File size: 2,693 Bytes
82da87f | 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.CommRing
import Mathlib.Algebra.MvPolynomial.Degrees
import Mathlib.RingTheory.Ideal.Quotient.Operations
import Mathlib.Basic.Complex.Basic
namespace OAI
/-!
# The four-variable complex polynomial
Variables are zero-indexed internally: `X 0, X 1, X 2, X 3` mean the
paper's `x₁, x₂, x₃, x₄`. The stabilization uses `Fin.castSucc`,
so its fifth independent variable is `X 4`; no subring-preservation condition
is imposed on its automorphism.
Affine fibers are encoded by their complex coordinate algebras, not by sets.
An algebra equivalence to the three-variable polynomial algebra is precisely
the algebraic affine-space conclusion. Rectifiability is encoded by the
ambient algebra automorphism carrying the defining ideal to `(X 0)`.
-/
namespace StableCoordinate
open MvPolynomial
abbrev R (n : ℕ) := MvPolynomial (Fin n) ℂ
noncomputable def Q : R 4 :=
X 1 ^ 2 - X 3 ^ 2 + X 0 * X 2
noncomputable def f : R 4 :=
X 0 - 2 * Q * (Q * (X 1 + X 3) + X 0 * X 3)
/-- A coordinate is sent to the first variable by some complex algebra automorphism. -/
def IsCoordinate {n : ℕ} (hn : 0 < n) (g : R n) : Prop :=
∃ φ : R n ≃ₐ[ℂ] R n, φ g = X ⟨0, hn⟩
/-- Adjoin one independent variable; the original variables retain their indices. -/
noncomputable def stabilize : R 4 →ₐ[ℂ] R 5 :=
rename (Fin.castSucc : Fin 4 → Fin 5)
/-- No requirement to preserve the original four-variable subring is imposed. -/
def IsOneStableCoordinate (g : R 4) : Prop :=
∃ φ : R 5 ≃ₐ[ℂ] R 5, φ (stabilize g) = X 0
noncomputable def fiberIdeal (t : ℂ) : Ideal (R 4) :=
Ideal.span ({f - C t} : Set (R 4))
abbrev FiberRing (t : ℂ) := R 4 ⧸ fiberIdeal t
/-- An isomorphism of affine coordinate algebras, not a set-theoretic bijection. -/
def IsAffineThreeSpaceFiber (t : ℂ) : Prop :=
Nonempty (FiberRing t ≃ₐ[ℂ] R 3)
/-- Algebraic rectifiability of the closed hypersurface embedding. -/
def IsRectifiableFiber (t : ℂ) : Prop :=
∃ φ : R 4 ≃ₐ[ℂ] R 4,
Ideal.map φ.toRingHom (fiberIdeal t) = Ideal.span ({X 0} : Set (R 4))
/-- Complete first principal group, with the advertised exact total degree. -/
def PolynomialPackage : Prop :=
f.totalDegree = 5 ∧
IsOneStableCoordinate f ∧
¬ IsCoordinate (by decide : 0 < 4) f
/-- Complete second principal group, universally including the zero fiber. -/
def FiberPackage : Prop :=
∀ t : ℂ, IsAffineThreeSpaceFiber t ∧ ¬ IsRectifiableFiber t
/-- Both principal groups must be proved together, without additional hypotheses. -/
def PrincipalPackage : Prop := PolynomialPackage ∧ FiberPackage
end StableCoordinate
end OAI
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