| import OAI.NumberTheory.PiExponent.Jets.TaylorRectangular | |
| namespace OAI | |
| noncomputable section | |
| namespace PiExponent.FrameTransverseMultiplicity | |
| open scoped BigOperators | |
| open PiExponentApprox | |
| open CommutingTaylor TaylorRectangular | |
| open TransverseMultiplicity | |
| variable {L K : Type*} [CommRing L] [Field K] | |
| def frameTaylor (m k : ℕ) (B : Fin k → Fin (m + 1)) | |
| [Algebra (FramePolynomial m) L] (rho : L →+* K) : | |
| FramePolynomial m →+* MvPowerSeries (Fin k) K := | |
| residueTaylorFin k (fun i => rationalFrame m (B i)) | |
| (rho.comp (algebraMap (FramePolynomial m) L)) | |
| @[simp] theorem frameTaylor_constant (m k : ℕ) (B : Fin k → Fin (m + 1)) | |
| [Algebra (FramePolynomial m) L] (rho : L →+* K) (a : FramePolynomial m) : | |
| MvPowerSeries.constantCoeff (frameTaylor m k B rho a) = | |
| rho (algebraMap (FramePolynomial m) L a) := by | |
| exact constantCoeff_residueTaylorFin _ _ _ _ | |
| def localizedFrameTaylor (m k : ℕ) (B : Fin k → Fin (m + 1)) | |
| [Algebra (FramePolynomial m) L] (M : Submonoid (FramePolynomial m)) [IsLocalization M L] | |
| (rho : L →+* K) : L →+* MvPowerSeries (Fin k) K := | |
| localizedTaylorMap M (frameTaylor m k B rho) rho (frameTaylor_constant m k B rho) | |
| @[simp] theorem localizedFrameTaylor_algebraMap (m k : ℕ) (B : Fin k → Fin (m + 1)) | |
| [Algebra (FramePolynomial m) L] (M : Submonoid (FramePolynomial m)) [IsLocalization M L] | |
| (rho : L →+* K) (a : FramePolynomial m) : | |
| localizedFrameTaylor m k B M rho (algebraMap (FramePolynomial m) L a) = | |
| frameTaylor m k B rho a := by | |
| exact localizedTaylorMap_algebraMap M _ rho _ a | |
| @[simp] theorem localizedFrameTaylor_constant (m k : ℕ) (B : Fin k → Fin (m + 1)) | |
| [Algebra (FramePolynomial m) L] (M : Submonoid (FramePolynomial m)) [IsLocalization M L] | |
| (rho : L →+* K) (a : L) : | |
| MvPowerSeries.constantCoeff (localizedFrameTaylor m k B M rho a) = rho a := by | |
| exact RingHom.congr_fun (localizedTaylorMap_constant M (frameTaylor m k B rho) rho | |
| (frameTaylor_constant m k B rho)) a | |
| @[simp] theorem frameTaylor_firstCoefficient (m k : ℕ) (B : Fin k → Fin (m + 1)) | |
| [Algebra (FramePolynomial m) L] (rho : L →+* K) (i : Fin k) (a : FramePolynomial m) : | |
| MvPowerSeries.coeff (Finsupp.single i 1) (frameTaylor m k B rho a) = | |
| rho (algebraMap (FramePolynomial m) L (polynomialFrame m (B i) a)) := by | |
| exact coeff_single_residueTaylorFin _ _ _ i a | |
| variable [Algebra ℚ K] | |
| theorem localizedFrameTaylor_ideal_le (m k : ℕ) (B : Fin k → Fin (m + 1)) | |
| [Algebra (FramePolynomial m) L] (M : Submonoid (FramePolynomial m)) [IsLocalization M L] | |
| (rho : L →+* K) (I : Ideal (FramePolynomial m)) | |
| (cost : Fin (m + 1) → ℝ) (n : Fin k → ℕ) (δ : ℝ) | |
| (hcost : ∀ i, 0 ≤ cost i) | |
| (hn : ∑ i, ((n i - 1 : ℕ) : ℝ) * cost (B i) ≤ δ) | |
| (hvanish : ∀ a ∈ I, ∀ word : List (Fin (m + 1)), frameWordCost cost word ≤ δ → | |
| rho (algebraMap (FramePolynomial m) L (polynomialFrameWord m word a)) = 0) : | |
| I.map (algebraMap (FramePolynomial m) L) ≤ | |
| (rectangularIdeal (K := K) n).comap (localizedFrameTaylor m k B M rho) := by | |
| apply localizedTaylorMap_ideal_le M (frameTaylor m k B rho) rho (frameTaylor_constant m k B rho) | |
| exact frame_residueTaylor_ideal_le m k B (rho.comp (algebraMap (FramePolynomial m) L)) | |
| I cost n δ hcost hn hvanish | |
| def frameRectangularHom (m k : ℕ) (B : Fin k → Fin (m + 1)) | |
| [Algebra ℚ L] [Algebra (FramePolynomial m) L] | |
| (M : Submonoid (FramePolynomial m)) [IsLocalization M L] | |
| (rho : L →+* K) (n : Fin k → ℕ) : | |
| L →ₐ[ℚ] (MvPowerSeries (Fin k) K ⧸ rectangularIdeal (K := K) n) := | |
| rectangularTaylorAlgHom n (localizedFrameTaylor m k B M rho) | |
| theorem frameRectangularHom_maps_residueIdeal (m k : ℕ) (B : Fin k → Fin (m + 1)) | |
| [Algebra ℚ L] [Algebra (FramePolynomial m) L] | |
| (M : Submonoid (FramePolynomial m)) [IsLocalization M L] | |
| (rho : L →+* K) (n : Fin k → ℕ) (hn : ∀ i, 0 < n i) | |
| (P : Ideal L) (hP : P ≤ RingHom.ker rho) : | |
| P ≤ (RingHom.ker (rectangularAugmentation (K := K) n hn)).comap | |
| (frameRectangularHom m k B M rho n) := | |
| rectangularTaylorAlgHom_maps_residueIdeal n hn _ rho | |
| (localizedFrameTaylor_constant m k B M rho) P hP | |
| theorem frame_rectangular_length_lower (m k : ℕ) (B : Fin k → Fin (m + 1)) | |
| [Algebra ℚ L] [Algebra (FramePolynomial m) L] | |
| (M : Submonoid (FramePolynomial m)) [IsLocalization M L] | |
| (rho : L →+* K) (hrho : Function.Surjective rho) | |
| (P : Ideal L) (hP : P ≤ RingHom.ker rho) (I : Ideal (FramePolynomial m)) | |
| (cost : Fin (m + 1) → ℝ) (n : Fin k → ℕ) (δ : ℝ) | |
| (hn : ∀ i, 0 < n i) (hcost : ∀ i, 0 ≤ cost i) | |
| (hbudget : ∑ i, ((n i - 1 : ℕ) : ℝ) * cost (B i) ≤ δ) | |
| (hCot : Function.Surjective | |
| (Ideal.mapCotangent P (RingHom.ker (rectangularAugmentation (K := K) n hn)) | |
| (frameRectangularHom m k B M rho n) | |
| (frameRectangularHom_maps_residueIdeal m k B M rho n hn P hP))) | |
| (hvanish : ∀ a ∈ I, ∀ word : List (Fin (m + 1)), frameWordCost cost word ≤ δ → | |
| rho (algebraMap (FramePolynomial m) L (polynomialFrameWord m word a)) = 0) : | |
| (∏ i, n i : ℕ) ≤ | |
| Module.length L (L ⧸ I.map (algebraMap (FramePolynomial m) L)) := by | |
| apply rectangularTaylor_length_lower n hn (localizedFrameTaylor m k B M rho) rho hrho | |
| (localizedFrameTaylor_constant m k B M rho) P _ hP hCot | |
| exact localizedFrameTaylor_ideal_le m k B M rho I cost n δ hcost hbudget hvanish | |
| end PiExponent.FrameTransverseMultiplicity | |
| end | |
| end OAI | |