| import OAI.NumberTheory.PiExponent.Jets.BranchContact | |
| import OAI.NumberTheory.PiExponent.Jets.DVRBranch | |
| import OAI.NumberTheory.PiExponent.Jets.OrdinaryBranchEvaluation | |
| namespace OAI | |
| noncomputable section | |
| namespace PiExponent.OrdinaryDVRBranchJet | |
| open OrdinaryAuxiliaryJet | |
| variable {A : Type*} [CommRing A] [IsDomain A] [IsDiscreteValuationRing A] | |
| [Algebra ℂ A] [Algebra.IsIntegral ℂ (IsLocalRing.ResidueField A)] | |
| theorem expansion_aeval {n : ℕ} (x : Fin n → A) (p : MvPolynomial (Fin n) ℂ) : | |
| DVRBranch.expansion ℂ A (MvPolynomial.aeval x p) = | |
| MvPolynomial.aeval (fun i => DVRBranch.expansion ℂ A (x i)) p := by | |
| have h : (DVRBranch.expansion ℂ A).comp (MvPolynomial.aeval x) = | |
| MvPolynomial.aeval (fun i => DVRBranch.expansion ℂ A (x i)) := by | |
| apply MvPolynomial.algHom_ext | |
| intro i | |
| simp | |
| exact AlgHom.congr_fun h p | |
| def coordinates {n : ℕ} (c : Fin n → ℂ) (x : Fin n → A) : | |
| Fin n → PowerSeries ℂ := | |
| branchCoordinates c (fun i => DVRBranch.expansion ℂ A (x i)) | |
| theorem coordinates_centered {n : ℕ} (c : Fin n → ℂ) (x : Fin n → A) | |
| (hx : ∀ i, CurveLocalOrder.residueAugmentation ℂ A (x i) = c i) : | |
| ∀ i, PowerSeries.constantCoeff (coordinates c x i) = 0 := by | |
| intro i | |
| change PowerSeries.constantCoeff | |
| (DVRBranch.expansion ℂ A (x i) - PowerSeries.C (c i)) = 0 | |
| rw [map_sub, DVRBranch.expansion_constantCoeff, hx i, | |
| PowerSeries.constantCoeff_C, sub_self] | |
| theorem coordinates_nonzero {n : ℕ} (c : Fin n → ℂ) (x : Fin n → A) | |
| (hnc : ∃ i, x i ≠ algebraMap ℂ A (c i)) : | |
| ∃ i, coordinates c x i ≠ 0 := by | |
| obtain ⟨i, hi⟩ := hnc | |
| refine ⟨i, ?_⟩ | |
| intro hz | |
| apply hi | |
| apply DVRBranch.expansion_injective ℂ A | |
| rw [AlgHom.commutes] | |
| change DVRBranch.expansion ℂ A (x i) = PowerSeries.C (c i) | |
| exact sub_eq_zero.mp hz | |
| def contact {n : ℕ} (v : Fin n → ℚ) (c : Fin n → ℂ) (x : Fin n → A) | |
| (hnc : ∃ i, x i ≠ algebraMap ℂ A (c i)) : ℚ := | |
| BranchContact.contact v (coordinates c x) (coordinates_nonzero c x hnc) | |
| theorem contact_pos {n : ℕ} (v : Fin n → ℚ) (hv : ∀ i, 0 < v i) | |
| (c : Fin n → ℂ) (x : Fin n → A) | |
| (hx : ∀ i, CurveLocalOrder.residueAugmentation ℂ A (x i) = c i) | |
| (hnc : ∃ i, x i ≠ algebraMap ℂ A (c i)) : | |
| 0 < contact v c x hnc := | |
| BranchContact.contact_pos v hv _ _ (coordinates_centered c x hx) | |
| theorem ordinaryWord_addVal_lower {n : ℕ} | |
| (v : Fin n → ℚ) (hv : ∀ i, 0 < v i) (H : ℚ) | |
| (c : Fin n → ℂ) (x : Fin n → A) | |
| (hx : ∀ i, CurveLocalOrder.residueAugmentation ℂ A (x i) = c i) | |
| (hnc : ∃ i, x i ≠ algebraMap ℂ A (c i)) | |
| (p : MvPolynomial (Fin n) ℂ) | |
| (hp : formalJet c p ∈ JetGeometry.rationalWeightedIdeal v (fun i => (hv i).le) H) | |
| (word : List (Fin n)) : | |
| (⌈contact v c x hnc * (H - (word.map v).sum)⌉₊ : ℕ∞) ≤ | |
| IsDiscreteValuationRing.addVal A | |
| (MvPolynomial.aeval x (OrdinaryDerivatives.word n word p)) := by | |
| have h := ordinaryWord_branch_order c (fun i => DVRBranch.expansion ℂ A (x i)) | |
| (fun i => by simpa using hx i) v (fun i => (hv i).le) | |
| (contact v c x hnc) H (contact_pos v hv c x hx hnc).le p hp | |
| (BranchContact.contact_bound v hv _ _) word | |
| rwa [← expansion_aeval, DVRBranch.expansion_order] at h | |
| theorem ordinaryWord_addVal_toNat_lower {n : ℕ} | |
| (v : Fin n → ℚ) (hv : ∀ i, 0 < v i) (H : ℚ) | |
| (c : Fin n → ℂ) (x : Fin n → A) | |
| (hx : ∀ i, CurveLocalOrder.residueAugmentation ℂ A (x i) = c i) | |
| (hnc : ∃ i, x i ≠ algebraMap ℂ A (c i)) | |
| (p : MvPolynomial (Fin n) ℂ) | |
| (hp : formalJet c p ∈ JetGeometry.rationalWeightedIdeal v (fun i => (hv i).le) H) | |
| (word : List (Fin n)) | |
| (hne : MvPolynomial.aeval x (OrdinaryDerivatives.word n word p) ≠ 0) : | |
| contact v c x hnc * (H - (word.map v).sum) ≤ | |
| ((IsDiscreteValuationRing.addVal A | |
| (MvPolynomial.aeval x (OrdinaryDerivatives.word n word p))).toNat : ℚ) := by | |
| have h := ordinaryWord_addVal_lower v hv H c x hx hnc p hp word | |
| have hn := IsDiscreteValuationRing.addVal_eq_top_iff.not.mpr hne | |
| have hh := ENat.toNat_le_toNat h hn | |
| exact Nat.ceil_le.mp (by simpa using hh) | |
| theorem ordinaryWord_colength_lower {n : ℕ} | |
| (v : Fin n → ℚ) (hv : ∀ i, 0 < v i) (H : ℚ) | |
| (c : Fin n → ℂ) (x : Fin n → A) | |
| (hx : ∀ i, CurveLocalOrder.residueAugmentation ℂ A (x i) = c i) | |
| (hnc : ∃ i, x i ≠ algebraMap ℂ A (c i)) | |
| (p : MvPolynomial (Fin n) ℂ) | |
| (hp : formalJet c p ∈ JetGeometry.rationalWeightedIdeal v (fun i => (hv i).le) H) | |
| (word : List (Fin n)) | |
| (hne : MvPolynomial.aeval x (OrdinaryDerivatives.word n word p) ≠ 0) : | |
| contact v c x hnc * (H - (word.map v).sum) ≤ | |
| ((Module.length A (A ⧸ Ideal.span { | |
| MvPolynomial.aeval x (OrdinaryDerivatives.word n word p)})).toNat : ℚ) := by | |
| rw [CurveLocalOrder.length_quotient_span_eq_addVal hne] | |
| exact ordinaryWord_addVal_toNat_lower v hv H c x hx hnc p hp word hne | |
| end PiExponent.OrdinaryDVRBranchJet | |
| end | |
| end OAI | |