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d9b3e9d | 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 | import Mathlib
namespace OAI
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)
def IsCoordinate {n : ℕ} (hn : 0 < n) (g : R n) : Prop :=
∃ φ : R n ≃ₐ[ℂ] R n, φ g = X ⟨0, hn⟩
noncomputable def stabilize : R 4 →ₐ[ℂ] R 5 :=
rename (Fin.castSucc : Fin 4 → Fin 5)
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
def IsAffineThreeSpaceFiber (t : ℂ) : Prop :=
Nonempty (FiberRing t ≃ₐ[ℂ] R 3)
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))
def PolynomialPackage : Prop :=
f.totalDegree = 5 ∧
IsOneStableCoordinate f ∧
¬ IsCoordinate (by decide : 0 < 4) f
def FiberPackage : Prop :=
∀ t : ℂ, IsAffineThreeSpaceFiber t ∧ ¬ IsRectifiableFiber t
def PrincipalPackage : Prop := PolynomialPackage ∧ FiberPackage
/-- The explicit degree-five stable noncoordinate and its nonrectifiable affine-space fibers. -/
theorem principal_package : PrincipalPackage := by
sorry
end StableCoordinate
end OAI
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