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7f5fab8 | 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 | 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
/-- In particular, no generator of a common kernel equal to `C[F_n]` can
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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