| 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 |
| |
