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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 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 | import OAI.NumberTheory.PiExponent.Analysis.FormalLogTruncation
import OAI.NumberTheory.PiExponent.Jets.AuxiliaryJetMap
import OAI.NumberTheory.PiExponent.Polynomials.NestedPolynomialSwap
namespace OAI
noncomputable section
namespace PiExponent.FormalMatrixBridge
open MvPowerSeries
open scoped BigOperators
def flatten (m : ℕ) :
MvPolynomial (Fin m) (Polynomial ℂ) →ₐ[ℂ] MvPowerSeries (Fin (m+1)) ℂ :=
(MvPolynomial.coeToMvPowerSeries.algHom ℂ).comp
((MvPolynomial.finSuccEquiv ℂ m).symm.toAlgHom.comp
(nestedPolynomialSwap ℂ m).toAlgHom)
@[simp] theorem coeff_flatten {m : ℕ}
(p : MvPolynomial (Fin m) (Polynomial ℂ)) (d : Fin (m+1) →₀ ℕ) :
coeff d (flatten m p) = (p.coeff d.tail).coeff (d 0) := by
change ((MvPolynomial.finSuccEquiv ℂ m).symm (nestedPolynomialSwap ℂ m p)).coeff d = _
rw [← Finsupp.cons_tail d, ← MvPolynomial.finSuccEquiv_coeff_coeff,
AlgEquiv.apply_symm_apply]
exact coeff_nestedPolynomialSwap p (d 0) d.tail
@[simp] theorem flatten_C {m : ℕ} (p : Polynomial ℂ) :
flatten m (MvPolynomial.C p) = FormalLogJet.liftSeries m (p : PowerSeries ℂ) := by
classical
ext d
rw [coeff_flatten, FormalLogJet.coeff_liftSeries]
simp only [MvPolynomial.coeff_C, Polynomial.coeff_coe]
split_ifs <;> simp_all
@[simp] theorem flatten_X {m : ℕ} (i : Fin m) :
flatten m (MvPolynomial.X i) = X i.succ := by
classical
ext d
rw [coeff_flatten, MvPolynomial.coeff_X, coeff_X]
have hd : d = Finsupp.single i.succ 1 ↔
d.tail = Finsupp.single i 1 ∧ d 0 = 0 := by
conv_lhs => rw [← Finsupp.cons_tail d]
exact Finsupp.cons_eq_single_succ_iff i
simp only [hd]
by_cases ht : Finsupp.single i 1 = d.tail
· rw [ite_eq_left ht]
have ht' : d.tail = Finsupp.single i 1 := ht.symm
simp only [ht', true_and, Polynomial.coeff_one]
· rw [ite_eq_right ht, Polynomial.coeff_zero]
have ht' : ¬ d.tail = Finsupp.single i 1 := Ne.symm ht
simp only [ht', false_and, ite_false]
theorem flatten_literalJetSubstitution {m : ℕ}
(r : Fin m → ℂ) (T : Fin m → ℕ) (j : ℕ)
(p : PiExponentApprox.FramePolynomial m) :
flatten m (literalJetSubstitution r
(fun i => InterpolationMatrix.truncatedLog (T i)) j p) =
FormalLogTruncation.truncatedFormalJet (fun i => (j : ℂ) * r i) T p := by
have hh : (flatten m).comp
(literalJetSubstitution r (fun i => InterpolationMatrix.truncatedLog (T i)) j) =
FormalLogTruncation.truncatedFormalJet (fun i => (j : ℂ) * r i) T := by
ext i
cases i using Fin.cases with
| zero =>
simp [literalJetSubstitution, FormalLogTruncation.truncatedFormalJet]
| succ i =>
simp [literalJetSubstitution, FormalLogTruncation.truncatedFormalJet,
InterpolationMatrix.truncatedLog, add_assoc, add_left_comm, add_comm]
exact DFunLike.congr_fun hh p
theorem coeff_truncatedFormalJet {m : ℕ}
(r : Fin m → ℂ) (T : Fin m → ℕ) (j : ℕ)
(p : PiExponentApprox.FramePolynomial m) (d : Fin (m+1) →₀ ℕ) :
coeff d (FormalLogTruncation.truncatedFormalJet
(fun i => (j : ℂ) * r i) T p) =
((literalJetSubstitution r (fun i => InterpolationMatrix.truncatedLog (T i)) j p).coeff
d.tail).coeff (d 0) := by
rw [← flatten_literalJetSubstitution, coeff_flatten]
@[simp] theorem exponentVector_tail {m : ℕ} (a : Fin (m+1) → ℕ) :
(InterpolationMatrix.exponentVector a).tail =
InterpolationMatrix.exponentVector (fun i => a i.succ) := by
ext i
simp [Finsupp.tail_apply]
theorem entry_eq_coeff_truncatedFormalJet {m : ℕ}
(r : Fin m → ℂ) (T : Fin m → ℕ) (j : ℕ)
(b e : Fin (m+1) → ℕ) :
InterpolationMatrix.entry r (fun i => InterpolationMatrix.truncatedLog (T i))
j (b 0) (fun i => b i.succ) (e 0) (fun i => e i.succ) =
coeff (InterpolationMatrix.exponentVector b)
(FormalLogTruncation.truncatedFormalJet (fun i => (j : ℂ) * r i) T
(MvPolynomial.monomial (InterpolationMatrix.exponentVector e) 1)) := by
rw [coeff_truncatedFormalJet, literalJetSubstitution_monomial]
simp [InterpolationMatrix.entry]
theorem truncatedLogMatrix_mulVec_eq_coeff {m : ℕ}
(K : ℕ) (w0 v0 θ : ℝ) (w : Fin m → ℝ) (H : ℝ)
(r : Fin m → ℂ) (T : Fin m → ℕ)
(x : InterpolationMatrix.Column w0 w H → ℂ)
(ρ : InterpolationMatrix.Row K v0 θ w H) :
(InterpolationMatrix.truncatedLogMatrix K w0 v0 θ w H r T).mulVecLin x ρ =
coeff (InterpolationMatrix.exponentVector ρ.2.val)
(FormalLogTruncation.truncatedFormalJet (fun i => (ρ.1.val : ℂ) * r i) T
(polynomialOfCoefficients
(realWeightedSimplex (InterpolationMatrix.columnWeights w0 w) H) x)) := by
rw [coeff_truncatedFormalJet, literalJetSubstitution_polynomialOfCoefficients]
simpa only [exponentVector_tail, InterpolationMatrix.exponentVector_apply,
InterpolationMatrix.linearEvaluation, InterpolationMatrix.truncatedLogMatrix] using
InterpolationMatrix.linearEvaluation_eq_coeff K w0 v0 θ w H r
(fun i => InterpolationMatrix.truncatedLog (T i)) x ρ
end PiExponent.FormalMatrixBridge
end
end OAI
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