| import OAI.NumberTheory.PiExponent.Approximation.GenericNormalRigidity |
| import OAI.NumberTheory.PiExponent.Geometry.CurveContact |
| import OAI.NumberTheory.PiExponent.Jets.PrimeNormalCotangent |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace PiExponent.CurveComponentRigidity |
|
|
| open scoped BigOperators |
| open PiExponentApprox NormalBasisRigidity |
|
|
| theorem residueY_ne_zero {m : ℕ} (Q : Ideal (FramePolynomial m)) [Q.IsPrime] |
| (hY : MvPolynomial.X (0 : Fin (m + 1)) ∉ Q) : |
| primeResidueMap Q (MvPolynomial.X (0 : Fin (m + 1))) ≠ 0 := by |
| intro h |
| exact hY (Ideal.algebraMap_residueField_eq_zero.mp h) |
|
|
| def NormalProductComparison {m : ℕ} (Q : Ideal (FramePolynomial m)) [Q.IsPrime] |
| (hY : MvPolynomial.X (0 : Fin (m + 1)) ∉ Q) |
| (w cost : Fin (m + 1) → ℝ) (M : ℝ) : Prop := |
| ∀ A B : Finset (Fin (m + 1)), |
| IsNormalBasis (K := Q.ResidueField) |
| (fun j => (polynomialTangent (primeResidueMap Q) Q).mkQ |
| (Pi.basisFun Q.ResidueField _ j)) A → |
| IsNormalBasis (K := Q.ResidueField) |
| (fun j => (polynomialTangent (primeResidueMap Q) Q).mkQ |
| (frameBasis m (primeResidueMap Q (MvPolynomial.X 0)) (residueY_ne_zero Q hY) j)) B → |
| (∏ j ∈ A, w j) ≤ M * ∏ j ∈ B, cost j |
|
|
| def PersistentNormalComparison {m : ℕ} (w cost : Fin (m + 1) → ℝ) |
| (M delta : ℝ) (F : FramePolynomial m) (P : Ideal (FramePolynomial m)) |
| (hY : MvPolynomial.X (0 : Fin (m + 1)) ∉ P) : Prop := |
| ∀ r : ℕ, r < m + 2 → ∀ Q : Ideal (FramePolynomial m), ∀ hQ : Q.IsPrime, |
| ∀ hQP : Q ≤ P, |
| Q ∈ (frameDerivativeIdeal cost ((r : ℝ) * delta) F).minimalPrimes → |
| Q ∈ (frameDerivativeIdeal cost (((r : ℝ) + 1) * delta) F).minimalPrimes → |
| 1 ≤ Q.height → Q.height ≤ m → |
| (∀ p ∈ frameDerivativeIdeal cost ((r : ℝ) * delta) F, |
| ∀ word : List (Fin (m + 1)), frameWordCost cost word ≤ delta → |
| polynomialFrameWord m word p ∈ Q) → |
| @NormalProductComparison m Q hQ (fun h => hY (hQP h)) w cost M |
|
|
| theorem coordinate_constant_of_persistent_comparison {m : ℕ} |
| (w cost : Fin (m + 1) → ℝ) (M delta : ℝ) |
| (hcost : ∀ i, 0 ≤ cost i) (hdelta : 0 ≤ delta) |
| (F : FramePolynomial m) (hF : F ≠ 0) |
| (P : Ideal (FramePolynomial m)) (hP : P.IsPrime) (hheight : P.height ≤ m) |
| (hY : MvPolynomial.X (0 : Fin (m + 1)) ∉ P) |
| (hvanish : ∀ word : List (Fin (m + 1)), |
| frameWordCost cost word ≤ ((m : ℝ) + 2) * delta → polynomialFrameWord m word F ∈ P) |
| (hcomparison : PersistentNormalComparison w cost M delta F P hY) |
| (hseparated : ∀ A B : Finset (Fin (m + 1)), A.card = B.card → |
| ∀ i, i ≠ 0 → i ∈ A → i ∉ B → |
| (∀ j, i < j → (j ∈ A ↔ j ∈ B)) → |
| M * (∏ j ∈ B, cost j) < ∏ j ∈ A, w j) : |
| ∃ c : ℂ, MvPolynomial.X (0 : Fin (m + 1)) - MvPolynomial.C c ∈ P := by |
| obtain ⟨r, hr, Q, hQr, hQnext, hQP, hQlo, hQhi, hpersist⟩ := |
| exists_persistent_frame_component cost hcost delta hdelta F hF P hP hheight hvanish |
| let : Q.IsPrime := hQr.isPrime |
| have hYQ : MvPolynomial.X (0 : Fin (m + 1)) ∉ Q := fun h => hY (hQP h) |
| have hFQ : F ∈ Q := hQr.le |
| (polynomial_mem_frameDerivativeIdeal cost ((r : ℝ) * delta) F |
| (mul_nonneg (Nat.cast_nonneg r) hdelta)) |
| have hQ0 : Q ≠ ⊥ := by |
| intro h |
| exact hF (by simpa only [h, Ideal.mem_bot] using hFQ) |
| have hcomp := hcomparison r hr Q inferInstance hQP hQr hQnext hQlo hQhi hpersist |
| let : Algebra.EssFiniteType ℂ Q.ResidueField := |
| Algebra.EssFiniteType.comp ℂ (FramePolynomial m) Q.ResidueField |
| have hker (p : FramePolynomial m) : primeResidueMap Q p = 0 ↔ p ∈ Q := |
| Ideal.algebraMap_residueField_eq_zero |
| obtain ⟨c, hc⟩ := constant_zeroth_coordinate_of_generic_normal_separation |
| (primeResidueMap Q) Q hker hQ0 (residueY_ne_zero Q hYQ) |
| (weighted_normalBasis_exclusion _ _ w cost M hcomp hseparated) |
| exact ⟨c, hQP hc⟩ |
|
|
| theorem coordinate_eq_one_of_persistent_comparison {m : ℕ} |
| (w cost : Fin (m + 1) → ℝ) (M delta : ℝ) |
| (hcost : ∀ i, 0 ≤ cost i) (hdelta : 0 ≤ delta) |
| (F : FramePolynomial m) (hF : F ≠ 0) |
| (P : Ideal (FramePolynomial m)) (hP : P.IsPrime) (hheight : P.height ≤ m) |
| (hY : MvPolynomial.X (0 : Fin (m + 1)) ∉ P) |
| (hvanish : ∀ word : List (Fin (m + 1)), |
| frameWordCost cost word ≤ ((m : ℝ) + 2) * delta → polynomialFrameWord m word F ∈ P) |
| (hcomparison : PersistentNormalComparison w cost M delta F P hY) |
| (hseparated : ∀ A B : Finset (Fin (m + 1)), A.card = B.card → |
| ∀ i, i ≠ 0 → i ∈ A → i ∉ B → |
| (∀ j, i < j → (j ∈ A ↔ j ∈ B)) → |
| M * (∏ j ∈ B, cost j) < ∏ j ∈ A, w j) |
| (a : Fin (m + 1) → ℂ) (ha : a 0 = 1) |
| (hcenter : ∀ p ∈ P, MvPolynomial.eval a p = 0) : |
| MvPolynomial.X (0 : Fin (m + 1)) - 1 ∈ P := |
| constant_coordinate_eq_one_of_center P a ha hcenter |
| (coordinate_constant_of_persistent_comparison w cost M delta hcost hdelta |
| F hF P hP hheight hY hvanish hcomparison hseparated) |
|
|
| end PiExponent.CurveComponentRigidity |
| end |
|
|
| end OAI |
|
|
