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b797a5b | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 | import OAI.NumberTheory.PiExponent.Polynomials.PowerCoverSubstitution
namespace OAI
namespace PiExponent.PowerCover
noncomputable section
variable {σ : Type*} [Fintype σ] {k : Type*} [CommSemiring k]
def decompositionAddEquiv (w : σ → ℕ) (hw : ∀ i, 0 < w i) :
MvPolynomial σ k ≃+ ((∀ i, Fin (w i)) →₀ MvPolynomial σ k) :=
(AddMonoidAlgebra.coeffAddEquiv : MvPolynomial σ k ≃+ ((σ →₀ ℕ) →₀ k)).trans <|
(Finsupp.domCongr ((exponentEquiv w hw).trans (Equiv.prodComm _ _))).trans <|
Finsupp.curryAddEquiv.trans <|
Finsupp.mapRange.addEquiv AddMonoidAlgebra.coeffAddEquiv.symm
@[simp] theorem decompositionAddEquiv_monomial (w : σ → ℕ) (hw : ∀ i, 0 < w i)
(a : σ →₀ ℕ) (c : k) :
decompositionAddEquiv w hw (MvPolynomial.monomial a c) =
Finsupp.single (residue w hw a) (MvPolynomial.monomial (quotient w a) c) := by
classical
simp [decompositionAddEquiv, MvPolynomial.monomial, Finsupp.curry_single]
theorem decompositionAddEquiv_power_mul (w : σ → ℕ) (hw : ∀ i, 0 < w i)
(p q : MvPolynomial σ k) :
decompositionAddEquiv w hw (powerSubstitution w p * q) =
p • decompositionAddEquiv w hw q := by
classical
induction p using MvPolynomial.induction_on' with
| monomial a c =>
induction q using MvPolynomial.induction_on' with
| monomial b d =>
simp only [powerSubstitution_monomial, MvPolynomial.monomial_mul_monomial,
decompositionAddEquiv_monomial]
rw [add_comm (scale w a) b, quotient_add_scale w hw, residue_add_scale]
simp [Finsupp.smul_single, smul_eq_mul, MvPolynomial.monomial_mul_monomial, add_comm]
| add q r hq hr =>
simp only [mul_add, map_add, smul_add, hq, hr]
| add p r hp hr =>
simp only [map_add, add_mul, add_smul, hp, hr]
def PowerModule (_weights : σ → ℕ) (k : Type*) [CommSemiring k] := MvPolynomial σ k
instance (w : σ → ℕ) : CommSemiring (PowerModule w k) :=
inferInstanceAs (CommSemiring (MvPolynomial σ k))
def toPolynomial (w : σ → ℕ) : PowerModule w k ≃+* MvPolynomial σ k :=
RingEquiv.refl _
instance powerAlgebra (w : σ → ℕ) : Algebra (MvPolynomial σ k) (PowerModule w k) :=
((toPolynomial w).symm.toRingHom.comp
(powerSubstitution w).toRingHom).toAlgebra
omit [Fintype σ] in
@[simp] theorem toPolynomial_smul (w : σ → ℕ) (p : MvPolynomial σ k)
(q : PowerModule w k) :
toPolynomial w (p • q) =
powerSubstitution w p * toPolynomial w q := rfl
def decomposition (w : σ → ℕ) (hw : ∀ i, 0 < w i) :
PowerModule w k ≃ₗ[MvPolynomial σ k] ((∀ i, Fin (w i)) →₀ MvPolynomial σ k) :=
((toPolynomial w).toAddEquiv.trans (decompositionAddEquiv w hw)).toLinearEquiv <| by
intro p q
exact decompositionAddEquiv_power_mul w hw p (toPolynomial w q)
def basis (w : σ → ℕ) (hw : ∀ i, 0 < w i) :
Module.Basis (∀ i, Fin (w i)) (MvPolynomial σ k) (PowerModule w k) where
repr := decomposition w hw
theorem basis_toPolynomial (w : σ → ℕ) (hw : ∀ i, 0 < w i)
(r : ∀ i, Fin (w i)) :
toPolynomial w (basis (k := k) w hw r) = MvPolynomial.monomial (combine w (0, r)) 1 := by
apply (decompositionAddEquiv w hw).injective
rw [decompositionAddEquiv_monomial, quotient_combine w hw, residue_combine]
change (basis (k := k) w hw).repr (basis w hw r) = _
simp
theorem module_free (w : σ → ℕ) (hw : ∀ i, 0 < w i) :
Module.Free (MvPolynomial σ k) (PowerModule w k) :=
Module.Free.of_basis (basis w hw)
theorem module_finite (w : σ → ℕ) (hw : ∀ i, 0 < w i) :
Module.Finite (MvPolynomial σ k) (PowerModule w k) := by
classical
exact Module.Finite.of_basis (basis w hw)
theorem finrank (w : σ → ℕ) (hw : ∀ i, 0 < w i)
[StrongRankCondition (MvPolynomial σ k)] :
Module.finrank (MvPolynomial σ k) (PowerModule w k) = ∏ i, w i := by
classical
rw [Module.finrank_eq_card_basis (basis w hw)]
simp [Fintype.card_pi]
end
noncomputable instance {σ k : Type*} [CommRing k] (w : σ → ℕ) :
CommRing (PowerModule w k) :=
inferInstanceAs (CommRing (MvPolynomial σ k))
end PiExponent.PowerCover
end OAI
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