| import Mathlib |
| import OAI.Algebra.FrobeniusObstruction.Obstruction |
| import OAI.Algebra.AlgebraicObstruction.Specialization |
| import OAI.Algebra.AlgebraicObstruction.IdealVanishing |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped BigOperators |
|
|
| namespace BoundaryOnly.FormalObstruction.CoefficientRing |
| open MvPowerSeries Matrix FormalCorrection |
| open scoped Classical |
| variable {R : Type*} [CommRing R] {d : ℕ} {n : Fin d → ℕ} |
|
|
| theorem plusCoordinates_centered (i : Fin d) : |
| ∀ j, constantCoeff (plusCoordinates (k := R) n i j) = 0 := by |
| intro j; cases j <;> simp [plusCoordinates] |
| theorem minusCoordinates_centered (i : Fin d) : |
| ∀ j, constantCoeff (minusCoordinates (k := R) n i j) = 0 := by |
| intro j; cases j <;> simp [minusCoordinates] |
|
|
| theorem potential_vanishes (P : (i : Fin d) → MvPowerSeries (WallVar n i) R) |
| (hP : ∀ i, P i ∈ (originIdeal R (WallVar n i))^3) : Vanishes 3 (potential n P) := by |
| apply Vanishes.sum |
| intro i _ |
| simpa only [sub_eq_add_neg] using ((origin_pow_vanishes (hP i)).subst _ (plusCoordinates_centered i)).add |
| (((origin_pow_vanishes (hP i)).subst _ (minusCoordinates_centered i)).neg) |
|
|
| theorem graphCoordinates_centered (Y : InternalVar n → MvPowerSeries (GraphVar n) R) |
| (hY : ∀ c, Y c ∈ originIdeal R (GraphVar n)) : |
| ∀ i, constantCoeff (graphCoordinates n Y i) = 0 := by |
| intro i |
| cases i with |
| | inl a => exact constantCoeff_X _ |
| | inr c => exact mem_origin_iff.mp (hY c) |
|
|
| def correctedGraph (Y : InternalVar n → MvPowerSeries (GraphVar n) R) |
| (δ : InternalVar n → MvPowerSeries (GraphVar n) R) := fun c => Y c + δ c |
|
|
| theorem graph_corrected (Y : InternalVar n → MvPowerSeries (GraphVar n) R) |
| (δ : InternalVar n → MvPowerSeries (GraphVar n) R) : |
| graphCoordinates n (correctedGraph Y δ) = shiftedGraph (graphCoordinates n Y) δ := by |
| funext i |
| cases i <;> rfl |
|
|
| def negativeAmbient (P : (i : Fin d) → MvPowerSeries (WallVar n i) R) (i : Fin d) : |
| MvPowerSeries (AmbientVar n) R := |
| subst (fun j => (X (.inr (false,⟨i,j⟩)) : MvPowerSeries (AmbientVar n) R)) (slopeDerivative n P i) |
|
|
| theorem subst_negativeAmbient (P : (i : Fin d) → MvPowerSeries (WallVar n i) R) |
| (Y : InternalVar n → MvPowerSeries (GraphVar n) R) |
| (hY : ∀ c, Y c ∈ originIdeal R (GraphVar n)) (i : Fin d) : |
| subst (graphCoordinates n Y) (negativeAmbient P i) = negativeSlopeValue n P Y i := by |
| let hx := hasSubst_of_constantCoeff_zero (graphCoordinates_centered Y hY) |
| let hi := hasSubst_of_constantCoeff_zero (fun j : Fin (n i) => |
| constantCoeff_X (R := R) (Sum.inr (false,⟨i,j⟩) : AmbientVar n)) |
| rw [negativeAmbient, subst_comp_subst_apply hi hx] |
| simp only [subst_X hx, graphCoordinates, negativeSlopeValue] |
|
|
| theorem complementaryMatrix_jet |
| (Y Z : InternalVar n → MvPowerSeries (GraphVar n) R) |
| (h : ∀ c, Jet 2 (Z c) (Y c)) : complementaryMatrix n Z = complementaryMatrix n Y := by |
| ext c bt |
| unfold complementaryMatrix tangentMatrix |
| split_ifs <;> exact Jet.pderiv_constant (h _) _ |
|
|
| structure QuadraticData where |
| P : (i : Fin d) → MvPowerSeries (WallVar n i) R |
| Y : InternalVar n → MvPowerSeries (GraphVar n) R |
| order_three : ∀ i, P i ∈ (originIdeal R (WallVar n i))^3 |
| centered : ∀ c, Y c ∈ originIdeal R (GraphVar n) |
| complementary : IsUnit (complementaryMatrix n Y).det |
| graph_square : subst (graphCoordinates n Y) (potential n P) ∈ (gradientIdeal n P Y)^2 |
| 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 |
|
|
| theorem QuadraticData.correct (Q : QuadraticData (R := R) (n := n)) : |
| ∃ D : FormalData (k := R) n, D.P = Q.P ∧ |
| (∀ c, Jet 2 (D.Y c) (Q.Y c)) := by |
| let f := potential n Q.P |
| let x := graphCoordinates n Q.Y |
| let v := gradientValue f x |
| have hf : Vanishes 3 f := potential_vanishes Q.P Q.order_three |
| have hx : ∀ i, constantCoeff (x i) = 0 := graphCoordinates_centered Q.Y Q.centered |
| have hv : ∀ c, Vanishes 2 (v c) := gradientValue_vanishes f hf x hx |
| have hid : vectorIdeal v = gradientIdeal n Q.P Q.Y := rfl |
| obtain ⟨b,hb⟩ := quadratic_witness v (subst x f) (hid ▸ Q.graph_square) |
| obtain ⟨B,hzero,hjet⟩ := exists_zero_correction f hf x hx b hb |
| let Z := correctedGraph Q.Y (B *ᵥ v) |
| have hZjet : ∀ c, Jet 2 (Z c) (Q.Y c) := fun c => hjet (.inr c) |
| have hZ : ∀ c, Z c ∈ originIdeal R (GraphVar n) := by |
| intro c |
| apply mem_origin_iff.mpr |
| have hh := (hZjet c).mono (by omega : 1 ≤ 2) |
| have hh' := Vanishes.one_iff.mp hh |
| simp only [map_sub, mem_origin_iff.mp (Q.centered c), sub_zero] at hh' |
| exact hh' |
| have hgrad : gradientIdeal n Q.P Z = gradientIdeal n Q.P Q.Y := by |
| change vectorIdeal (gradientValue f (graphCoordinates n Z)) = vectorIdeal v |
| rw [graph_corrected] |
| exact gradient_ideal_transport f hf x hx B |
| have hdiff : ∀ i, negativeSlopeValue n Q.P Z i - negativeSlopeValue n Q.P Q.Y i ∈ |
| gradientIdeal n Q.P Q.Y := by |
| intro i |
| rw [← subst_negativeAmbient Q.P Z hZ, ← subst_negativeAmbient Q.P Q.Y Q.centered, graph_corrected] |
| exact scalar_transport_mem (negativeAmbient Q.P i) x hx v hv B |
| refine ⟨⟨Q.P, Z, Q.order_three, hZ, ?_, ?_, ?_, ?_⟩, rfl, hZjet⟩ |
| · rw [complementaryMatrix_jet Q.Y Z hZjet] |
| exact Q.complementary |
| · rw [graph_corrected] |
| exact hzero |
| · intro i j l |
| rw [hgrad] |
| exact Q.positive_cubic i j l |
| · intro i j l |
| rw [hgrad] |
| exact triple_transport_mem _ (hdiff i) (hdiff j) (hdiff l) (Q.negative_cubic i j l) |
|
|
| end BoundaryOnly.FormalObstruction.CoefficientRing |
| namespace BoundaryOnly.FormalObstruction.CoefficientRing |
|
|
| theorem no_integral_quadratic_data (R : Type*) [CommRing R] [IsDomain R] [CharZero R] |
| [Algebra.FiniteType ℤ R] (d : ℕ) (hd : 5 ≤ d) (n : Fin d → ℕ) : |
| ¬ Nonempty (QuadraticData (R := R) (n := n)) := by |
| rintro ⟨Q⟩ |
| obtain ⟨D,_,_⟩ := Q.correct |
| exact no_integral_formal_data R d hd n ⟨D⟩ |
| end BoundaryOnly.FormalObstruction.CoefficientRing |
|
|
| end |
|
|
| end OAI |
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