature":"theorem Polynomial.eval_surjective {R : Type u} [Semiring R] (x : R) :Function.Surjective (eval x)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L362-L362"},"Polynomial.map_contract":{"signature":"theorem Polynomial.map_contract {R : Type u} [CommSemiring R] {S : Type v} [CommSemiring S] {p : \u2115} (hp : p \u2260 0) {f : R \u2192+* S} {q : Polynomial R} :map f (contract p q) = contract p (map f q)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Expand.lean#L197-L199"},"Polynomial.algHom_ext'_iff":{"signature":"theorem Polynomial.algHom_ext'_iff {R : Type u} {A : Type z} {B : Type u_2} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] {f g : Polynomial A \u2192\u2090[R] B} :f = g \u2194 f.comp CAlgHom = g.comp CAlgHom \u2227 f X = g X","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L101-L101"},"Polynomial.comp_neg_X_comp_neg_X":{"signature":"theorem Polynomial.comp_neg_X_comp_neg_X {R : Type u_3} [CommRing R] (p : Polynomial R) :(p.comp (-X)).comp (-X) = p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L374-L377"},"Polynomial.contentIdeal_eq_top_of_contentIdeal_mul_eq_top":{"signature":"theorem Polynomial.contentIdeal_eq_top_of_contentIdeal_mul_eq_top {R : Type u_3} [CommSemiring R] {p q : Polynomial R} (h : (p * q).contentIdeal = \u22a4) :p.contentIdeal = \u22a4","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ContentIdeal.lean#L158-L164"},"Polynomial.cyclotomic_prime_mul_X_sub_one":{"signature":"theorem Polynomial.cyclotomic_prime_mul_X_sub_one (R : Type u_1) [Ring R] (p : \u2115) [hn : Fact (Nat.Prime p)] :cyclotomic p R * (X - 1) = X ^ p - 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean#L375-L376"},"Polynomial.aevalTower_id":{"signature":"theorem Polynomial.aevalTower_id {S : Type v} [CommSemiring S] :aevalTower (AlgHom.id S S) = aeval","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L552-L555"},"Polynomial.contDiff_aeval":{"signature":"theorem Polynomial.contDiff_aeval {R : Type u_1} {\ud835\udd5c : Type u_2} [CommSemiring R] [NontriviallyNormedField \ud835\udd5c] [Algebra R \ud835\udd5c] (f : Polynomial R) (n : WithTop \u2115\u221e) :ContDiff \ud835\udd5c n fun (x : \ud835\udd5c) => (aeval x) f","doc":"Polynomials are smooth","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Calculus/ContDiff/Polynomial.lean#L21-L26"},"Polynomial.eraseLead_X":{"signature":"theorem Polynomial.eraseLead_X {R : Type u_1} [Semiring R] :X.eraseLead = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/EraseLead.lean#L142-L144"},"Polynomial.toFn_zero":{"signature":"theorem Polynomial.toFn_zero {R : Type u_1} [Semiring R] (n : \u2115) :(toFn n) 0 = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/OfFn.lean#L36-L36"},"Polynomial.unique_int_coeff_of_cycl":{"signature":"theorem Polynomial.unique_int_coeff_of_cycl {K : Type u_2} [CommRing K] [IsDomain K] [CharZero K] {\u03b6 : K} {n : \u2115+} (h : IsPrimitiveRoot \u03b6 \u2191n) :\u2203! P : Polynomial \u2124, map (Int.castRingHom K) P = cyclotomic' (\u2191n) K","doc":"If K is of characteristic 0 and there is a primitive n-th root of unity in K, then cyclotomic n K comes from a unique polynomial with integer coefficients.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean#L213-L221"},"Polynomial.bernoulli_eval_neg":{"signature":"theorem Polynomial.bernoulli_eval_neg (n : \u2115) (x : \u211a) :eval (-x) (bernoulli n) = (-1) ^ n * (eval x (bernoulli n) + \u2191n * x ^ (n - 1))","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/NumberTheory/BernoulliPolynomials.lean#L243-L245"},"Polynomial.smeval_mul_X_pow":{"signature":"theorem Polynomial.smeval_mul_X_pow (R : Type u_1) [Semiring R] (p : Polynomial R) {S : Type u_2} [NonAssocSemiring S] [Module R S] [Pow S \u2115] (x : S) [NatPowAssoc S] [IsScalarTower R S S] (n : \u2115) :(p * X ^ n).smeval x = p.smeval x * x ^ n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Smeval.lean#L257-L262"},"Polynomial.aeval_derivative_of_splits":{"signature":"theorem Polynomial.aeval_derivative_of_splits {R : Type u_1} [CommRing R] {f : Polynomial R} [IsDomain R] [DecidableEq R] (hf : f.Splits) (x : R) :eval x (derivative f) = f.leadingCoeff * (Multiset.map (fun (a : R) => (Multiset.map (fun (x_1 : R) => x - x_1) (f.roots.erase a)).prod) f.roots).sum","doc":"Alias of Polynomial.Splits.eval_derivative.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L868-L869"},"Polynomial.exists_max_root":{"signature":"theorem Polynomial.exists_max_root {R : Type u} [CommRing R] [IsDomain R] [LinearOrder R] (p : Polynomial R) (hp : p \u2260 0) :\u2203 (x\u2080 : R), \u2200 (x : R), p.IsRoot x \u2192 x \u2264 x\u2080","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L148-L149"},"Polynomial.update_eq_add_sub_coeff":{"signature":"theorem Polynomial.update_eq_add_sub_coeff {R : Type u_1} [Ring R] (p : Polynomial R) (n : \u2115) (a : R) :p.update n a = p + C (a - p.coeff n) * X ^ n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L332-L336"},"Polynomial.supDegree_eq_degree":{"signature":"theorem Polynomial.supDegree_eq_degree {R : Type u} [Semiring R] (p : Polynomial R) :AddMonoidAlgebra.supDegree WithBot.some p.toFinsupp = p.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L39-L40"},"Polynomial.degree_leadingCoeff_inv":{"signature":"theorem Polynomial.degree_leadingCoeff_inv {K : Type u_1} [DivisionRing K] {p : Polynomial K} (hp0 : p \u2260 0) :(C p.leadingCoeff\u207b\u00b9).degree = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L454-L456"},"Polynomial.trailingDegree_lt_wf":{"signature":"theorem Polynomial.trailingDegree_lt_wf {R : Type u} [Semiring R] :WellFounded fun (p q : Polynomial R) => p.trailingDegree < q.trailingDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L51-L52"},"Polynomial.coeff_mul_intCast":{"signature":"theorem Polynomial.coeff_mul_intCast {S : Type v} [Ring S] {p : Polynomial S} {a : \u2124} {k : \u2115} :(p * \u2191a).coeff k = p.coeff k * \u2191a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L179-L180"},"Polynomial.abs_tendsto_atTop_iff":{"signature":"theorem Polynomial.abs_tendsto_atTop_iff {\ud835\udd5c : Type u_1} [NormedField \ud835\udd5c] [LinearOrder \ud835\udd5c] [IsStrictOrderedRing \ud835\udd5c] (P : Polynomial \ud835\udd5c) [OrderTopology \ud835\udd5c] :Filter.Tendsto (fun (x : \ud835\udd5c) => |eval x P|) Filter.atTop Filter.atTop \u2194 0 < P.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/Basic.lean#L94-L96"},"Polynomial.exists_root_of_splits":{"signature":"theorem Polynomial.exists_root_of_splits {R : Type u_1} [CommRing R] {f : Polynomial R} (hf : f.Splits) (hf0 : f.degree \u2260 0) :\u2203 (a : R), eval a f = 0","doc":"Alias of Polynomial.Splits.exists_eval_eq_zero.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L769-L770"},"Polynomial.restriction_zero":{"signature":"theorem Polynomial.restriction_zero {R : Type u} [Ring R] :restriction 0 = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Basic.lean#L342-L344"},"Polynomial.C_smul_derivation_apply":{"signature":"theorem Polynomial.C_smul_derivation_apply {R : Type u_1} {A : Type u_2} [CommSemiring R] [AddCommMonoid A] [Module R A] [Module (Polynomial R) A] (D : Derivation R (Polynomial R) A) (a : R) (f : Polynomial R) :C a \u2022 D f = a \u2022 D f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivation.lean#L47-L51"},"Polynomial.natDegree_lt_coeff_mul":{"signature":"theorem Polynomial.natDegree_lt_coeff_mul {R : Type u} {m n : \u2115} [Semiring R] {p q : Polynomial R} (h : p.natDegree + q.natDegree < m + n) :(p * q).coeff (m + n) = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L147-L149"},"Polynomial.splits_of_splits_gcd_right":{"signature":"theorem Polynomial.splits_of_splits_gcd_right {K : Type v} [Field K] [DecidableEq K] {f g : Polynomial K} (hg0 : g \u2260 0) (hg : g.Splits) :(EuclideanDomain.gcd f g).Splits","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L761-L764"},"Polynomial.valuation_eq_valuation_X_pow_natDegree_of_one_lt_valuation_X":{"signature":"theorem Polynomial.valuation_eq_valuation_X_pow_natDegree_of_one_lt_valuation_X (K : Type u_1) [Field K] {\u0393 : Type u_2} [LinearOrderedCommGroupWithZero \u0393] {v : Valuation (RatFunc K) \u0393} (hv : \u2200 (a : K), a \u2260 0 \u2192 v \u2191(C a) = 1) (hlt : 1 < v RatFunc.X) {p : Polynomial K} (hp : p \u2260 0) :v \u2191p = v RatFunc.X ^ p.natDegree","doc":"If a valuation v is trivial on constants and 1 < v RatFunc.X then for every polynomial p, v p = v RatFunc.X ^ p.natDegree. Note: The condition 1 < v RatFunc.X is typically satisfied by the valuation at infinity.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/RatFunc/AsPolynomial.lean#L307-L317"},"Polynomial.iterate_derivative_eq_zero_of_degree_lt":{"signature":"theorem Polynomial.iterate_derivative_eq_zero_of_degree_lt {R : Type u} [CommRing R] {k : \u2115} {P : Polynomial R} (h : P.degree < \u2191k) :(\u21d1derivative)^[k] P = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L679-L695"},"Polynomial.dvd_C_mul":{"signature":"theorem Polynomial.dvd_C_mul {R : Type u} {a : R} [Field R] {p q : Polynomial R} (ha : a \u2260 0) :p \u2223 C a * q \u2194 p \u2223 q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L584-L589"},"Polynomial.root_or_root_of_root_mul":{"signature":"theorem Polynomial.root_or_root_of_root_mul {R : Type u} {a : R} [CommSemiring R] {p q : Polynomial R} [NoZeroDivisors R] (h : (p * q).IsRoot a) :p.IsRoot a \u2228 q.IsRoot a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L697-L698"},"Polynomial.algEquivOfCompEqX_eq_iff":{"signature":"theorem Polynomial.algEquivOfCompEqX_eq_iff {R : Type u} [CommSemiring R] (p q p' q' : Polynomial R) (hpq : p.comp q = X) (hqp : q.comp p = X) (hpq' : p'.comp q' = X) (hqp' : q'.comp p' = X) :p.algEquivOfCompEqX q hpq hqp = p'.algEquivOfCompEqX q' hpq' hqp' \u2194 p = p'","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L329-L333"},"Polynomial.algEquivAevalXAddC_eq_iff":{"signature":"theorem Polynomial.algEquivAevalXAddC_eq_iff {R : Type u_3} [CommRing R] (t t' : R) :algEquivAevalXAddC t = algEquivAevalXAddC t' \u2194 t = t'","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L359-L362"},"Polynomial.prod_roots_eq_coeff_zero_of_monic_of_splits":{"signature":"theorem Polynomial.prod_roots_eq_coeff_zero_of_monic_of_splits {R : Type u_1} [CommRing R] {f : Polynomial R} [IsDomain R] (hf : f.Splits) (hm : f.Monic) :f.coeff 0 = (-1) ^ f.natDegree * f.roots.prod","doc":"Alias of Polynomial.Splits.coeff_zero_eq_prod_roots_of_monic. Alias of Polynomial.Splits.coeff_zero_eq_prod_roots_of_monic. If f is a monic polynomial that splits, then coeff f 0 equals the product of the roots.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L897-L898"},"Polynomial.resultant_one_left":{"signature":"theorem Polynomial.resultant_one_left {R : Type u_1} [CommRing R] (g : Polynomial R) (m n : \u2115) :resultant 1 g m n = (-1) ^ (m * n) * g.coeff n ^ m","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L372-L373"},"Polynomial.coeff_hermite_of_odd_add":{"signature":"theorem Polynomial.coeff_hermite_of_odd_add {n k : \u2115} (hnk : Odd (n + k)) :(hermite n).coeff k = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Hermite/Basic.lean#L119-L133"},"Polynomial.hilbertPoly_add_left":{"signature":"theorem Polynomial.hilbertPoly_add_left {F : Type u_1} [Field F] (p q : Polynomial F) (d : \u2115) :(p + q).hilbertPoly d = p.hilbertPoly d + q.hilbertPoly d","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/HilbertPoly.lean#L128-L136"},"Polynomial.adjoin_rootSet_eq_range":{"signature":"theorem Polynomial.adjoin_rootSet_eq_range {R : Type u_1} [CommRing R] {f : Polynomial R} {A : Type u_2} {B : Type u_3} [CommRing A] [CommRing B] [IsDomain A] [IsDomain B] [Algebra R A] [Algebra R B] [IsSimpleRing A] (hf : (map (algebraMap R A) f).Splits) (g : A \u2192\u2090[R] B) :Algebra.adjoin R (f.rootSet B) = g.range \u2194 Algebra.adjoin R (f.rootSet A) = \u22a4","doc":"Alias of Polynomial.Splits.adjoin_rootSet_eq_range.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L796-L797"},"Polynomial.natDegree_sub_eq_left_of_natDegree_lt":{"signature":"theorem Polynomial.natDegree_sub_eq_left_of_natDegree_lt {R : Type u} [Ring R] {p q : Polynomial R} (h : q.natDegree < p.natDegree) :(p - q).natDegree = p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L581-L583"},"Polynomial.eval\u2082_list_prod":{"signature":"theorem Polynomial.eval\u2082_list_prod {R : Type u} {S : Type v} [Semiring R] [CommSemiring S] (f : R \u2192+* S) (l : List (Polynomial R)) (x : S) :eval\u2082 f x l.prod = (List.map (eval\u2082 f x) l).prod","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L231-L232"},"Polynomial.smeval_X_pow":{"signature":"theorem Polynomial.smeval_X_pow (R : Type u_1) [Semiring R] {S : Type u_2} [AddCommMonoid S] [Pow S \u2115] [MulActionWithZero R S] (x : S) {n : \u2115} :(X ^ n).smeval x = x ^ n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Smeval.lean#L96-L99"},"Polynomial.rootMultiplicity_expand":{"signature":"theorem Polynomial.rootMultiplicity_expand {R : Type u} [CommRing R] {p : \u2115} [ExpChar R p] {f : Polynomial R} {r : R} :rootMultiplicity r ((expand R p) f) = p * rootMultiplicity (r ^ p) f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Expand.lean#L302-L304"},"Polynomial.orderOf_root_cyclotomic_dvd":{"signature":"theorem Polynomial.orderOf_root_cyclotomic_dvd {n : \u2115} (hpos : 0 < n) {p : \u2115} [Fact (Nat.Prime p)] {a : \u2115} (hroot : (cyclotomic n (ZMod p)).IsRoot ((Nat.castRingHom (ZMod p)) a)) :orderOf (ZMod.unitOfCoprime a \u22ef) \u2223 n","doc":"If (a : \u2115) is a root of cyclotomic n (ZMod p), then the multiplicative order of a modulo p divides n.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean#L583-L595"},"Polynomial.comp_one":{"signature":"theorem Polynomial.comp_one {R : Type u} [Semiring R] {p : Polynomial R} :p.comp 1 = C (eval 1 p)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L403-L404"},"Polynomial.resultant_C_zero_left":{"signature":"theorem Polynomial.resultant_C_zero_left {R : Type u_1} [CommRing R] (g : Polynomial R) (r : R) (m : \u2115) :(C r).resultant g 0 m = r ^ m","doc":"For polynomial f and constant a, Res(f, a) = a ^ m.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L152-L153"},"Polynomial.sumIDeriv_apply_of_lt":{"signature":"theorem Polynomial.sumIDeriv_apply_of_lt {R : Type u_1} [Semiring R] {p : Polynomial R} {n : \u2115} (hn : p.natDegree < n) :sumIDeriv p = \u2211 i \u2208 Finset.range n, (\u21d1derivative)^[i] p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/SumIteratedDerivative.lean#L70-L73"},"Polynomial.monomial_zero_one":{"signature":"theorem Polynomial.monomial_zero_one {R : Type u} [Semiring R] :(monomial 0) 1 = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L395-L396"},"Polynomial.exists_irreducible_of_natDegree_ne_zero":{"signature":"theorem Polynomial.exists_irreducible_of_natDegree_ne_zero {R : Type u_1} [CommSemiring R] [NoZeroDivisors R] [WfDvdMonoid R] {f : Polynomial R} [Nontrivial R] (hf : f.natDegree \u2260 0) :\u2203 (g : Polynomial R), Irreducible g \u2227 g \u2223 f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/UniqueFactorization.lean#L79-L81"},"Polynomial.differentiableWithinAt":{"signature":"theorem Polynomial.differentiableWithinAt {\ud835\udd5c : Type u} [NontriviallyNormedField \ud835\udd5c] {x : \ud835\udd5c} {s : Set \ud835\udd5c} (p : Polynomial \ud835\udd5c) :DifferentiableWithinAt \ud835\udd5c (fun (x : \ud835\udd5c) => eval x p) s x","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Calculus/Deriv/Polynomial.lean#L87-L88"},"Polynomial.natTrailingDegree_le_natTrailingDegree":{"signature":"theorem Polynomial.natTrailingDegree_le_natTrailingDegree {R : Type u} [Semiring R] {p q : Polynomial R} (hq : q \u2260 0) (hpq : p.trailingDegree \u2264 q.trailingDegree) :p.natTrailingDegree \u2264 q.natTrailingDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L165-L167"},"Polynomial.support_C_mul_X'":{"signature":"theorem Polynomial.support_C_mul_X' {R : Type u} [Semiring R] (c : R) :(C c * X).support \u2286 {1}","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L757-L758"},"Polynomial.derivation_ext_iff":{"signature":"theorem Polynomial.derivation_ext_iff {R : Type u_1} {A : Type u_2} [CommSemiring R] [AddCommMonoid A] [Module R A] [Module (Polynomial R) A] {D\u2081 D\u2082 : Derivation R (Polynomial R) A} :D\u2081 = D\u2082 \u2194 D\u2081 X = D\u2082 X","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivation.lean#L53-L53"},"Polynomial.modByMonic_eq_zero_iff_quotient_eq_zero":{"signature":"theorem Polynomial.modByMonic_eq_zero_iff_quotient_eq_zero {R : Type u_1} [CommRing R] (p q : Polynomial R) (hq : q.Monic) :p %\u2098 q = 0 \u2194 (Ideal.Quotient.mk (Ideal.span {q})) p = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Quotient.lean#L67-L69"},"Polynomial.splits_iff":{"signature":"theorem Polynomial.splits_iff {R : Type u_1} [Field R] {f : Polynomial R} :f.Splits \u2194 f = 0 \u2228 \u2200 {g : Polynomial R}, Irreducible g \u2192 g \u2223 f \u2192 g.degree = 1","doc":"This lemma is for polynomials over a field.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L742-L744"},"Polynomial.resultant_eq_zero_of_lt_lt":{"signature":"theorem Polynomial.resultant_eq_zero_of_lt_lt {R : Type u_1} [CommRing R] (f g : Polynomial R) (m n : \u2115) (hf : f.natDegree < m) (hg : g.natDegree < n) :f.resultant g m n = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L355-L360"},"Polynomial.gaussNorm_nonneg":{"signature":"theorem Polynomial.gaussNorm_nonneg {R : Type u_1} {F : Type u_2} [Semiring R] [FunLike F R \u211d] (v : F) {c : \u211d} (p : Polynomial R) (hc : 0 \u2264 c) [NonnegHomClass F R \u211d] :0 \u2264 gaussNorm v c p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/GaussNorm.lean#L117-L119"},"Polynomial.degree_prod_of_monic":{"signature":"theorem Polynomial.degree_prod_of_monic {R : Type u} {\u03b9 : Type w} (s : Finset \u03b9) [CommSemiring R] (f : \u03b9 \u2192 Polynomial R) [Nontrivial R] (h : \u2200 i \u2208 s, (f i).Monic) :(\u220f i \u2208 s, f i).degree = \u2211 i \u2208 s, (f i).degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/BigOperators.lean#L218-L220"},"Polynomial.mkDerivationEquiv_symm_apply":{"signature":"theorem Polynomial.mkDerivationEquiv_symm_apply (R : Type u_1) {A : Type u_2} [CommSemiring R] [AddCommMonoid A] [Module R A] [Module (Polynomial R) A] [IsScalarTower R (Polynomial R) A] (D : Derivation R (Polynomial R) A) :(mkDerivationEquiv R).symm D = D X","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivation.lean#L96-L97"},"Polynomial.restriction_one":{"signature":"theorem Polynomial.restriction_one {R : Type u} [Ring R] :restriction 1 = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Basic.lean#L346-L348"},"Polynomial.separable_prod":{"signature":"theorem Polynomial.separable_prod {R : Type u} [CommRing R] {\u03b9 : Type u_1} [Fintype \u03b9] {f : \u03b9 \u2192 Polynomial R} (h1 : Pairwise (Function.onFun IsCoprime f)) (h2 : \u2200 (x : \u03b9), (f x).Separable) :(\u220f x : \u03b9, f x).Separable","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Separable.lean#L221-L223"},"Polynomial.evalEval_pow":{"signature":"theorem Polynomial.evalEval_pow {R : Type u_1} [CommSemiring R] (x y : R) (p : Polynomial (Polynomial R)) (n : \u2115) :evalEval x y (p ^ n) = evalEval x y p ^ n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Bivariate.lean#L135-L137"},"Polynomial.le_gaussNorm":{"signature":"theorem Polynomial.le_gaussNorm {R : Type u_1} {F : Type u_2} [Semiring R] [FunLike F R \u211d] (v : F) {c : \u211d} (p : Polynomial R) [ZeroHomClass F R \u211d] [NonnegHomClass F R \u211d] (hc : 0 \u2264 c) (i : \u2115) :v (p.coeff i) * c ^ i \u2264 gaussNorm v c p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/GaussNorm.lean#L121-L125"},"Polynomial.sumIDeriv_C":{"signature":"theorem Polynomial.sumIDeriv_C {R : Type u_1} [Semiring R] (a : R) :sumIDeriv (C a) = C a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/SumIteratedDerivative.lean#L80-L82"},"Polynomial.integralNormalization_eval\u2082_eq_zero":{"signature":"theorem Polynomial.integralNormalization_eval\u2082_eq_zero {R : Type u} {S : Type v} [Semiring R] [CommSemiring S] {p : Polynomial R} (f : R \u2192+* S) {z : S} (hz : eval\u2082 f z p = 0) (inj : \u2200 (x : R), f x = 0 \u2192 x = 0) :eval\u2082 f (f p.leadingCoeff * z) p.integralNormalization = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/IntegralNormalization.lean#L165-L168"},"Polynomial.toFinsuppIsoLinear_apply":{"signature":"theorem Polynomial.toFinsuppIsoLinear_apply (R : Type u) [Semiring R] (a\u271d : Polynomial R) :(toFinsuppIsoLinear R) a\u271d = a\u271d.toFinsupp","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L339-L339"},"Polynomial.of_irreducible_expand_pow":{"signature":"theorem Polynomial.of_irreducible_expand_pow {R : Type u} [CommRing R] [IsDomain R] {p : \u2115} (hp : p \u2260 0) {f : Polynomial R} {n : \u2115} :Irreducible ((expand R (p ^ n)) f) \u2192 Irreducible f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Expand.lean#L325-L330"},"Polynomial.cyclotomic_dvd_geom_sum_of_dvd":{"signature":"theorem Polynomial.cyclotomic_dvd_geom_sum_of_dvd (R : Type u_1) [Ring R] {d n : \u2115} (hdn : d \u2223 n) (hd : d \u2260 1) :cyclotomic d R \u2223 \u2211 i \u2208 Finset.range n, X ^ i","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean#L385-L394"},"Polynomial.leadingCoeff_add_of_degree_eq":{"signature":"theorem Polynomial.leadingCoeff_add_of_degree_eq {R : Type u} [Semiring R] {p q : Polynomial R} (h : p.degree = q.degree) (hlc : p.leadingCoeff + q.leadingCoeff \u2260 0) :(p + q).leadingCoeff = p.leadingCoeff + q.leadingCoeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L274-L280"},"Polynomial.leadingCoeff_sub_of_degree_eq":{"signature":"theorem Polynomial.leadingCoeff_sub_of_degree_eq {R : Type u} [Ring R] {p q : Polynomial R} (h : p.degree = q.degree) (hlc : p.leadingCoeff \u2260 q.leadingCoeff) :(p - q).leadingCoeff = p.leadingCoeff - q.leadingCoeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L565-L571"},"Polynomial.disc_of_degree_eq_two":{"signature":"theorem Polynomial.disc_of_degree_eq_two {R : Type u_1} [CommRing R] {f : Polynomial R} (hf : f.degree = 2) :f.discr = f.coeff 1 ^ 2 - 4 * f.coeff 0 * f.coeff 2","doc":"Alias of Polynomial.discr_of_degree_eq_two. Standard formula for the discriminant of a quadratic polynomial.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Resultant/Basic.lean#L976-L976"},"Polynomial.forall_iff_forall_finsupp":{"signature":"theorem Polynomial.forall_iff_forall_finsupp {R : Type u} [Semiring R] (P : Polynomial R \u2192 Prop) :(\u2200 (p : Polynomial R), P p) \u2194 \u2200 (q : AddMonoidAlgebra R \u2115), P { toFinsupp := q }","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L90-L92"},"Polynomial.linearIndependent_powers_iff_aeval":{"signature":"theorem Polynomial.linearIndependent_powers_iff_aeval {R : Type u} {M : Type w} [CommRing R] [AddCommGroup M] [Module R M] (f : M \u2192\u2097[R] M) (v : M) :(LinearIndependent R fun (n : \u2115) => (f ^ n) v) \u2194 \u2200 (p : Polynomial R), ((aeval f) p) v = 0 \u2192 p = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Basic.lean#L900-L905"},"Polynomial.taylor_X_pow":{"signature":"theorem Polynomial.taylor_X_pow {R : Type u_1} [Semiring R] (r : R) (n : \u2115) :(taylor r) (X ^ n) = (X + C r) ^ n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Taylor.lean#L49-L50"},"Polynomial.natSepDegree_X_pow":{"signature":"theorem Polynomial.natSepDegree_X_pow {F : Type u} [Field F] {n : \u2115} :(X ^ n).natSepDegree = if n = 0 then 0 else 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/SeparableDegree.lean#L411-L412"},"Polynomial.eraseLead_add_of_natDegree_lt_left":{"signature":"theorem Polynomial.eraseLead_add_of_natDegree_lt_left {R : Type u_1} [Semiring R] {p q : Polynomial R} (pq : q.natDegree < p.natDegree) :(p + q).eraseLead = p.eraseLead + q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/EraseLead.lean#L167-L169"},"Polynomial.eval\u2082RingHom_eval\u2082RingHom":{"signature":"theorem Polynomial.eval\u2082RingHom_eval\u2082RingHom {R : Type u_1} {S : Type u_2} [CommSemiring R] [CommSemiring S] (f : R \u2192+* S) (x y : S) :eval\u2082RingHom (eval\u2082RingHom f x) y = (evalEvalRingHom x y).comp (mapRingHom (mapRingHom f))","doc":"Two equivalent ways to express the evaluation of a bivariate polynomial over R at a point in the affine plane over an R-algebra S.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Bivariate.lean#L181-L186"},"Polynomial.eval\u2082_eval\u2082RingHom_apply":{"signature":"theorem Polynomial.eval\u2082_eval\u2082RingHom_apply {R : Type u_1} {S : Type u_2} [CommSemiring R] [CommSemiring S] (f : R \u2192+* S) (x y : S) (p : Polynomial (Polynomial R)) :eval\u2082 (eval\u2082RingHom f x) y p = evalEval x y (map (mapRingHom f) p)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Bivariate.lean#L188-L190"},"Polynomial.ne_zero_of_mem_nthRootsFinset":{"signature":"theorem Polynomial.ne_zero_of_mem_nthRootsFinset {R : Type u} {n : \u2115} [CommRing R] [IsDomain R] {\u03b7 a : R} (ha : a \u2260 0) (h\u03b7 : \u03b7 \u2208 nthRootsFinset n a) :\u03b7 \u2260 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L403-L411"},"Polynomial.roots_eq_of_degree_le_card":{"signature":"theorem Polynomial.roots_eq_of_degree_le_card {R : Type u} [CommRing R] [IsDomain R] {p : Polynomial R} {S : Finset R} (hS : \u2200 x \u2208 S, eval x p = 0) (hcard : \u2191S.card = p.degree) :p.roots = S.val","doc":"Alias of Polynomial.roots_eq_of_degree_eq_card.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L311-L311"},"Polynomial.eval_eq_prod_roots_sub_of_splits_id":{"signature":"theorem Polynomial.eval_eq_prod_roots_sub_of_splits_id {R : Type u_1} [CommRing R] {f : Polynomial R} [IsDomain R] (hf : f.Splits) (x : R) :eval x f = f.leadingCoeff * (Multiset.map (fun (x_1 : R) => x - x_1) f.roots).prod","doc":"Alias of Polynomial.Splits.eval_eq_prod_roots.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L808-L809"},"Polynomial.CAlgHom_apply":{"signature":"theorem Polynomial.CAlgHom_apply {R : Type u} {A : Type z} [CommSemiring R] [Semiring A] [Algebra R A] (a\u271d : A) :CAlgHom a\u271d = C a\u271d","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L94-L94"},"Polynomial.splits_of_isUnit":{"signature":"theorem Polynomial.splits_of_isUnit {R : Type u_1} [Semiring R] [NoZeroDivisors R] {f : Polynomial R} (hf : IsUnit f) :f.Splits","doc":"Alias of IsUnit.splits.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L99-L100"},"Polynomial.int_eval\u2082_eq":{"signature":"theorem Polynomial.int_eval\u2082_eq {K : Type u_1} [Field K] (R : Subring K) (P : Polynomial K) (hP : \u2200 (n : \u2115), P.coeff n \u2208 R) {L : Type u_2} [Field L] [Algebra K L] (x : L) :eval\u2082 (algebraMap (\u21a5R) L) x (int R P hP) = (aeval x) P","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Ring/Subring/IntPolynomial.lean#L57-L61"},"Polynomial.toFinsupp_ofNat":{"signature":"theorem Polynomial.toFinsupp_ofNat {R : Type u} [Semiring R] (n : \u2115) [n.AtLeastTwo] :(OfNat.ofNat n).toFinsupp = OfNat.ofNat n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L264-L265"},"Polynomial.degree_add_div":{"signature":"theorem Polynomial.degree_add_div {R : Type u} [Field R] {p q : Polynomial R} (hq0 : q \u2260 0) (hpq : q.degree \u2264 p.degree) :q.degree + (p / q).degree = p.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L387-L394"},"Polynomial.logMahlerMeasure_monomial":{"signature":"theorem Polynomial.logMahlerMeasure_monomial (n : \u2115) (z : \u2102) :((monomial n) z).logMahlerMeasure = Real.log \u2016z\u2016","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/MahlerMeasure.lean#L66-L68"},"Polynomial.mapAlgEquiv_coe_ringHom":{"signature":"theorem Polynomial.mapAlgEquiv_coe_ringHom {R : Type u} {A : Type z} {B : Type u_2} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A \u2243\u2090[R] B) :\u2191(mapAlgEquiv f) = mapRingHom \u2191f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L217-L220"},"Polynomial.map_surjective_iff":{"signature":"theorem Polynomial.map_surjective_iff {R : Type u} {S : Type v} [Semiring R] [Semiring S] (f : R \u2192+* S) :Function.Surjective (map f) \u2194 Function.Surjective \u21d1f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Coeff.lean#L116-L117"},"Polynomial.isPrimitive_iff_forall_gaussNorm_eq_one":{"signature":"theorem Polynomial.isPrimitive_iff_forall_gaussNorm_eq_one {R : Type u_2} [CommRing R] [IsDomain R] [IsPrincipalIdealRing R] (hR : \u00acIsField R) {b : NNReal} (hb : 1 < b) (p : Polynomial R) :p.IsPrimitive \u2194 \u2200 (v : IsDedekindDomain.HeightOneSpectrum R), gaussNorm (v.intAdicAbv hb) 1 p = 1","doc":"In case R is PID, given a polynomial p in R[X], p is primitive if and only if the v-adic Gauss norms of p are equal to 1 for all v.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/DedekindDomain/GaussLemma.lean#L66-L70"},"Polynomial.natDegree_mul_C_eq_of_mul_ne_zero":{"signature":"theorem Polynomial.natDegree_mul_C_eq_of_mul_ne_zero {R : Type u} {a : R} [Semiring R] {p : Polynomial R} (h : p.leadingCoeff * a \u2260 0) :(p * C a).natDegree = p.natDegree","doc":"Although not explicitly stated, the assumptions of lemma natDegree_mul_C_eq_of_mul_ne_zero force the polynomial p to be non-zero, via p.leadingCoeff \u2260 0.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L122-L129"},"Polynomial.dvd_comp_X_add_C_iff":{"signature":"theorem Polynomial.dvd_comp_X_add_C_iff {R : Type u} [CommRing R] (p q : Polynomial R) (a : R) :p \u2223 q.comp (X + C a) \u2194 p.comp (X - C a) \u2223 q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L672-L674"},"Polynomial.leadingCoeff_of_injective":{"signature":"theorem Polynomial.leadingCoeff_of_injective {R : Type u} {S : Type v} [Semiring R] [Semiring S] {f : R \u2192+* S} (hf : Function.Injective \u21d1f) (p : Polynomial R) :(map f p).leadingCoeff = f p.leadingCoeff","doc":"Alias of Polynomial.leadingCoeff_map_of_injective.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Monic.lean#L415-L416"},"Polynomial.dvd_C_mul_X_sub_one_pow_add_one":{"signature":"theorem Polynomial.dvd_C_mul_X_sub_one_pow_add_one {R : Type u_1} [CommRing R] {p : \u2115} (hpri : Nat.Prime p) (hp : p \u2260 2) (a r : R) (h\u2081 : r \u2223 a ^ p) (h\u2082 : r \u2223 \u2191p * a) :C r \u2223 (C a * X - 1) ^ p + 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean#L605-L610"},"Polynomial.natDegree_divX_le":{"signature":"theorem Polynomial.natDegree_divX_le {R : Type u} [Semiring R] {p : Polynomial R} :p.divX.natDegree \u2264 p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Inductions.lean#L102-L103"},"Polynomial.coeff_divByMonic_mem_pow_natDegree_mul":{"signature":"theorem Polynomial.coeff_divByMonic_mem_pow_natDegree_mul {R : Type u_2} {S : Type u_3} [CommRing R] [Ring S] [Algebra R S] (p q : Polynomial S) (Mp : Submodule R S) (hp : \u2200 (i : \u2115), p.coeff i \u2208 Mp) (hp' : 1 \u2208 Mp) (Mq : Submodule R S) (hq : \u2200 (i : \u2115), q.coeff i \u2208 Mq) (hq' : 1 \u2208 Mq) (i : \u2115) :(p /\u2098 q).coeff i \u2208 Mq ^ p.natDegree * Mp","doc":"For polynomials p q : R[X], the coefficients of p /\u2098 q can be written as sums of products of coefficients of p and q. Precisely, each summand needs at most one coefficient of p and deg p coefficients of q.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/CoeffMem.lean#L97-L109"},"Polynomial.isMonicOfDegree_monomial_one":{"signature":"theorem Polynomial.isMonicOfDegree_monomial_one {R : Type u_1} [Semiring R] [Nontrivial R] (n : \u2115) :((monomial n) 1).IsMonicOfDegree n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/IsMonicOfDegree.lean#L173-L174"},"Polynomial.roots_of_cyclotomic":{"signature":"theorem Polynomial.roots_of_cyclotomic (n : \u2115) (R : Type u_2) [CommRing R] [IsDomain R] :(cyclotomic' n R).roots = (primitiveRoots n R).val","doc":"The roots of cyclotomic' n R are the primitive n-th roots of unity.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean#L112-L115"},"Polynomial.isRoot_of_isRoot_iff_dvd_derivative_mul":{"signature":"theorem Polynomial.isRoot_of_isRoot_iff_dvd_derivative_mul {K : Type u_1} [Field K] [IsAlgClosed K] [CharZero K] {f g : Polynomial K} (hf0 : f \u2260 0) :(\u2200 (x : K), f.IsRoot x \u2192 g.IsRoot x) \u2194 f \u2223 derivative f * g","doc":"Over an algebraically closed field of characteristic zero a necessary and sufficient condition for the set of roots of a nonzero polynomial f to be a subset of the set of roots of g is that f divides f.derivative * g. Over an integral domain, this is a sufficient but not necessary condition. See isRoot_of_isRoot_of_dvd_derivative_mul","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/IsAlgClosed/Basic.lean#L549-L574"},"Polynomial.dickson_of_two_le":{"signature":"theorem Polynomial.dickson_of_two_le {R : Type u_1} [CommRing R] (k : \u2115) (a : R) {n : \u2115} (h : 2 \u2264 n) :dickson k a n = X * dickson k a (n - 1) - C a * dickson k a (n - 2)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Dickson.lean#L83-L87"},"Polynomial.instFinitePresentationUniversalFactorizationRing":{"signature":"instance Polynomial.instFinitePresentationUniversalFactorizationRing {R : Type u_1} [CommRing R] {n : \u2115} (m k : \u2115) (hn : n = m + k) (p : MonicDegreeEq R n) :Algebra.FinitePresentation R (UniversalFactorizationRing m k hn p)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/UniversalFactorizationRing.lean#L492-L497"},"Polynomial.roots_ne_zero_of_splits":{"signature":"theorem Polynomial.roots_ne_zero_of_splits {R : Type u_1} [CommRing R] {f : Polynomial R} [IsDomain R] (hf : f.Splits) (hf0 : f.natDegree \u2260 0) :f.roots \u2260 0","doc":"Alias of Polynomial.Splits.roots_ne_zero.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L772-L773"},"Polynomial.ringHom_eval\u2082_intCastRingHom":{"signature":"theorem Polynomial.ringHom_eval\u2082_intCastRingHom {R : Type u_3} {S : Type u_4} [Ring R] [Ring S] (p : Polynomial \u2124) (f : R \u2192+* S) (r : R) :f (eval\u2082 (Int.castRingHom R) r p) = eval\u2082 (Int.castRingHom S) (f r) p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L144-L147"},"Polynomial.map_evalRingHom_eval":{"signature":"theorem Polynomial.map_evalRingHom_eval {R : Type u_1} [CommSemiring R] (x y : R) (p : Polynomial (Polynomial R)) :eval y (map (evalRingHom x) p) = evalEval x y p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Bivariate.lean#L156-L158"},"Polynomial.X_pow_smul_rTensor_monomial":{"signature":"theorem Polynomial.X_pow_smul_rTensor_monomial {R : Type u} {S : Type v} [CommSemiring R] [CommSemiring S] [Algebra R S] {N : Type u_3} [AddCommMonoid N] [Module R N] (k : \u2115) (sn : TensorProduct R S N) :X ^ k \u2022 (LinearMap.rTensor N (\u2191R (monomial 0))) sn = (LinearMap.rTensor N (\u2191R (monomial k))) sn","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L565-L574"},"Polynomial.splits_of_degree_le_one":{"signature":"theorem Polynomial.splits_of_degree_le_one {R : Type u_1} [DivisionSemiring R] {f : Polynomial R} (hf : f.degree \u2264 1) :f.Splits","doc":"Alias of Polynomial.Splits.of_degree_le_one.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L666-L667"},"Polynomial.taylor_inj":{"signature":"theorem Polynomial.taylor_inj {R : Type u_1} [Ring R] {r : R} {p q : Polynomial R} :(taylor r) p = (taylor r) q \u2194 p = q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Taylor.lean#L133-L134"},"Polynomial.aeval_apply_smul_mem_of_le_comap":{"signature":"theorem Polynomial.aeval_apply_smul_mem_of_le_comap {R : Type u} {M : Type u_3} [CommSemiring R] [AddCommMonoid M] [Module R M] {q : Submodule R M} {m : M} (hm : m \u2208 q) (p : Polynomial R) (f : Module.End R M) (hq : q \u2264 Submodule.comap f q) :((aeval f) p) m \u2208 q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L707-L710"},"Polynomial.mirror_smul":{"signature":"theorem Polynomial.mirror_smul {R : Type u_1} [Ring R] (p : Polynomial R) [NoZeroDivisors R] (a : R) :(a \u2022 p).mirror = a \u2022 p.mirror","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Mirror.lean#L186-L187"},"Polynomial.aeval_X_left_eq_map":{"signature":"theorem Polynomial.aeval_X_left_eq_map {R : Type u} {S : Type v} [CommSemiring R] [CommSemiring S] [Algebra R S] (p : Polynomial R) :(aeval X) p = map (algebraMap R S) p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L389-L391"},"Polynomial.ofFn_eq_sum_monomial":{"signature":"theorem Polynomial.ofFn_eq_sum_monomial {R : Type u_1} [Semiring R] [DecidableEq R] {n : \u2115} (v : Fin n \u2192 R) :(ofFn n) v = \u2211 i : Fin n, (monomial \u2191i) (v i)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/OfFn.lean#L91-L97"},"Polynomial.supDegree_eq_natDegree":{"signature":"theorem Polynomial.supDegree_eq_natDegree {R : Type u} [Semiring R] (p : Polynomial R) :AddMonoidAlgebra.supDegree id p.toFinsupp = p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Support.lean#L38-L44"},"Polynomial.eval_one_cyclotomic_prime":{"signature":"theorem Polynomial.eval_one_cyclotomic_prime {R : Type u_1} [CommRing R] {p : \u2115} [hn : Fact (Nat.Prime p)] :eval 1 (cyclotomic p R) = \u2191p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Eval.lean#L30-L34"},"Polynomial.eval\u2082_one_cyclotomic_prime":{"signature":"theorem Polynomial.eval\u2082_one_cyclotomic_prime {R : Type u_1} {S : Type u_2} [CommRing R] [Semiring S] (f : R \u2192+* S) {p : \u2115} [Fact (Nat.Prime p)] :eval\u2082 f 1 (cyclotomic p R) = \u2191p","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Cyclotomic/Eval.lean#L36-L37"},"Polynomial.self_mul_modByMonic":{"signature":"theorem Polynomial.self_mul_modByMonic {R : Type u} [Ring R] {p q : Polynomial R} (hq : q.Monic) :q * p %\u2098 q = 0","doc":"See Polynomial.mul_self_modByMonic for the other multiplication order. That version, unlike this one, requires commutativity.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L403-L408"},"Polynomial.hermite_eq_iterate":{"signature":"theorem Polynomial.hermite_eq_iterate (n : \u2115) :hermite n = (fun (p : Polynomial \u2124) => X * p - derivative p)^[n] 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Hermite/Basic.lean#L57-L60"},"Polynomial.int_coeff_eq":{"signature":"theorem Polynomial.int_coeff_eq {K : Type u_1} [Field K] (R : Subring K) (P : Polynomial K) (hP : \u2200 (n : \u2115), P.coeff n \u2208 R) (n : \u2115) :\u2191((int R P hP).coeff n) = P.coeff n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Ring/Subring/IntPolynomial.lean#L42-L43"},"Polynomial.splits_id_iff_splits":{"signature":"theorem Polynomial.splits_id_iff_splits {K : Type v} {L : Type w} [CommRing K] [Field L] (i : K \u2192+* L) {f : Polynomial K} :(map (RingHom.id L) (map i f)).Splits \u2194 (map i f).Splits","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L707-L710"},"Polynomial.natDegree_add_le_iff_right":{"signature":"theorem Polynomial.natDegree_add_le_iff_right {R : Type u} [Semiring R] {n : \u2115} (p q : Polynomial R) (pn : p.natDegree \u2264 n) :(p + q).natDegree \u2264 n \u2194 q.natDegree \u2264 n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L89-L92"},"Polynomial.splits_of_degree_eq_two":{"signature":"theorem Polynomial.splits_of_degree_eq_two {R : Type u_1} [Field R] {f : Polynomial R} {x : R} (h\u2081 : f.degree = 2) (h\u2082 : eval x f = 0) :f.Splits","doc":"Alias of Polynomial.Splits.of_degree_eq_two.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L829-L830"},"Polynomial.base_mul_mem_lifts":{"signature":"theorem Polynomial.base_mul_mem_lifts {R : Type u} [Semiring R] {S : Type v} [Semiring S] {f : R \u2192+* S} {p : Polynomial S} (r : R) (hp : p \u2208 lifts f) :C (f r) * p \u2208 lifts f","doc":"If p lifts and (r : R) then r * p lifts.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Lifts.lean#L109-L114"},"Polynomial.coeff_mul_ofNat":{"signature":"theorem Polynomial.coeff_mul_ofNat {R : Type u} [Semiring R] {p : Polynomial R} {a k : \u2115} [a.AtLeastTwo] :(p * OfNat.ofNat a).coeff k = p.coeff k * OfNat.ofNat a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Coeff.lean#L173-L174"},"Polynomial.reverse_trailingCoeff":{"signature":"theorem Polynomial.reverse_trailingCoeff {R : Type u_1} [Semiring R] (f : Polynomial R) :f.reverse.trailingCoeff = f.leadingCoeff","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Reverse.lean#L268-L269"},"Polynomial.instIsJacobsonRing":{"signature":"instance Polynomial.instIsJacobsonRing {R : Type u_1} [CommRing R] [IsJacobsonRing R] :IsJacobsonRing (Polynomial R)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Jacobson/Ring.lean#L425-L426"},"Polynomial.mul_self_modByMonic":{"signature":"theorem Polynomial.mul_self_modByMonic {R : Type u} [CommRing R] {p q : Polynomial R} (hq : q.Monic) :p * q %\u2098 q = 0","doc":"See Polynomial.self_mul_modByMonic for the other multiplication order. This version, unlike that one, requires commutativity.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Div.lean#L657-L662"},"Polynomial.natTrailingDegree_le_of_mem_supp":{"signature":"theorem Polynomial.natTrailingDegree_le_of_mem_supp {R : Type u} [Semiring R] {p : Polynomial R} (a : \u2115) :a \u2208 p.support \u2192 p.natTrailingDegree \u2264 a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L259-L260"},"Polynomial.taylorEquiv_symm":{"signature":"theorem Polynomial.taylorEquiv_symm {R : Type u_1} [CommRing R] (r : R) :(taylorEquiv r).symm = taylorEquiv (-r)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Taylor.lean#L193-L194"},"Polynomial.degree_list_prod":{"signature":"theorem Polynomial.degree_list_prod {R : Type u} [Semiring R] [NoZeroDivisors R] [Nontrivial R] (l : List (Polynomial R)) :l.prod.degree = (List.map degree l).sum","doc":"The degree of a product of polynomials is equal to the sum of the degrees, where the degree of the zero polynomial is \u22a5. [Nontrivial R] is needed, otherwise for l = [] we have \u22a5 in the LHS and 0 in the RHS.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/BigOperators.lean#L307-L312"},"Polynomial.content_X":{"signature":"theorem Polynomial.content_X {R : Type u_1} [CommRing R] [NormalizedGCDMonoid R] :X.content = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Content.lean#L142-L143"},"Polynomial.eq_neg_mul_add_of_aroots_quadratic_eq_pair":{"signature":"theorem Polynomial.eq_neg_mul_add_of_aroots_quadratic_eq_pair {T : Type u_2} {S : Type u_3} [CommRing T] [CommRing S] [IsDomain S] [Algebra T S] {a b c : T} {x1 x2 : S} (haroots : (C a * X ^ 2 + C b * X + C c).aroots S = {x1, x2}) :(algebraMap T S) b = -(algebraMap T S) a * (x1 + x2)","doc":"Vieta's formula for quadratics (aroots version).","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/SmallDegreeVieta.lean#L53-L60"},"Polynomial.le_trailingDegree_X_pow":{"signature":"theorem Polynomial.le_trailingDegree_X_pow {R : Type u} [Semiring R] (n : \u2115) :\u2191n \u2264 (X ^ n).trailingDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L236-L237"},"Polynomial.coeff_pow_eq_ite_of_natDegree_le_of_le":{"signature":"theorem Polynomial.coeff_pow_eq_ite_of_natDegree_le_of_le {R : Type u} {m n : \u2115} [Semiring R] {p : Polynomial R} {o : \u2115} (pn : p.natDegree \u2264 n) (mno : m * n \u2264 o) :(p ^ m).coeff o = if o = m * n then p.coeff n ^ m else 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean#L163-L169"},"Polynomial.natDegree_C_add":{"signature":"theorem Polynomial.natDegree_C_add {R : Type u} [Semiring R] {p : Polynomial R} {a : R} :(C a + p).natDegree = p.natDegree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L211-L212"},"Polynomial.monic_zero_iff_subsingleton'":{"signature":"theorem Polynomial.monic_zero_iff_subsingleton' {R : Type u} [Semiring R] :Monic 0 \u2194 (\u2200 (f g : Polynomial R), f = g) \u2227 \u2200 (a b : R), a = b","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Monic.lean#L43-L48"},"Polynomial.mapAlgEquiv_comp":{"signature":"theorem Polynomial.mapAlgEquiv_comp {R : Type u} {A : Type z} {B : Type u_2} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (C : Type u_3) [Semiring C] [Algebra R C] (f : A \u2243\u2090[R] B) (g : B \u2243\u2090[R] C) :(mapAlgEquiv f).trans (mapAlgEquiv g) = mapAlgEquiv (f.trans g)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L226-L230"},"Polynomial.eval\u2082_mul_eq_zero_of_right":{"signature":"theorem Polynomial.eval\u2082_mul_eq_zero_of_right {R : Type u} {S : Type v} [Semiring R] {q : Polynomial R} [CommSemiring S] (f : R \u2192+* S) (x : S) (p : Polynomial R) (hq : eval\u2082 f x q = 0) :eval\u2082 f x (p * q) = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L206-L208"},"Polynomial.degree_mono":{"signature":"theorem Polynomial.degree_mono {R : Type u} {S : Type v} [Semiring R] [Semiring S] {f : Polynomial R} {g : Polynomial S} (h : f.support \u2286 g.support) :f.degree \u2264 g.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Defs.lean#L128-L130"},"Polynomial.eval\u2082_finset_prod":{"signature":"theorem Polynomial.eval\u2082_finset_prod {R : Type u} {S : Type v} {\u03b9 : Type y} [CommSemiring R] [CommSemiring S] (f : R \u2192+* S) (s : Finset \u03b9) (g : \u03b9 \u2192 Polynomial R) (x : S) :eval\u2082 f x (\u220f i \u2208 s, g i) = \u220f i \u2208 s, eval\u2082 f x (g i)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Eval/Defs.lean#L645-L647"},"Polynomial.mul_mod":{"signature":"theorem Polynomial.mul_mod {R : Type u} [Field R] (p\u2081 p\u2082 q : Polynomial R) :p\u2081 * p\u2082 % q = p\u2081 % q * (p\u2082 % q) % q","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L448-L450"},"Polynomial.natTrailingDegree_X_le":{"signature":"theorem Polynomial.natTrailingDegree_X_le {R : Type u} [Semiring R] :X.natTrailingDegree \u2264 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean#L242-L243"},"Polynomial.dickson_two_zero":{"signature":"theorem Polynomial.dickson_two_zero {R : Type u_1} [CommRing R] (n : \u2115) :dickson 2 0 n = X ^ n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Dickson.lean#L99-L107"},"Polynomial.splits_of_splits_mul":{"signature":"theorem Polynomial.splits_of_splits_mul {R : Type u_1} [CommRing R] {f g : Polynomial R} [IsDomain R] (hf\u2080 : f \u2260 0) (hg\u2080 : g \u2260 0) :(f * g).Splits \u2194 f.Splits \u2227 g.Splits","doc":"Alias of Polynomial.splits_mul_iff.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L750-L751"},"Polynomial.instAlgebraSplittingField":{"signature":"instance Polynomial.instAlgebraSplittingField {K : Type u_1} [Field K] (f : Polynomial K) :Algebra K f.SplittingField","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/SplittingField/Construction.lean#L218-L218"},"Polynomial.splits_of_splits_mul'":{"signature":"theorem Polynomial.splits_of_splits_mul' {R : Type u_1} [CommRing R] {f g : Polynomial R} [IsDomain R] (hf\u2080 : f \u2260 0) (hg\u2080 : g \u2260 0) :(f * g).Splits \u2194 f.Splits \u2227 g.Splits","doc":"Alias of Polynomial.splits_mul_iff.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L681-L682"},"Polynomial.coe_aevalAeval_eq_evalEval":{"signature":"theorem Polynomial.coe_aevalAeval_eq_evalEval {A : Type u_2} [CommSemiring A] (x y : A) :\u21d1(aevalAeval x y) = evalEval x y","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Bivariate.lean#L218-L219"},"Polynomial.finite_mahlerMeasure_le":{"signature":"theorem Polynomial.finite_mahlerMeasure_le (n : \u2115) (B : NNReal) :{p : Polynomial \u2124 | p.natDegree \u2264 n \u2227 (map (Int.castRingHom \u2102) p).mahlerMeasure \u2264 \u2191B}.Finite","doc":"Northcott's Theorem: the set of integer polynomials of degree at most n and Mahler measure at most B is finite.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/NumberTheory/MahlerMeasure.lean#L110-L114"},"Polynomial.monomial_add":{"signature":"theorem Polynomial.monomial_add {R : Type u} [Semiring R] (n : \u2115) (r s : R) :(monomial n) (r + s) = (monomial n) r + (monomial n) s","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Basic.lean#L398-L400"},"Polynomial.fderiv":{"signature":"theorem Polynomial.fderiv {\ud835\udd5c : Type u} [NontriviallyNormedField \ud835\udd5c] {x : \ud835\udd5c} (p : Polynomial \ud835\udd5c) :fderiv \ud835\udd5c (fun (x : \ud835\udd5c) => eval x p) x = ContinuousLinearMap.smulRight 1 (eval x (derivative p))","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Calculus/Deriv/Polynomial.lean#L141-L144"},"Polynomial.continuousAt_aeval":{"signature":"theorem Polynomial.continuousAt_aeval {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] [TopologicalSpace A] [IsTopologicalSemiring A] (p : Polynomial R) {a : A} :ContinuousAt (fun (x : A) => (aeval x) p) a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Topology/Algebra/Polynomial.lean#L83-L85"},"Polynomial.leadingCoeff_mul_C_of_isUnit":{"signature":"theorem Polynomial.leadingCoeff_mul_C_of_isUnit {R : Type u} {a : R} [Semiring R] (ha : IsUnit a) (p : Polynomial R) :(p * C a).leadingCoeff = p.leadingCoeff * a","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Degree/Operations.lean#L408-L410"},"Polynomial.degree_derivative_le":{"signature":"theorem Polynomial.degree_derivative_le {R : Type u} [Semiring R] {p : Polynomial R} :(derivative p).degree \u2264 p.degree","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Derivative.lean#L168-L170"},"Polynomial.bernoulli_one":{"signature":"theorem Polynomial.bernoulli_one :bernoulli 1 = X - C 2\u207b\u00b9","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/NumberTheory/BernoulliPolynomials.lean#L77-L79"},"Polynomial.injective_ofFn":{"signature":"theorem Polynomial.injective_ofFn {R : Type u_1} [Semiring R] [DecidableEq R] (n : \u2115) :Function.Injective \u21d1(ofFn n)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/OfFn.lean#L102-L103"},"Polynomial.isOpenMap_comap_C":{"signature":"theorem Polynomial.isOpenMap_comap_C {R : Type u_1} [CommRing R] :IsOpenMap (PrimeSpectrum.comap C)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Spectrum/Prime/Polynomial.lean#L158-L166"},"Polynomial.card_roots_X_pow_sub_C":{"signature":"theorem Polynomial.card_roots_X_pow_sub_C {R : Type u} [CommRing R] [IsDomain R] {n : \u2115} (hn : 0 < n) (a : R) :(X ^ n - C a).roots.card \u2264 n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L293-L299"},"Polynomial.hilbertPoly_zero_right":{"signature":"theorem Polynomial.hilbertPoly_zero_right {F : Type u_1} [Field F] (p : Polynomial F) :p.hilbertPoly 0 = 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/HilbertPoly.lean#L119-L119"},"Polynomial.hilbertPoly_linearMap":{"signature":"noncomputable def Polynomial.hilbertPoly_linearMap (F : Type u_1) [Field F] (d : \u2115) :Polynomial F \u2192\u2097[F] Polynomial F","doc":"The function that sends any p : F[X] to Polynomial.hilbertPoly p d is an F-linear map from F[X] to F[X].","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/HilbertPoly.lean#L150-L157"},"Polynomial.coeff_hermite":{"signature":"theorem Polynomial.coeff_hermite (n k : \u2115) :(hermite n).coeff k = if Even (n + k) then (-1) ^ ((n - k) / 2) * \u2191(n - k - 1).doubleFactorial * \u2191(n.choose k) else 0","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/Hermite/Basic.lean#L193-L198"},"Polynomial.roots_eq_of_natDegree_le_card_of_ne_zero":{"signature":"theorem Polynomial.roots_eq_of_natDegree_le_card_of_ne_zero {R : Type u} [CommRing R] [IsDomain R] {p : Polynomial R} {S : Finset R} (hS : \u2200 x \u2208 S, eval x p = 0) (hcard : p.natDegree \u2264 S.card) (hp : p \u2260 0) :p.roots = S.val","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L313-L315"},"Polynomial.unique_separable_of_irreducible":{"signature":"theorem Polynomial.unique_separable_of_irreducible {F : Type u} [Field F] (p : \u2115) [HF : CharP F p] {f : Polynomial F} (hf : Irreducible f) (hp : 0 < p) (n\u2081 : \u2115) (g\u2081 : Polynomial F) (hg\u2081 : g\u2081.Separable) (hgf\u2081 : (expand F (p ^ n\u2081)) g\u2081 = f) (n\u2082 : \u2115) (g\u2082 : Polynomial F) (hg\u2082 : g\u2082.Separable) (hgf\u2082 : (expand F (p ^ n\u2082)) g\u2082 = f) :n\u2081 = n\u2082 \u2227 g\u2081 = g\u2082","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/FieldTheory/Separable.lean#L405-L424"},"Polynomial.homogenize_neg":{"signature":"theorem Polynomial.homogenize_neg {R : Type u_1} [CommRing R] (p : Polynomial R) (n : \u2115) :(-p).homogenize n = -p.homogenize n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Homogenize.lean#L197-L199"},"Polynomial.card_mahlerMeasure_le_prod":{"signature":"theorem Polynomial.card_mahlerMeasure_le_prod (n : \u2115) (B : NNReal) :{p : Polynomial \u2124 | p.natDegree \u2264 n \u2227 (map (Int.castRingHom \u2102) p).mahlerMeasure \u2264 \u2191B}.ncard \u2264 \u220f i : Fin (n + 1), (2 * \u230a\u2191(n.choose \u2191i) * B\u230b\u208a + 1)","doc":"An upper bound on the number of integer polynomials of degree at most n and Mahler measure at most B.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/NumberTheory/MahlerMeasure.lean#L116-L120"},"Polynomial.differentiableOn_aeval":{"signature":"theorem Polynomial.differentiableOn_aeval {\ud835\udd5c : Type u} [NontriviallyNormedField \ud835\udd5c] {s : Set \ud835\udd5c} {R : Type u_1} [CommSemiring R] [Algebra R \ud835\udd5c] (q : Polynomial R) :DifferentiableOn \ud835\udd5c (fun (x : \ud835\udd5c) => (aeval x) q) s","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Calculus/Deriv/Polynomial.lean#L103-L104"},"Polynomial.coe_mapAlgEquiv":{"signature":"theorem Polynomial.coe_mapAlgEquiv {R : Type u} {A : Type z} {B : Type u_2} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A \u2243\u2090[R] B) :\u21d1(mapAlgEquiv f) = map \u2191f","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L209-L211"},"Polynomial.toLaurent_C_mul_X_pow":{"signature":"theorem Polynomial.toLaurent_C_mul_X_pow {R : Type u_1} [Semiring R] (n : \u2115) (r : R) :toLaurent (C r * X ^ n) = LaurentPolynomial.C r * LaurentPolynomial.T \u2191n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Laurent.lean#L217-L219"},"Polynomial.splits_of_map_eq_C":{"signature":"theorem Polynomial.splits_of_map_eq_C {K : Type v} {L : Type w} [CommRing K] [Field L] (i : K \u2192+* L) {f : Polynomial K} {a : L} (h : map i f = C a) :(map i f).Splits","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L656-L658"},"Polynomial.pow_scaleRoots_of_isReduced":{"signature":"theorem Polynomial.pow_scaleRoots_of_isReduced {R : Type u_1} [CommSemiring R] [IsReduced R] (p : Polynomial R) (r : R) (n : \u2115) :(p ^ n).scaleRoots r = p.scaleRoots r ^ n","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/Polynomial/ScaleRoots.lean#L234-L240"},"Polynomial.map_under_lt_comap_of_quasiFiniteAt":{"signature":"theorem Polynomial.map_under_lt_comap_of_quasiFiniteAt {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [Algebra R S] (f : Polynomial R \u2192\u2090[R] S) (P : Ideal S) [P.IsPrime] [Algebra.QuasiFiniteAt R P] :Ideal.map C (Ideal.under R P) < Ideal.comap (\u2191f) P","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/QuasiFinite/Basic.lean#L513-L526"},"Polynomial.instIsScalarTowerElemRootSet":{"signature":"instance Polynomial.instIsScalarTowerElemRootSet {R : Type u} {S : Type v} [CommRing R] [IsDomain R] [CommRing S] [Algebra S R] (G : Type u_1) (H : Type u_2) [Monoid G] [MulSemiringAction G R] [SMulCommClass G S R] [Monoid H] [MulSemiringAction H R] [SMulCommClass H S R] [SMul G H] [IsScalarTower G H R] (f : Polynomial S) :IsScalarTower G H \u2191(f.rootSet R)","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Roots.lean#L682-L686"},"Polynomial.trdeg_of_isDomain":{"signature":"theorem Polynomial.trdeg_of_isDomain {R : Type u_1} [CommRing R] [IsDomain R] :Algebra.trdeg R (Polynomial R) = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/RingTheory/AlgebraicIndependent/TranscendenceBasis.lean#L450-L452"},"Polynomial.toFinsuppIsoAlg_symm_apply_toFinsupp":{"signature":"theorem Polynomial.toFinsuppIsoAlg_symm_apply_toFinsupp (R : Type u) [CommSemiring R] (toFinsupp : AddMonoidAlgebra R \u2115) :((toFinsuppIsoAlg R).symm toFinsupp).toFinsupp = toFinsupp","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/AlgebraMap.lean#L111-L111"},"Polynomial.roots_normalize":{"signature":"theorem Polynomial.roots_normalize {R : Type u_1} [CommRing R] [IsDomain R] [NormalizationMonoid R] {p : Polynomial R} :(normalize p).roots = p.roots","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/FieldDivision.lean#L237-L239"},"Polynomial.logMahlerMeasure_of_degree_eq_one":{"signature":"theorem Polynomial.logMahlerMeasure_of_degree_eq_one {p : Polynomial \u2102} (h : p.degree = 1) :p.logMahlerMeasure = Real.log \u2016p.coeff 1\u2016 + \u2016(p.coeff 1)\u207b\u00b9 * p.coeff 0\u2016.posLog","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/MahlerMeasure.lean#L172-L175"},"Polynomial.exists_root_of_splits'":{"signature":"theorem Polynomial.exists_root_of_splits' {R : Type u_1} [CommRing R] {f : Polynomial R} (hf : f.Splits) (hf0 : f.degree \u2260 0) :\u2203 (a : R), eval a f = 0","doc":"Alias of Polynomial.Splits.exists_eval_eq_zero.","src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L719-L720"},"Polynomial.cauchyBound_X":{"signature":"theorem Polynomial.cauchyBound_X {K : Type u_1} [NormedDivisionRing K] :X.cauchyBound = 1","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Analysis/Polynomial/CauchyBound.lean#L51-L53"},"Polynomial.splits_of_degree_le_one_of_monic":{"signature":"theorem Polynomial.splits_of_degree_le_one_of_monic {R : Type u_1} [Semiring R] {f : Polynomial R} (hf : f.degree \u2264 1) (h : f.Monic) :f.Splits","doc":null,"src":"https://github.com/leanprover-community/mathlib4/blob/534cf0b8f5267c3f20bf52f932ad5f9834187c35/Mathlib/Algebra/Polynomial/Splits.lean#L119-L121"}}