File size: 2,393 Bytes
dae740b | 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 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 | 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
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