| import Mathlib.RingTheory.PowerSeries.Derivative |
| import Mathlib.RingTheory.PowerSeries.Order |
| import Mathlib.RingTheory.Polynomial.Wronskian |
| import Mathlib.RingTheory.PrincipalIdealDomain |
| import Mathlib.Algebra.Polynomial.FieldDivision |
| import Mathlib.Tactic.Ring |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Polynomial |
| namespace NCHitting |
|
|
| variable {F : Type*} [Field F] [CharZero F] |
|
|
| omit [CharZero F] in |
| theorem wronskian_mul_common (g p q : F[X]) : |
| Polynomial.wronskian (g*p) (g*q) = g^2 * Polynomial.wronskian p q := by |
| simp only [Polynomial.wronskian, Polynomial.derivative_mul] |
| ring |
|
|
| theorem eq_smul_of_wronskian_eq_zero (p q : F[X]) (hp : p ≠ 0) |
| (hw : Polynomial.wronskian p q = 0) : ∃ c : F, q = c • p := by |
| classical |
| let : GCDMonoid F[X] := IsBezout.toGCDDomain _ |
| obtain ⟨a, b, ha, hb, hab⟩ := extract_gcd p q |
| let g := gcd p q |
| change p = g * a at ha |
| change q = g * b at hb |
| have hg : g ≠ 0 := by |
| intro h |
| apply hp |
| simpa [h] using ha |
| have ha0 : a ≠ 0 := by |
| intro h |
| apply hp |
| simpa [h] using ha |
| have hwab : Polynomial.wronskian a b = 0 := by |
| have hfactor : Polynomial.wronskian p q = g^2 * Polynomial.wronskian a b := |
| (congrArg₂ Polynomial.wronskian ha hb).trans (wronskian_mul_common g a b) |
| rw [hfactor] at hw |
| exact (mul_eq_zero.mp hw).resolve_left (pow_ne_zero _ hg) |
| have hcop : IsCoprime a b := (gcd_isUnit_iff a b).mp hab |
| obtain ⟨had, hbd⟩ := hcop.wronskian_eq_zero_iff.mp hwab |
| have hac : a = C (a.coeff 0) := Polynomial.eq_C_of_derivative_eq_zero had |
| have hbc : b = C (b.coeff 0) := Polynomial.eq_C_of_derivative_eq_zero hbd |
| have hac0 : a.coeff 0 ≠ 0 := by |
| intro h |
| apply ha0 |
| simpa [h] using hac |
| refine ⟨b.coeff 0 / a.coeff 0, ?_⟩ |
| rw [Polynomial.smul_eq_C_mul] |
| calc |
| q = g * C (b.coeff 0) := hb.trans (congrArg (g * ·) hbc) |
| _ = C (b.coeff 0 / a.coeff 0) * (g * C (a.coeff 0)) := by |
| rw [mul_left_comm, ← C_mul, div_mul_cancel₀ _ hac0] |
| _ = C (b.coeff 0 / a.coeff 0) * p := |
| congrArg (C (b.coeff 0 / a.coeff 0) * ·) (ha.trans (congrArg (g * ·) hac)).symm |
|
|
| section FormalResidue |
| open PowerSeries |
|
|
| theorem coeff_X_mul_derivative (v : PowerSeries F) (k : ℕ) : |
| PowerSeries.coeff k (PowerSeries.X * PowerSeries.derivative (R := F) v) = |
| (k : F) * PowerSeries.coeff k v := by |
| cases k with |
| | zero => simp |
| | succ k => |
| have hs := PowerSeries.coeff_X_pow_mul (PowerSeries.derivative (R := F) v) 1 k |
| simp only [pow_one] at hs |
| rw [hs, PowerSeries.coeff_derivative] |
| simp only [Nat.cast_add, Nat.cast_one] |
| ring |
|
|
| theorem X_mul_derivative_X_pow (k : ℕ) : |
| PowerSeries.X * PowerSeries.derivative (R := F) ((PowerSeries.X : PowerSeries F)^k) = |
| PowerSeries.C (k : F) * PowerSeries.X^k := by |
| ext r |
| rw [coeff_X_mul_derivative, PowerSeries.coeff_C_mul, PowerSeries.coeff_X_pow] |
| split_ifs with hr |
| · subst r; rfl |
| · simp |
| |
| private theorem formal_wronskian_factor (A v Q S : PowerSeries F) (k : ℕ) : |
| (PowerSeries.X * Q) * |
| ((PowerSeries.X^k * A) * PowerSeries.derivative (R := F) (A*v) - |
| PowerSeries.derivative (R := F) (PowerSeries.X^k * A) * (A*v)) + |
| (PowerSeries.X^k * A)^2 * S = |
| PowerSeries.X^k * A^2 * |
| (Q * (PowerSeries.X * PowerSeries.derivative (R := F) v - PowerSeries.C (k : F) * v) + |
| PowerSeries.X^k * S) := by |
| simp only [Derivation.leibniz, smul_eq_mul] |
| calc |
| _ = PowerSeries.X^k * A^2 * (Q * (PowerSeries.X * PowerSeries.derivative (R := F) v) + |
| PowerSeries.X^k * S) - |
| (PowerSeries.X * PowerSeries.derivative (R := F) (PowerSeries.X^k)) * A^2 * Q * v := by |
| ring |
| _ = _ := by rw [X_mul_derivative_X_pow]; ring |
| |
| theorem formal_residue_obstruction (p0 p Q S : PowerSeries F) |
| (hp0 : p0 ≠ 0) (hQ : PowerSeries.constantCoeff Q ≠ 0) |
| (h : (PowerSeries.X * Q) * |
| (p0 * PowerSeries.derivative (R := F) p - PowerSeries.derivative (R := F) p0 * p) + |
| p0^2 * S = 0) : PowerSeries.constantCoeff S = 0 := by |
| let k := p0.order.toNat |
| let A := p0.divXPowOrder |
| let v := A⁻¹ * p |
| have hA : PowerSeries.constantCoeff A ≠ 0 := by |
| exact mt PowerSeries.constantCoeff_divXPowOrder_eq_zero_iff.mp hp0 |
| have hA0 : A ≠ 0 := by |
| intro he |
| exact hA (by simp [he]) |
| have hpa : PowerSeries.X^k * A = p0 := PowerSeries.X_pow_order_mul_divXPowOrder |
| have hav : A*v = p := by |
| dsimp [v] |
| rw [← mul_assoc, MvPowerSeries.mul_inv_cancel A hA, one_mul] |
| have hf : PowerSeries.X^k * A^2 * |
| (Q * (PowerSeries.X * PowerSeries.derivative (R := F) v - PowerSeries.C (k : F) * v) + |
| PowerSeries.X^k * S) = 0 := by |
| rw [← formal_wronskian_factor] |
| simpa only [hpa, hav] using h |
| have hn : PowerSeries.X^k * A^2 ≠ 0 := |
| mul_ne_zero (pow_ne_zero _ PowerSeries.X_ne_zero) (pow_ne_zero _ hA0) |
| have he := (mul_eq_zero.mp hf).resolve_left hn |
| have he' : |
| (PowerSeries.X * PowerSeries.derivative (R := F) v - PowerSeries.C (k : F) * v) + |
| PowerSeries.X^k * (Q⁻¹*S) = 0 := by |
| calc |
| _ = Q⁻¹ * |
| (Q * (PowerSeries.X * PowerSeries.derivative (R := F) v - PowerSeries.C (k : F) * v) + |
| PowerSeries.X^k * S) := by |
| simp only [mul_add, ← mul_assoc, MvPowerSeries.inv_mul_cancel Q hQ, one_mul] |
| ring |
| _ = 0 := by rw [he, mul_zero] |
| have hk := congrArg (PowerSeries.coeff k) he' |
| have heuler : PowerSeries.coeff k |
| (PowerSeries.X * PowerSeries.derivative (R := F) v - PowerSeries.C (k : F) * v) = 0 := by |
| rw [map_sub, coeff_X_mul_derivative, PowerSeries.coeff_C_mul, sub_self] |
| have hshift : PowerSeries.coeff k (PowerSeries.X^k * (Q⁻¹*S)) = |
| PowerSeries.constantCoeff (Q⁻¹*S) := by |
| simpa only [zero_add, PowerSeries.coeff_zero_eq_constantCoeff] using |
| PowerSeries.coeff_X_pow_mul (Q⁻¹*S) k 0 |
| rw [map_add, heuler, zero_add, hshift, map_zero] at hk |
| rw [map_mul] at hk |
| change (MvPowerSeries.constantCoeff Q⁻¹) * PowerSeries.constantCoeff S = 0 at hk |
| rw [MvPowerSeries.constantCoeff_inv] at hk |
| exact (mul_eq_zero.mp hk).resolve_left (inv_ne_zero hQ) |
|
|
| theorem polynomial_residue_obstruction (a : F) (p0 p Q S : F[X]) |
| (hp0 : p0 ≠ 0) (hQ : Q.eval a ≠ 0) |
| (h : ((Polynomial.X - Polynomial.C a) * Q) * |
| (p0 * p.derivative - p0.derivative * p) + p0^2 * S = 0) : S.eval a = 0 := by |
| let shift := Polynomial.X + Polynomial.C a |
| have hzero : (p0.comp shift : PowerSeries F) ≠ 0 := by |
| simpa [shift, Polynomial.comp_X_add_C_eq_zero_iff] using hp0 |
| have hQzero : PowerSeries.constantCoeff (Q.comp shift : PowerSeries F) ≠ 0 := by |
| simpa [Polynomial.constantCoeff_coe, Polynomial.coeff_zero_eq_eval_zero, |
| shift, Polynomial.eval_comp] using hQ |
| have heq : (PowerSeries.X * (Q.comp shift : PowerSeries F)) * |
| ((p0.comp shift : PowerSeries F) * PowerSeries.derivative (R := F) (p.comp shift : PowerSeries F) - |
| PowerSeries.derivative (R := F) (p0.comp shift : PowerSeries F) * (p.comp shift : PowerSeries F)) + |
| (p0.comp shift : PowerSeries F)^2 * (S.comp shift : PowerSeries F) = 0 := by |
| have hh := congrArg (fun r : F[X] => (r.comp shift : PowerSeries F)) h |
| simpa [Polynomial.add_comp, Polynomial.mul_comp, Polynomial.sub_comp, Polynomial.pow_comp, |
| PowerSeries.derivative_coe, Polynomial.derivative_comp, shift] using hh |
| have hs := formal_residue_obstruction _ _ _ _ hzero hQzero heq |
| simpa [Polynomial.constantCoeff_coe, Polynomial.coeff_zero_eq_eval_zero, |
| shift, Polynomial.eval_comp] using hs |
|
|
| end FormalResidue |
|
|
| section DistinctPoles |
| variable {ι : Type*} [Fintype ι] [DecidableEq ι] |
|
|
| def poleProduct (a : ι → F) : F[X] := ∏ i, (Polynomial.X - Polynomial.C (a i)) |
|
|
| def puncturedPoleProduct (a : ι → F) (i : ι) : F[X] := |
| ∏ j ∈ Finset.univ.erase i, (Polynomial.X - Polynomial.C (a j)) |
|
|
| omit [CharZero F] in |
| theorem poleProduct_eq (a : ι → F) (i : ι) : |
| poleProduct a = (Polynomial.X - Polynomial.C (a i)) * puncturedPoleProduct a i := by |
| exact (Finset.mul_prod_erase Finset.univ (fun j => Polynomial.X - Polynomial.C (a j)) |
| (Finset.mem_univ i)).symm |
|
|
| omit [CharZero F] in |
| theorem eval_puncturedPoleProduct_self_ne_zero (a : ι → F) (ha : Function.Injective a) (i : ι) : |
| (puncturedPoleProduct a i).eval (a i) ≠ 0 := by |
| simp only [puncturedPoleProduct, Polynomial.eval_prod, Polynomial.eval_sub, |
| Polynomial.eval_X, Polynomial.eval_C] |
| apply Finset.prod_ne_zero_iff.mpr |
| intro j hj |
| apply sub_ne_zero.mpr |
| intro he |
| have hji := (Finset.mem_erase.mp hj).1 |
| exact hji (ha he).symm |
|
|
| omit [CharZero F] in |
| theorem eval_puncturedPoleProduct_other (a : ι → F) (i j : ι) (hij : i ≠ j) : |
| (puncturedPoleProduct a j).eval (a i) = 0 := by |
| simp only [puncturedPoleProduct, Polynomial.eval_prod, Polynomial.eval_sub, |
| Polynomial.eval_X, Polynomial.eval_C] |
| apply Finset.prod_eq_zero (i := i) |
| · simp [hij] |
| · simp |
|
|
| theorem no_polynomial_log_primitive (a : ι → F) (ha : Function.Injective a) |
| (p0 p : F[X]) (hp0 : p0 ≠ 0) (c : ι → F) |
| (h : poleProduct a * (p0 * p.derivative - p0.derivative * p) + |
| p0^2 * (∑ i, Polynomial.C (c i) * puncturedPoleProduct a i) = 0) : |
| ∀ i, c i = 0 := by |
| intro i |
| have hr := polynomial_residue_obstruction (a i) p0 p (puncturedPoleProduct a i) |
| (∑ j, Polynomial.C (c j) * puncturedPoleProduct a j) hp0 |
| (eval_puncturedPoleProduct_self_ne_zero a ha i) (by simpa [poleProduct_eq a i] using h) |
| have heval : (∑ j, Polynomial.C (c j) * puncturedPoleProduct a j).eval (a i) = |
| c i * (puncturedPoleProduct a i).eval (a i) := by |
| rw [Polynomial.eval_finsetSum] |
| simp only [Polynomial.eval_mul, Polynomial.eval_C] |
| apply Finset.sum_eq_single i |
| · intro j hj hji |
| simp [eval_puncturedPoleProduct_other a i j (Ne.symm hji)] |
| · simp |
| rw [heval] at hr |
| exact (mul_eq_zero.mp hr).resolve_right (eval_puncturedPoleProduct_self_ne_zero a ha i) |
|
|
| end DistinctPoles |
|
|
| end NCHitting |
|
|
| end |
|
|
| end OAI |
|
|
