| import OAI.NumberTheory.PiExponent.Analysis.FormalLogTruncation | |
| import OAI.NumberTheory.PiExponent.Jets.PolynomialJetPackets | |
| namespace OAI | |
| noncomputable section | |
| namespace PiExponent.AlgebraicJetPackets | |
| open MvPolynomial | |
| variable {R : Type*} [CommRing R] {m : ℕ} | |
| def triangularMap (c : Fin m → R) (G : Fin m → Polynomial R) : | |
| MvPolynomial (Fin (m+1)) R →ₐ[R] MvPolynomial (Fin (m+1)) R := | |
| MvPolynomial.aeval (Fin.cases (1 + X 0) | |
| (fun i => C (c i) + X i.succ + Polynomial.aeval (X 0) (G i))) | |
| def inverseTriangularMap (c : Fin m → R) (G : Fin m → Polynomial R) : | |
| MvPolynomial (Fin (m+1)) R →ₐ[R] MvPolynomial (Fin (m+1)) R := | |
| MvPolynomial.aeval (Fin.cases (X 0 - 1) | |
| (fun i => X i.succ - C (c i) - Polynomial.aeval (X 0 - 1) (G i))) | |
| @[simp] theorem triangularMap_X_zero (c : Fin m → R) (G : Fin m → Polynomial R) : | |
| triangularMap c G (X 0) = 1 + X 0 := by simp [triangularMap] | |
| @[simp] theorem triangularMap_X_succ (c : Fin m → R) (G : Fin m → Polynomial R) | |
| (i : Fin m) : | |
| triangularMap c G (X i.succ) = C (c i) + X i.succ + Polynomial.aeval (X 0) (G i) := by | |
| simp [triangularMap] | |
| @[simp] theorem inverseTriangularMap_X_zero (c : Fin m → R) (G : Fin m → Polynomial R) : | |
| inverseTriangularMap c G (X 0) = X 0 - 1 := by simp [inverseTriangularMap] | |
| @[simp] theorem inverseTriangularMap_X_succ (c : Fin m → R) (G : Fin m → Polynomial R) | |
| (i : Fin m) : | |
| inverseTriangularMap c G (X i.succ) = | |
| X i.succ - C (c i) - Polynomial.aeval (X 0 - 1) (G i) := by | |
| simp [inverseTriangularMap] | |
| theorem triangularMap_comp_inverse (c : Fin m → R) (G : Fin m → Polynomial R) : | |
| (triangularMap c G).comp (inverseTriangularMap c G) = AlgHom.id R _ := by | |
| apply MvPolynomial.algHom_ext | |
| intro i | |
| cases i using Fin.cases with | |
| | zero => simp | |
| | succ i => | |
| simp only [AlgHom.comp_apply, inverseTriangularMap_X_succ, map_sub, | |
| triangularMap_X_succ, MvPolynomial.algHom_C, MvPolynomial.algebraMap_eq, | |
| ← Polynomial.aeval_algHom_apply, triangularMap_X_zero, map_one, AlgHom.id_apply] | |
| rw [show (1 + X (0 : Fin (m+1)) - 1 : MvPolynomial (Fin (m+1)) R) = X 0 by ring] | |
| ring | |
| theorem inverseTriangularMap_comp (c : Fin m → R) (G : Fin m → Polynomial R) : | |
| (inverseTriangularMap c G).comp (triangularMap c G) = AlgHom.id R _ := by | |
| apply MvPolynomial.algHom_ext | |
| intro i | |
| cases i using Fin.cases with | |
| | zero => simp | |
| | succ i => | |
| simp only [AlgHom.comp_apply, triangularMap_X_succ, map_add, | |
| inverseTriangularMap_X_succ, MvPolynomial.algHom_C, MvPolynomial.algebraMap_eq, | |
| ← Polynomial.aeval_algHom_apply, inverseTriangularMap_X_zero, AlgHom.id_apply] | |
| ring | |
| def triangularEquiv (c : Fin m → R) (G : Fin m → Polynomial R) : | |
| MvPolynomial (Fin (m+1)) R ≃ₐ[R] MvPolynomial (Fin (m+1)) R := | |
| AlgEquiv.ofAlgHom (triangularMap c G) (inverseTriangularMap c G) | |
| (triangularMap_comp_inverse c G) (inverseTriangularMap_comp c G) | |
| theorem liftSeries_polynomial (p : Polynomial R) : | |
| FormalLogJet.liftSeries m (p : PowerSeries R) = | |
| Polynomial.aeval (MvPowerSeries.X (0 : Fin (m+1))) p := by | |
| have h : (FormalLogJet.liftSeries (R := R) m).comp Polynomial.coeToPowerSeries.ringHom = | |
| (Polynomial.aeval (MvPowerSeries.X (0 : Fin (m+1)))).toRingHom := by | |
| apply Polynomial.ringHom_ext | |
| · intro r | |
| simp [MvPowerSeries.algebraMap_apply] | |
| · simp | |
| exact RingHom.congr_fun h p | |
| theorem truncatedFormalJet_eq_polynomial (c : Fin m → ℂ) (T : Fin m → ℕ) | |
| (P : PiExponentApprox.FramePolynomial m) : | |
| FormalLogTruncation.truncatedFormalJet c T P = | |
| ((triangularMap c (fun i => PowerSeries.trunc (T i) (PowerSeries.log ℂ)) P) : | |
| MvPowerSeries (Fin (m+1)) ℂ) := by | |
| have h : FormalLogTruncation.truncatedFormalJet c T = | |
| (MvPolynomial.coeToMvPowerSeries.algHom ℂ).comp | |
| (triangularMap c (fun i => PowerSeries.trunc (T i) (PowerSeries.log ℂ))) := by | |
| apply MvPolynomial.algHom_ext | |
| intro i | |
| cases i using Fin.cases with | |
| | zero => | |
| simp [FormalLogTruncation.truncatedFormalJet] | |
| | succ i => | |
| simp only [FormalLogTruncation.truncatedFormalJet, MvPolynomial.aeval_X, | |
| Fin.cases_succ, AlgHom.comp_apply, triangularMap_X_succ, map_add] | |
| rw [liftSeries_polynomial, ← Polynomial.aeval_algHom_apply] | |
| simp [MvPowerSeries.algebraMap_apply] | |
| simpa using AlgHom.congr_fun h P | |
| theorem truncatedFormalJet_inverse (c : Fin m → ℂ) (T : Fin m → ℕ) | |
| (P : MvPolynomial (Fin (m+1)) ℂ) : | |
| FormalLogTruncation.truncatedFormalJet c T | |
| (inverseTriangularMap c (fun i => PowerSeries.trunc (T i) (PowerSeries.log ℂ)) P) = | |
| (P : MvPowerSeries (Fin (m+1)) ℂ) := by | |
| rw [truncatedFormalJet_eq_polynomial] | |
| have h := AlgHom.congr_fun | |
| (triangularMap_comp_inverse c (fun i => PowerSeries.trunc (T i) (PowerSeries.log ℂ))) P | |
| simpa using congrArg (fun Q : MvPolynomial (Fin (m+1)) ℂ => | |
| (Q : MvPowerSeries (Fin (m+1)) ℂ)) h | |
| theorem quotient_surjective_of_packet {A : Type*} [CommRing A] [Algebra R A] | |
| (v : Fin (m+1) → ℚ) (hv : ∀ i, 0 ≤ v i) (H : ℚ) | |
| (f : A →ₐ[R] MvPowerSeries (Fin (m+1)) R) | |
| (hf : Function.Surjective (fun p => JetGeometry.rationalCoefficientPacket v H (f p))) : | |
| Function.Surjective ((Ideal.Quotient.mkₐ R (JetGeometry.rationalWeightedIdeal v hv H)).comp f) := by | |
| intro q | |
| obtain ⟨F, rfl⟩ := Ideal.Quotient.mk_surjective q | |
| obtain ⟨p, hp⟩ := hf (JetGeometry.rationalCoefficientPacket v H F) | |
| refine ⟨p, ?_⟩ | |
| change Ideal.Quotient.mk _ (f p) = Ideal.Quotient.mk _ F | |
| apply Ideal.Quotient.eq.mpr | |
| exact (FormalLogTruncation.rationalCoefficientPacket_eq_iff v hv H _ _).mp hp | |
| def algebraicJetMap (c : Fin m → ℂ) (T : Fin m → ℕ) | |
| (v : Fin (m+1) → ℚ) (hv : ∀ i, 0 ≤ v i) (H : ℚ) : | |
| PiExponentApprox.FramePolynomial m →ₐ[ℂ] | |
| MvPowerSeries (Fin (m+1)) ℂ ⧸ JetGeometry.rationalWeightedIdeal (R := ℂ) v hv H := | |
| (Ideal.Quotient.mkₐ ℂ (JetGeometry.rationalWeightedIdeal (R := ℂ) v hv H)).comp | |
| (FormalLogTruncation.truncatedFormalJet c T) | |
| def algebraicJetIdeal (c : Fin m → ℂ) (T : Fin m → ℕ) | |
| (v : Fin (m+1) → ℚ) (hv : ∀ i, 0 ≤ v i) (H : ℚ) : | |
| Ideal (PiExponentApprox.FramePolynomial m) := | |
| RingHom.ker (algebraicJetMap c T v hv H) | |
| @[simp] theorem mem_algebraicJetIdeal (c : Fin m → ℂ) (T : Fin m → ℕ) | |
| (v : Fin (m+1) → ℚ) (hv : ∀ i, 0 ≤ v i) (H : ℚ) | |
| (P : PiExponentApprox.FramePolynomial m) : | |
| P ∈ algebraicJetIdeal c T v hv H ↔ | |
| FormalLogTruncation.truncatedFormalJet c T P ∈ JetGeometry.rationalWeightedIdeal (R := ℂ) v hv H := by | |
| change Ideal.Quotient.mk _ _ = 0 ↔ _ | |
| exact Ideal.Quotient.eq_zero_iff_mem | |
| theorem truncatedFormalJet_packet_surjective (c : Fin m → ℂ) (T : Fin m → ℕ) | |
| (v : Fin (m+1) → ℚ) (hv : ∀ i, 0 < v i) (H : ℚ) : | |
| Function.Surjective (fun P : PiExponentApprox.FramePolynomial m => | |
| JetGeometry.rationalCoefficientPacket v H (FormalLogTruncation.truncatedFormalJet c T P)) := by | |
| intro packet | |
| obtain ⟨P, hP⟩ := PolynomialJetPackets.polynomial_packet_surjective v hv H packet | |
| refine ⟨inverseTriangularMap c (fun i => PowerSeries.trunc (T i) (PowerSeries.log ℂ)) P, ?_⟩ | |
| simpa only [truncatedFormalJet_inverse] using hP | |
| theorem algebraicJetMap_surjective (c : Fin m → ℂ) (T : Fin m → ℕ) | |
| (v : Fin (m+1) → ℚ) (hv : ∀ i, 0 < v i) (H : ℚ) : | |
| Function.Surjective (algebraicJetMap c T v (fun i => (hv i).le) H) := | |
| quotient_surjective_of_packet v (fun i => (hv i).le) H _ | |
| (truncatedFormalJet_packet_surjective c T v hv H) | |
| def algebraicJetQuotientEquiv (c : Fin m → ℂ) (T : Fin m → ℕ) | |
| (v : Fin (m+1) → ℚ) (hv : ∀ i, 0 < v i) (H : ℚ) : | |
| (PiExponentApprox.FramePolynomial m ⧸ algebraicJetIdeal c T v (fun i => (hv i).le) H) ≃ₐ[ℂ] | |
| (MvPowerSeries (Fin (m+1)) ℂ ⧸ JetGeometry.rationalWeightedIdeal (R := ℂ) v (fun i => (hv i).le) H) := | |
| Ideal.quotientKerAlgEquivOfSurjective (algebraicJetMap_surjective c T v hv H) | |
| @[simp] theorem algebraicJetQuotientEquiv_mk (c : Fin m → ℂ) (T : Fin m → ℕ) | |
| (v : Fin (m+1) → ℚ) (hv : ∀ i, 0 < v i) (H : ℚ) | |
| (P : PiExponentApprox.FramePolynomial m) : | |
| algebraicJetQuotientEquiv c T v hv H (Ideal.Quotient.mk _ P) = | |
| Ideal.Quotient.mk _ (FormalLogTruncation.truncatedFormalJet c T P) := | |
| Ideal.quotientKerAlgEquivOfSurjective_mk _ P | |
| theorem independent_truncatedFormalJet_packets_surjective {J : Type*} | |
| (c : J → Fin m → ℂ) (T : Fin m → ℕ) | |
| (v : Fin (m+1) → ℚ) (hv : ∀ i, 0 < v i) (H : ℚ) : | |
| Function.Surjective (fun P : J → PiExponentApprox.FramePolynomial m => fun j => | |
| JetGeometry.rationalCoefficientPacket v H (FormalLogTruncation.truncatedFormalJet (c j) T (P j))) := by | |
| intro packets | |
| choose P hP using fun j => truncatedFormalJet_packet_surjective (c j) T v hv H (packets j) | |
| exact ⟨P, funext hP⟩ | |
| theorem formalJet_packet_surjective (c : Fin m → ℂ) | |
| (v : Fin (m+1) → ℚ) (hv : ∀ i, 0 < v i) (H : ℚ) : | |
| Function.Surjective (fun P : PiExponentApprox.FramePolynomial m => | |
| JetGeometry.rationalCoefficientPacket v H (FormalLogJet.formalJet c P)) := by | |
| let T : Fin m → ℕ := fun i => Nat.ceil (v i.succ / v 0) | |
| have hT : ∀ i, v i.succ ≤ (T i : ℚ) * v 0 := by | |
| intro i | |
| exact (div_le_iff₀ (hv 0)).mp (Nat.le_ceil _) | |
| have ht : Function.Surjective (fun P : PiExponentApprox.FramePolynomial m => | |
| fun _ : Unit => JetGeometry.rationalCoefficientPacket v H | |
| (FormalLogTruncation.truncatedFormalJet c T P)) := by | |
| intro packets | |
| obtain ⟨P, hP⟩ := truncatedFormalJet_packet_surjective c T v hv H (packets ()) | |
| refine ⟨P, ?_⟩ | |
| funext j | |
| cases j | |
| exact hP | |
| have hs := (FormalLogTruncation.formalLog_packets_surjective_iff_truncated | |
| v (fun i => (hv i).le) H T hT (fun _ : Unit => c) (fun P => P)).mpr ht | |
| intro packet | |
| obtain ⟨P, hP⟩ := hs (fun _ => packet) | |
| exact ⟨P, congrFun hP ()⟩ | |
| end PiExponent.AlgebraicJetPackets | |
| end | |
| end OAI | |