| import Mathlib |
| import OAI.Algebra.FrobeniusObstruction.LinearChange |
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| namespace OAI |
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| noncomputable section |
| open scoped BigOperators |
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| namespace BoundaryOnly.FormalObstruction |
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| abbrev SlopeVar (d : ℕ) := Fin d × Fin 3 |
| abbrev TangentVar {d : ℕ} (n : Fin d → ℕ) := (i : Fin d) × Fin (n i) |
| abbrev WallVar {d : ℕ} (n : Fin d → ℕ) (i : Fin d) := Fin 3 ⊕ Fin (n i) |
| abbrev InternalVar {d : ℕ} (n : Fin d → ℕ) := Bool × TangentVar n |
| abbrev GraphVar {d : ℕ} (n : Fin d → ℕ) := SlopeVar d ⊕ TangentVar n |
| abbrev AmbientVar {d : ℕ} (n : Fin d → ℕ) := SlopeVar d ⊕ InternalVar n |
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| def originIdeal (k : Type*) [CommRing k] (ι : Type*) : Ideal (MvPowerSeries ι k) := |
| Ideal.span (Set.range (MvPowerSeries.X : ι → MvPowerSeries ι k)) |
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| variable {k : Type*} [Field k] {d : ℕ} (n : Fin d → ℕ) |
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| def plusCoordinates (i : Fin d) : WallVar n i → MvPowerSeries (AmbientVar n) k |
| | .inl j => MvPowerSeries.X (.inl (i, j)) |
| | .inr y => MvPowerSeries.X (.inr (true, ⟨i, y⟩)) |
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| def minusCoordinates (i : Fin d) : WallVar n i → MvPowerSeries (AmbientVar n) k |
| | .inl _ => 0 |
| | .inr y => MvPowerSeries.X (.inr (false, ⟨i, y⟩)) |
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| def zeroSlopes (i : Fin d) : WallVar n i → MvPowerSeries (Fin (n i)) k |
| | .inl _ => 0 |
| | .inr y => MvPowerSeries.X y |
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| def potential (P : (i : Fin d) → MvPowerSeries (WallVar n i) k) : |
| MvPowerSeries (AmbientVar n) k := |
| ∑ i, (MvPowerSeries.subst (plusCoordinates n i) (P i) - |
| MvPowerSeries.subst (minusCoordinates n i) (P i)) |
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| def slopeDerivative (P : (i : Fin d) → MvPowerSeries (WallVar n i) k) |
| (i : Fin d) : MvPowerSeries (Fin (n i)) k := |
| MvPowerSeries.subst (zeroSlopes n i) (MvPowerSeries.pderiv (.inl 0) (P i)) |
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| def graphCoordinates (Y : InternalVar n → MvPowerSeries (GraphVar n) k) : |
| AmbientVar n → MvPowerSeries (GraphVar n) k |
| | .inl a => MvPowerSeries.X (.inl a) |
| | .inr c => Y c |
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| def restrictedGradient (P : (i : Fin d) → MvPowerSeries (WallVar n i) k) |
| (Y : InternalVar n → MvPowerSeries (GraphVar n) k) (c : InternalVar n) : |
| MvPowerSeries (GraphVar n) k := |
| MvPowerSeries.subst (graphCoordinates n Y) |
| (MvPowerSeries.pderiv (.inr c) (potential n P)) |
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| def gradientIdeal (P : (i : Fin d) → MvPowerSeries (WallVar n i) k) |
| (Y : InternalVar n → MvPowerSeries (GraphVar n) k) : |
| Ideal (MvPowerSeries (GraphVar n) k) := |
| Ideal.span (Set.range (restrictedGradient n P Y)) |
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| def negativeSlopeValue (P : (i : Fin d) → MvPowerSeries (WallVar n i) k) |
| (Y : InternalVar n → MvPowerSeries (GraphVar n) k) (i : Fin d) : |
| MvPowerSeries (GraphVar n) k := |
| MvPowerSeries.subst (fun j => Y (false, ⟨i, j⟩)) (slopeDerivative n P i) |
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| def slope (i : Fin d) : MvPowerSeries (GraphVar n) k := |
| MvPowerSeries.X (.inl (i, 0)) |
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| def swapBlocks (c : InternalVar n) : InternalVar n := (!c.1, c.2) |
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| def tangentMatrix (Y : InternalVar n → MvPowerSeries (GraphVar n) k) : |
| Matrix (InternalVar n) (TangentVar n) k := |
| fun c t => MvPowerSeries.constantCoeff (MvPowerSeries.pderiv (.inr t) (Y c)) |
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| |
| def complementaryMatrix (Y : InternalVar n → MvPowerSeries (GraphVar n) k) : |
| Matrix (InternalVar n) (InternalVar n) k := |
| fun c bt => if bt.1 then tangentMatrix n Y (swapBlocks n c) bt.2 |
| else tangentMatrix n Y c bt.2 |
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| |
| structure FormalData where |
| P : (i : Fin d) → MvPowerSeries (WallVar n i) k |
| Y : InternalVar n → MvPowerSeries (GraphVar n) k |
| order_three : ∀ i, P i ∈ (originIdeal k (WallVar n i)) ^ 3 |
| centered : ∀ c, Y c ∈ originIdeal k (GraphVar n) |
| complementary : (complementaryMatrix n Y).det ≠ 0 |
| graph_zero : MvPowerSeries.subst (graphCoordinates n Y) (potential n P) = 0 |
| positive_cubic : ∀ i j l : Fin d, |
| slope n i * slope n j * slope n l ∈ gradientIdeal n P Y |
| negative_cubic : ∀ i j l : Fin d, |
| negativeSlopeValue n P Y i * negativeSlopeValue n P Y j * |
| negativeSlopeValue n P Y l ∈ gradientIdeal n P Y |
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| end BoundaryOnly.FormalObstruction |
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| end |
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| end OAI |
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