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b797a5b | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 | 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
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