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a72760f | 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 | import OAI.NumberTheory.PiExponent.Approximation.FormalMatrixBridge
import OAI.NumberTheory.PiExponent.Jets.FormalJetIndices
namespace OAI
noncomputable section
namespace PiExponent.FormalMatrixBridge
open MvPowerSeries
theorem truncatedLogMatrix_surjective_of_rational_packets {m : ℕ} {α : Type*}
(K : ℕ) (w0 v0 θ : ℝ) (w : Fin m → ℝ) (H : ℚ)
(hw0 : 0 < w0) (hw : ∀ i, 0 < w i)
(V : Fin (m+1) → ℚ) (hV : ∀ i, 0 < V i)
(hrow : ∀ i, (V i : ℝ) = InterpolationMatrix.rowWeights v0 θ w i)
(r : Fin m → ℂ) (T : Fin m → ℕ)
(P : α → PiExponentApprox.FramePolynomial m)
(hdegree : ∀ a, PiExponentApprox.HasWeightedDegreeLE
(InterpolationMatrix.columnWeights w0 w) (H : ℝ) (P a))
(hpacket : Function.Surjective (fun a => fun j : Fin K =>
JetGeometry.rationalCoefficientPacket V H
(FormalLogTruncation.truncatedFormalJet (fun i => (j.val : ℂ) * r i) T (P a)))) :
Function.Surjective
(InterpolationMatrix.truncatedLogMatrix K w0 v0 θ w (H : ℝ) r T).mulVecLin := by
classical
have hrow' : (fun i => (V i : ℝ)) = InterpolationMatrix.rowWeights v0 θ w := funext hrow
have hW : ∀ i, 0 < InterpolationMatrix.columnWeights w0 w i := by
intro i
exact Fin.cases hw0 hw i
intro y
let e := rationalJetIndexEquiv V hV H
let z : Fin K → JetGeometry.RationalCoefficientPacket (R := ℂ) V H :=
fun j d => y (j, ⟨(e d).val, by
simpa only [← hrow'] using (e d).property⟩)
obtain ⟨a, ha⟩ := hpacket z
let x : InterpolationMatrix.Column w0 w (H : ℝ) → ℂ :=
fun c => (P a).coeff (InterpolationMatrix.exponentVector c.val)
refine ⟨x, ?_⟩
funext ρ
rw [truncatedLogMatrix_mulVec_eq_coeff]
change coeff (InterpolationMatrix.exponentVector ρ.2.val)
(FormalLogTruncation.truncatedFormalJet (fun i => (ρ.1.val : ℂ) * r i) T
(polynomialOfCoefficients
(realWeightedSimplex (InterpolationMatrix.columnWeights w0 w) (H : ℝ))
(fun c => (P a).coeff (InterpolationMatrix.exponentVector c.val)))) = y ρ
rw [polynomialOfCoefficients_of_weighted _ hW _ (P a) (hdegree a)]
let b : ↥(strictWeightedSimplex (fun i => (V i : ℝ)) (H : ℝ)) :=
⟨ρ.2.val, by simpa only [hrow'] using ρ.2.property⟩
have hh := congrFun (congrFun ha ρ.1) (e.symm b)
change coeff (e.symm b).val
(FormalLogTruncation.truncatedFormalJet (fun i => (ρ.1.val : ℂ) * r i) T (P a)) =
z ρ.1 (e.symm b) at hh
simpa only [e, rationalJetIndexEquiv_symm_val, b, z, Equiv.apply_symm_apply] using hh
theorem truncatedLogMatrix_surjective_of_formalLog_packets {m : ℕ} {α : Type*}
(K : ℕ) (w0 v0 θ : ℝ) (w : Fin m → ℝ) (H : ℚ)
(hw0 : 0 < w0) (hw : ∀ i, 0 < w i)
(V : Fin (m+1) → ℚ) (hV : ∀ i, 0 < V i)
(hrow : ∀ i, (V i : ℝ) = InterpolationMatrix.rowWeights v0 θ w i)
(r : Fin m → ℂ) (T : Fin m → ℕ)
(hT : ∀ i, V i.succ ≤ (T i : ℚ) * V 0)
(P : α → PiExponentApprox.FramePolynomial m)
(hdegree : ∀ a, PiExponentApprox.HasWeightedDegreeLE
(InterpolationMatrix.columnWeights w0 w) (H : ℝ) (P a))
(hpacket : Function.Surjective (fun a => fun j : Fin K =>
JetGeometry.rationalCoefficientPacket V H
(FormalLogJet.formalJet (fun i => (j.val : ℂ) * r i) (P a)))) :
Function.Surjective
(InterpolationMatrix.truncatedLogMatrix K w0 v0 θ w (H : ℝ) r T).mulVecLin := by
apply truncatedLogMatrix_surjective_of_rational_packets
K w0 v0 θ w H hw0 hw V hV hrow r T P hdegree
exact (FormalLogTruncation.formalLog_packets_surjective_iff_truncated
V (fun i => (hV i).le) H T hT (fun j : Fin K => fun i => (j.val : ℂ) * r i) P).mp hpacket
end PiExponent.FormalMatrixBridge
end
end OAI
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