| import OAI.NumberTheory.PiExponent.Jets.BranchContact |
| import OAI.NumberTheory.PiExponent.Jets.DVRBranch |
| import OAI.NumberTheory.PiExponent.Jets.FormalBranchEvaluation |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace PiExponent.DVRBranchJet |
| open PiExponentApprox FormalBranchEvaluation |
|
|
| 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 {m : ℕ} (c : Fin m → ℂ) (y : A) (x : Fin m → A) : |
| Fin (m+1) → PowerSeries ℂ := |
| branchCoordinates c (DVRBranch.expansion ℂ A y) |
| (fun i => DVRBranch.expansion ℂ A (x i)) |
|
|
| theorem coordinates_centered {m : ℕ} (c : Fin m → ℂ) (y : A) (x : Fin m → A) |
| (hy : CurveLocalOrder.residueAugmentation ℂ A y = 1) |
| (hx : ∀ i, CurveLocalOrder.residueAugmentation ℂ A (x i) = c i) : |
| ∀ i, PowerSeries.constantCoeff (coordinates c y x i) = 0 := by |
| intro i |
| have ht : MvPowerSeries.constantCoeff (DVRBranch.expansion ℂ A y - 1) = 0 := by |
| change PowerSeries.constantCoeff (DVRBranch.expansion ℂ A y - 1) = 0 |
| simp [hy] |
| cases i using Fin.cases with |
| | zero => exact ht |
| | succ i => |
| change PowerSeries.constantCoeff |
| (DVRBranch.expansion ℂ A (x i) - PowerSeries.C (c i) - |
| PowerSeries.subst (DVRBranch.expansion ℂ A y - 1) (PowerSeries.log ℂ)) = 0 |
| change MvPowerSeries.constantCoeff |
| (DVRBranch.expansion ℂ A (x i) - PowerSeries.C (c i) - |
| PowerSeries.subst (DVRBranch.expansion ℂ A y - 1) (PowerSeries.log ℂ)) = 0 |
| rw [map_sub, PowerSeries.constantCoeff_subst_eq_zero ht _ PowerSeries.constantCoeff_log, |
| sub_zero, map_sub] |
| change PowerSeries.constantCoeff (DVRBranch.expansion ℂ A (x i)) - |
| PowerSeries.constantCoeff (PowerSeries.C (c i)) = 0 |
| rw [DVRBranch.expansion_constantCoeff, hx i, PowerSeries.constantCoeff_C, sub_self] |
|
|
| theorem coordinates_nonzero {m : ℕ} (c : Fin m → ℂ) (y : A) (x : Fin m → A) |
| (hnc : y ≠ 1 ∨ ∃ i, x i ≠ algebraMap ℂ A (c i)) : |
| ∃ i, coordinates c y x i ≠ 0 := by |
| classical |
| by_contra hn |
| push Not at hn |
| have hy : y = 1 := by |
| apply DVRBranch.expansion_injective ℂ A |
| have h0 := hn 0 |
| change DVRBranch.expansion ℂ A y - 1 = 0 at h0 |
| simpa only [map_one] using sub_eq_zero.mp h0 |
| rcases hnc with hnc | ⟨i, hi⟩ |
| · exact hnc hy |
| · apply hi |
| apply DVRBranch.expansion_injective ℂ A |
| have hs := hn i.succ |
| change DVRBranch.expansion ℂ A (x i) - PowerSeries.C (c i) - |
| PowerSeries.subst (DVRBranch.expansion ℂ A y - 1) (PowerSeries.log ℂ) = 0 at hs |
| rw [hy, map_one, sub_self, |
| PowerSeries.subst_zero_of_constantCoeff_zero PowerSeries.constantCoeff_log, sub_zero] at hs |
| rw [AlgHom.commutes] |
| change DVRBranch.expansion ℂ A (x i) = PowerSeries.C (c i) |
| exact sub_eq_zero.mp hs |
|
|
| def contact {m : ℕ} (v : Fin (m+1) → ℚ) (c : Fin m → ℂ) (y : A) (x : Fin m → A) |
| (hnc : y ≠ 1 ∨ ∃ i, x i ≠ algebraMap ℂ A (c i)) : ℚ := |
| BranchContact.contact v (coordinates c y x) (coordinates_nonzero c y x hnc) |
|
|
| theorem contact_pos {m : ℕ} (v : Fin (m+1) → ℚ) (hv : ∀ i, 0 < v i) |
| (c : Fin m → ℂ) (y : A) (x : Fin m → A) |
| (hy : CurveLocalOrder.residueAugmentation ℂ A y = 1) |
| (hx : ∀ i, CurveLocalOrder.residueAugmentation ℂ A (x i) = c i) |
| (hnc : y ≠ 1 ∨ ∃ i, x i ≠ algebraMap ℂ A (c i)) : |
| 0 < contact v c y x hnc := |
| BranchContact.contact_pos v hv _ _ (coordinates_centered c y x hy hx) |
|
|
| theorem frameWord_addVal_lower {m : ℕ} |
| (v : Fin (m+1) → ℚ) (hv : ∀ i, 0 < v i) (H : ℚ) |
| (c : Fin m → ℂ) (y : A) (x : Fin m → A) |
| (hy : CurveLocalOrder.residueAugmentation ℂ A y = 1) |
| (hx : ∀ i, CurveLocalOrder.residueAugmentation ℂ A (x i) = c i) |
| (hnc : y ≠ 1 ∨ ∃ i, x i ≠ algebraMap ℂ A (c i)) |
| (p : FramePolynomial m) |
| (hp : FormalLogJet.formalJet c p ∈ |
| JetGeometry.rationalWeightedIdeal v (fun i => (hv i).le) H) |
| (word : List (Fin (m+1))) |
| (hne : MvPolynomial.aeval (Fin.cases y x) (polynomialFrameWord m word p) ≠ 0) : |
| contact v c y x hnc * (H - (word.map v).sum) ≤ |
| ((IsDiscreteValuationRing.addVal A |
| (MvPolynomial.aeval (Fin.cases y x) (polynomialFrameWord m word p))).toNat : ℚ) := by |
| have heval (q : FramePolynomial m) : |
| DVRBranch.expansion ℂ A (MvPolynomial.aeval (Fin.cases y x) q) = |
| MvPolynomial.aeval (Fin.cases (DVRBranch.expansion ℂ A y) |
| (fun i => DVRBranch.expansion ℂ A (x i))) q := by |
| rw [expansion_aeval] |
| apply congrArg (fun z => MvPolynomial.aeval z q) |
| funext i |
| cases i using Fin.cases <;> rfl |
| have hne' : MvPolynomial.aeval (Fin.cases (DVRBranch.expansion ℂ A y) |
| (fun i => DVRBranch.expansion ℂ A (x i))) (polynomialFrameWord m word p) ≠ 0 := by |
| rw [← heval] |
| exact (map_ne_zero_iff _ (DVRBranch.expansion_injective ℂ A)).mpr hne |
| have hb := polynomialFrameWord_branch_order_toNat c (DVRBranch.expansion ℂ A y) |
| (fun i => DVRBranch.expansion ℂ A (x i)) |
| (by simpa using hy) (fun i => by simpa using hx i) |
| v (fun i => (hv i).le) (contact v c y x hnc) H |
| (contact_pos v hv c y x hy hx hnc).le p hp |
| (BranchContact.contact_bound v hv _ _) word hne' |
| rw [← heval, DVRBranch.expansion_order] at hb |
| exact hb |
|
|
| theorem frameWord_colength_lower {m : ℕ} |
| (v : Fin (m+1) → ℚ) (hv : ∀ i, 0 < v i) (H : ℚ) |
| (c : Fin m → ℂ) (y : A) (x : Fin m → A) |
| (hy : CurveLocalOrder.residueAugmentation ℂ A y = 1) |
| (hx : ∀ i, CurveLocalOrder.residueAugmentation ℂ A (x i) = c i) |
| (hnc : y ≠ 1 ∨ ∃ i, x i ≠ algebraMap ℂ A (c i)) |
| (p : FramePolynomial m) |
| (hp : FormalLogJet.formalJet c p ∈ |
| JetGeometry.rationalWeightedIdeal v (fun i => (hv i).le) H) |
| (word : List (Fin (m+1))) |
| (hne : MvPolynomial.aeval (Fin.cases y x) (polynomialFrameWord m word p) ≠ 0) : |
| contact v c y x hnc * (H - (word.map v).sum) ≤ |
| ((Module.length A (A ⧸ Ideal.span { |
| MvPolynomial.aeval (Fin.cases y x) (polynomialFrameWord m word p)})).toNat : ℚ) := by |
| rw [CurveLocalOrder.length_quotient_span_eq_addVal hne] |
| exact frameWord_addVal_lower v hv H c y x hy hx hnc p hp word hne |
|
|
| end PiExponent.DVRBranchJet |
|
|
| end |
|
|
| end OAI |
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|
