| import OAI.AlgebraicGeometry.AbhyankarSathaye.Stabilization |
| import Mathlib.Algebra.Polynomial.AlgebraMap |
|
|
| namespace OAI |
|
|
| /-! Every element of the proposed common kernel has a critical point. |
| This proves the coordinate obstruction for all such elements, independently |
| of whether they generate that subalgebra. It does not assert kernel equality. -/ |
| noncomputable section |
| namespace AbhyankarSathaye.CommutingDerivations |
| open MvPolynomial |
|
|
| theorem critical_aeval {σ B : Type*} [CommRing B] |
| (f : MvPolynomial σ B) (point : σ → B) |
| (hc : ∀ i, eval point (pderiv i f) = 0) (q : Polynomial B) (i : σ) : |
| eval point (pderiv i (Polynomial.aeval f q)) = 0 := by |
| induction q using Polynomial.induction_on' with |
| | add q r hq hr => simp only [map_add, hq, hr, add_zero] |
| | monomial n a => |
| rw [← Polynomial.C_mul_X_pow_eq_monomial] |
| simp [Polynomial.aeval_C, MvPolynomial.algebraMap_eq, hc] |
|
|
| theorem no_coordinate_aeval_of_critical {σ B : Type*} [Fintype σ] [CommRing B] |
| [Nontrivial B] (f : MvPolynomial σ B) (point : σ → B) |
| (hc : ∀ i, eval point (pderiv i f) = 0) (q : Polynomial B) : |
| ¬ ∃ (e : MvPolynomial σ B ≃ₐ[B] MvPolynomial σ B) (i : σ), |
| e (X i) = Polynomial.aeval f q := |
| no_coordinate_of_critical _ point (critical_aeval f point hc q) |
|
|
| theorem polynomial_in_extendedF_not_coordinate {n : ℕ} (hn : 4 ≤ n) |
| (q : Polynomial ℂ) : |
| ¬ ∃ (e : MvPolynomial (Fin n) ℂ ≃ₐ[ℂ] MvPolynomial (Fin n) ℂ) (i : Fin n), |
| e (X i) = Polynomial.aeval (extendedF hn) q := |
| no_coordinate_aeval_of_critical (extendedF hn) (extendedPoint hn) |
| (extendedF_critical hn) q |
|
|
| / |
| be a coordinate. The explicit polynomial-expression hypothesis suffices. -/ |
| theorem element_of_extendedF_subring_not_coordinate {n : ℕ} (hn : 4 ≤ n) |
| (g : MvPolynomial (Fin n) ℂ) |
| (hg : ∃ q : Polynomial ℂ, g = Polynomial.aeval (extendedF hn) q) : |
| ¬ ∃ (e : MvPolynomial (Fin n) ℂ ≃ₐ[ℂ] MvPolynomial (Fin n) ℂ) (i : Fin n), |
| e (X i) = g := by |
| obtain ⟨q, rfl⟩ := hg |
| exact polynomial_in_extendedF_not_coordinate hn q |
|
|
| end AbhyankarSathaye.CommutingDerivations |
|
|
| end |
|
|
| end OAI |
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|
