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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 99 100 101 102 103 104 | import OAI.NumberTheory.PiExponent.Geometry.CurveCenters
import OAI.NumberTheory.PiExponent.Geometry.WeightedCurveRigidity
namespace OAI
noncomputable section
namespace PiExponent.CurveFieldRigidity
open scoped BigOperators
open PiExponentApprox PersistentWeightComparison CurveCenters CurveValuationCenter
variable {E : Type*} [Field E] [Algebra ℂ E]
def coordinateKernel {n : ℕ} (x : Fin n → E) : Ideal (MvPolynomial (Fin n) ℂ) :=
RingHom.ker (MvPolynomial.aeval x).toRingHom
instance coordinateKernel_isPrime {n : ℕ} (x : Fin n → E) : (coordinateKernel x).IsPrime :=
RingHom.ker_isPrime _
@[simp] theorem mem_coordinateKernel {n : ℕ} (x : Fin n → E)
(p : MvPolynomial (Fin n) ℂ) : p ∈ coordinateKernel x ↔ MvPolynomial.aeval x p = 0 :=
Iff.rfl
theorem logarithmic_coordinate_eq_one {m : ℕ}
(x : Fin (m+1) → E) (a : Fin (m+1) → ℂ) (ha : a 0 = 1)
(p : NormalizedPlace ℂ E) (hp : Centered x a p)
(hheight : (coordinateKernel x).height ≤ m)
(rho : Fin (m+1) → ℚ) (hrho : ∀ i, 0 < rho i)
(cost : Fin (m+1) → ℝ) (hcost : ∀ i, 0 < cost i)
(sigma N : ℝ) (hsigma : 0 < sigma) (hN : 0 < N)
(hrect : UniformRectangles (m+1) cost (sigma / ((m : ℝ)+2)) N)
(F : FramePolynomial m) (hF0 : F ≠ 0)
(hF : HasWeightedDegreeLE (fun i => (rho i : ℝ)) N F)
(hvanish : ∀ word : List (Fin (m+1)), frameWordCost cost word ≤ sigma*N →
MvPolynomial.aeval x (polynomialFrameWord m word F) = 0)
(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)) →
comparisonConstant m sigma * (∏ j ∈ B, cost j) < ∏ j ∈ A, (rho j : ℝ)) :
x 0 = 1 := by
have hx0 : x 0 ≠ 0 := by
intro hz
have hc := hp.constant_coordinate 0 0 (by simpa using hz)
rw [ha] at hc
exact zero_ne_one hc
have hY : MvPolynomial.X (0 : Fin (m+1)) ∉ coordinateKernel x := by
simpa only [mem_coordinateKernel, MvPolynomial.aeval_X] using hx0
have hepsilon : 0 < sigma / ((m : ℝ)+2) := by positivity
have hbound : ((m : ℝ)+2) * ((sigma / ((m : ℝ)+2))*N) = sigma*N := by
have hm : (m : ℝ)+2 ≠ 0 := by positivity
field_simp
obtain ⟨c, hc⟩ := CurveComponentRigidity.coordinate_constant_of_persistent_comparison
(fun i => (rho i : ℝ)) cost (comparisonConstant m sigma)
((sigma / ((m : ℝ)+2))*N) (fun i => (hcost i).le) (by positivity)
F hF0 (coordinateKernel x) inferInstance hheight hY
(fun word hword => hvanish word (by simpa only [hbound] using hword))
(logarithmic_persistent_comparison rho hrho cost hcost sigma N hsigma hN hrect
F hF (coordinateKernel x) hY) hseparated
have hxc : x 0 = algebraMap ℂ E c := by
simpa only [mem_coordinateKernel, map_sub, MvPolynomial.aeval_X,
MvPolynomial.aeval_C, sub_eq_zero] using hc
have hc1 := hp.constant_coordinate 0 c hxc
rw [ha] at hc1
simpa only [hc1, map_one] using hxc
theorem ordinary_coordinate_constant {m n : ℕ} (hn : n ≤ m+2)
(x : Fin n → E) (d : ℕ) (hd : d ≤ m)
(hheight : (coordinateKernel x).height ≤ d)
(rho : Fin n → ℚ) (hrho : ∀ i, 0 < rho i)
(cost : Fin n → ℝ) (hcost : ∀ i, 0 < cost i)
(sigma N : ℝ) (hsigma : 0 < sigma) (hN : 0 < N)
(hrect : UniformRectangles n cost (sigma / ((m : ℝ)+2)) N)
(F : OrdinaryDerivatives.Polynomial n) (hF0 : F ≠ 0)
(hF : WeightedSliceDegree.SupportBound (fun i => (rho i : ℝ)) N F)
(hvanish : ∀ word : List (Fin n), DerivativeIdeals.wordCost cost word ≤ sigma*N →
MvPolynomial.aeval x (OrdinaryDerivatives.word n word F) = 0)
(hseparated : ∀ A B : Finset (Fin n), A.card = B.card →
∀ i, i ∈ A → i ∉ B → (∀ j, i < j → (j ∈ A ↔ j ∈ B)) →
comparisonConstant m sigma * (∏ j ∈ B, cost j) < ∏ j ∈ A, (rho j : ℝ)) :
∃ i : Fin n, ∃ c : ℂ, x i = algebraMap ℂ E c := by
obtain ⟨i,c,hc⟩ := WeightedCurveRigidity.ordinary_coordinate_constant hn
rho hrho cost hcost sigma N hsigma hN hrect F hF0 hF d hd
(coordinateKernel x) inferInstance hheight hvanish hseparated
refine ⟨i,c,?_⟩
simpa only [mem_coordinateKernel, map_sub, MvPolynomial.aeval_X,
MvPolynomial.aeval_C, sub_eq_zero] using hc
theorem fibre_generates {m : ℕ} (x : Fin m → E)
(hx : IntermediateField.adjoin ℂ
(Set.range (Fin.cases (1 : E) x : Fin (m+1) → E)) = ⊤) :
IntermediateField.adjoin ℂ (Set.range x) = ⊤ := by
apply top_unique
rw [← hx]
apply IntermediateField.adjoin_le_iff.mpr
rintro _ ⟨i, rfl⟩
cases i using Fin.cases with
| zero => exact one_mem _
| succ i => exact IntermediateField.subset_adjoin ℂ _ ⟨i,rfl⟩
end PiExponent.CurveFieldRigidity
end
end OAI
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