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b797a5b | 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 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 | 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
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