| import OAI.AlgebraicGeometry.CommutingDerivations.DerivationOperations | |
| import Mathlib.Algebra.MvPolynomial.PDeriv | |
| import Mathlib.Algebra.MvPolynomial.Degrees | |
| import Mathlib.LinearAlgebra.LinearIndependent.Defs | |
| namespace OAI | |
| /-! The actual polynomial coordinate partials: commuting, locally nilpotent, | |
| linearly independent over the polynomial ring, and with exact constant common | |
| kernel in characteristic zero. No localization equivalence is assumed here. -/ | |
| noncomputable section | |
| namespace AbhyankarSathaye.CommutingDerivations | |
| open MvPolynomial | |
| variable {B σ : Type*} [CommRing B] | |
| theorem coordinate_partials_commute (i j : σ) (p : MvPolynomial σ B) : | |
| pderiv i (pderiv j p) = pderiv j (pderiv i p) := by | |
| classical | |
| by_cases hij : i = j | |
| · subst j | |
| rfl | |
| ext m | |
| simp [coeff_pderiv, Finsupp.add_apply, hij, Ne.symm hij, | |
| add_comm, add_left_comm, mul_comm, mul_left_comm] | |
| /-- Vanishing of all coefficients at exponent at least n implies annihilation | |
| by the nth partial iterate. This works over every commutative ring. -/ | |
| theorem coordinate_partial_iterate_eq_zero (i : σ) (n : ℕ) (p : MvPolynomial σ B) | |
| (hp : ∀ m : σ →₀ ℕ, n ≤ m i → p.coeff m = 0) : | |
| (pderiv i : MvPolynomial σ B → MvPolynomial σ B)^[n] p = 0 := by | |
| classical | |
| induction n generalizing p with | |
| | zero => | |
| apply MvPolynomial.ext | |
| intro m | |
| simpa using hp m (Nat.zero_le _) | |
| | succ n ih => | |
| rw [Function.iterate_succ_apply] | |
| apply ih | |
| intro m hm | |
| rw [coeff_pderiv] | |
| have hz : p.coeff (m + Finsupp.single i 1) = 0 := by | |
| apply hp | |
| simpa using Nat.succ_le_succ hm | |
| rw [hz, zero_mul] | |
| /-- An explicit pointwise bound: one more than the largest exponent of i | |
| among the finitely many supported monomials. -/ | |
| theorem coordinate_partial_nilpotence_bound (i : σ) (p : MvPolynomial σ B) : | |
| (pderiv i : MvPolynomial σ B → MvPolynomial σ B)^[ | |
| p.support.sup (fun m => m i) + 1] p = 0 := by | |
| classical | |
| apply coordinate_partial_iterate_eq_zero | |
| intro m hm | |
| by_contra hc | |
| have hmem : m ∈ p.support := mem_support_iff.mpr hc | |
| have hle : m i ≤ p.support.sup (fun a => a i) := | |
| Finset.le_sup (f := fun a : σ →₀ ℕ => a i) hmem | |
| exact (Nat.not_succ_le_self _) (hm.trans hle) | |
| theorem coordinate_partial_locallyNilpotent (i : σ) : | |
| LocallyNilpotent (pderiv i : Derivation B (MvPolynomial σ B) (MvPolynomial σ B)) := by | |
| intro p | |
| exact ⟨p.support.sup (fun m => m i) + 1, coordinate_partial_nilpotence_bound i p⟩ | |
| /-- Exact coefficient consequence of the common-kernel condition. -/ | |
| theorem coefficient_eq_zero_of_all_partials_zero [IsDomain B] [CharZero B] | |
| (p : MvPolynomial σ B) (hp : ∀ i, pderiv i p = 0) | |
| (m : σ →₀ ℕ) (hm : m ≠ 0) : p.coeff m = 0 := by | |
| classical | |
| have hex : ∃ i, m i ≠ 0 := by | |
| by_contra hh | |
| apply hm | |
| ext i | |
| change m i = 0 | |
| by_contra hi | |
| exact hh ⟨i, hi⟩ | |
| obtain ⟨i, hi⟩ := hex | |
| have he := congrArg (fun polynomial : MvPolynomial σ B => | |
| polynomial.coeff (m - Finsupp.single i 1)) (hp i) | |
| rw [coeff_pderiv, Finsupp.sub_add_single_one_cancel hi, | |
| AddMonoidAlgebra.coeff_zero] at he | |
| have hfactor : (((m - Finsupp.single i 1 : σ →₀ ℕ) i : ℕ) : B) + 1 ≠ 0 := by | |
| rw [← Nat.cast_succ] | |
| exact Nat.cast_ne_zero.mpr (Nat.succ_ne_zero _) | |
| exact (mul_eq_zero.mp he).resolve_right hfactor | |
| /-- The common kernel is literally the coefficient ring, with witness equal | |
| to the constant coefficient. In particular this applies to Laurent coefficients | |
| over C, once that coefficient algebra's domain and characteristic are installed. -/ | |
| theorem coordinate_partials_common_kernel [IsDomain B] [CharZero B] | |
| (p : MvPolynomial σ B) : | |
| (∀ i, pderiv i p = 0) ↔ ∃ b : B, p = C b := by | |
| classical | |
| constructor | |
| · intro hp | |
| refine ⟨p.coeff 0, ?_⟩ | |
| ext m | |
| by_cases hm : m = 0 | |
| · subst m | |
| simp | |
| · rw [coefficient_eq_zero_of_all_partials_zero p hp m hm, coeff_C_of_ne_zero hm] | |
| · rintro ⟨b, rfl⟩ i | |
| exact pderiv_C | |
| theorem coordinate_partials_linearIndependent [Fintype σ] : | |
| LinearIndependent (MvPolynomial σ B) | |
| (fun i : σ => (pderiv i : Derivation B (MvPolynomial σ B) (MvPolynomial σ B))) := by | |
| classical | |
| apply Fintype.linearIndependent_iff.mpr | |
| intro a ha j | |
| let ev : Derivation B (MvPolynomial σ B) (MvPolynomial σ B) →+ MvPolynomial σ B := | |
| { toFun := fun D => D (X j) | |
| map_zero' := rfl | |
| map_add' := fun _ _ => rfl } | |
| have he := congrArg ev ha | |
| rw [map_sum, map_zero] at he | |
| simpa [ev, Derivation.smul_apply, smul_eq_mul, pderiv_X, Pi.single_apply] using he | |
| end AbhyankarSathaye.CommutingDerivations | |
| end | |
| end OAI | |