| import Mathlib.Algebra.MvPolynomial.PDeriv |
| import Mathlib.Basic.Complex.Basic |
| import Mathlib.LinearAlgebra.LinearIndependent.Defs |
| import Mathlib.Logic.Equiv.Fin.Basic |
| import Mathlib.Logic.Function.Iterate |
| import Mathlib.RingTheory.Adjoin.Basic |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace AbhyankarSathaye |
|
|
| def S {B : Type*} [CommRing B] (h u v w : B) : B := |
| 2*u^3*v + 3*u^4*w + h*(v^2 - 3*u^2*w^2) + h^2*w^3 |
|
|
| def P {B : Type*} [CommRing B] (x y s : B) : B := |
| -2*s^2*x + 3*s*y^2 - 3*s^3*y |
|
|
| abbrev R := MvPolynomial (Fin 4) ℂ |
|
|
| def h : R := MvPolynomial.X 0 |
| def u : R := MvPolynomial.X 1 |
| def v : R := MvPolynomial.X 2 |
| def w : R := MvPolynomial.X 3 |
| def x : R := u^3 + h*v |
| def y : R := -u^2 + h*w |
| def s : R := S h u v w |
| def p : R := P x y s |
| def F : R := h - p - 1 |
|
|
| def fourInclusion {n : ℕ} (hn : 4 ≤ n) (i : Fin 4) : Fin n := |
| ⟨i.val, lt_of_lt_of_le i.isLt hn⟩ |
|
|
| def extendedF {n : ℕ} (hn : 4 ≤ n) : MvPolynomial (Fin n) ℂ := |
| MvPolynomial.rename (fourInclusion hn) F |
|
|
| def blocksToAmbient {n : ℕ} (hn : 4 ≤ n) : Fin (n-4) ⊕ Fin 4 ≃ Fin n := |
| (Equiv.sumComm _ _).trans (finSumFinEquiv.trans (finCongr (by omega))) |
|
|
| def blocksToOutput {n : ℕ} (hn : 4 ≤ n) : Fin (n-4) ⊕ Fin 3 ≃ Fin (n-1) := |
| (Equiv.sumComm _ _).trans (finSumFinEquiv.trans (finCongr (by omega))) |
|
|
| namespace CommutingDerivations |
|
|
| variable {K A B : Type*} [CommRing K] [CommRing A] [CommRing B] |
| [Algebra K A] [Algebra K B] |
|
|
| def LocallyNilpotent (D : Derivation K A A) : Prop := |
| ∀ a : A, ∃ n : ℕ, (D : A → A)^[n] a = 0 |
|
|
| open MvPolynomial |
|
|
| theorem commuting_derivations_with_ordinary_extras (n : ℕ) (hn : 4 ≤ n) : |
| ∃ d : Fin (n-1) → Derivation ℂ (MvPolynomial (Fin n) ℂ) (MvPolynomial (Fin n) ℂ), |
| (∀ i, d i (extendedF hn + 1) = 0) ∧ |
| (∀ i j p, d i (d j p) = d j (d i p)) ∧ |
| (∀ i, LocallyNilpotent (d i)) ∧ |
| LinearIndependent (MvPolynomial (Fin n) ℂ) d ∧ |
| (∀ p, (∀ i, d i p = 0) ↔ |
| p ∈ Algebra.adjoin ℂ ({extendedF hn} : Set (MvPolynomial (Fin n) ℂ))) ∧ |
| (∀ g, (∀ i, d i g = 0) → |
| ¬ ∃ (e : MvPolynomial (Fin n) ℂ ≃ₐ[ℂ] MvPolynomial (Fin n) ℂ) (j : Fin n), |
| e (X j) = g) ∧ |
| (∀ i : Fin (n-4), d (blocksToOutput hn (Sum.inl i)) = |
| pderiv (blocksToAmbient hn (Sum.inl i))) := by |
| sorry |
|
|
| end CommutingDerivations |
|
|
| end AbhyankarSathaye |
|
|
| end |
|
|
| end OAI |
|
|
