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import OAI.NumberTheory.PiExponent.Jets.FormalJetDerivatives
import OAI.NumberTheory.PiExponent.Polynomials.PolynomialFrame
namespace OAI
noncomputable section
namespace PiExponent.FormalLogJet
open MvPowerSeries
open PiExponent.FormalJetDerivatives
variable {R : Type*} [CommRing R]
def liftSeries (m : ℕ) : PowerSeries R →+* MvPowerSeries (Fin (m + 1)) R :=
(finSuccEquiv R m).symm.toRingHom.comp (PowerSeries.map MvPowerSeries.C)
@[simp] theorem coeff_liftSeries (m : ℕ) (f : PowerSeries R)
(d : Fin (m + 1) →₀ ℕ) :
coeff d (liftSeries m f) = if d.tail = 0 then PowerSeries.coeff (d 0) f else 0 := by
have h := coeff_coeff_finSuccEquiv (liftSeries m f) (k := d 0) (x := d.tail)
rw [Finsupp.cons_tail] at h
rw [← h]
simp [liftSeries, PowerSeries.coeff_map, coeff_C]
@[simp] theorem liftSeries_X (m : ℕ) : liftSeries (R := R) m PowerSeries.X = X 0 := by
apply (finSuccEquiv R m).injective
simp [liftSeries]
@[simp] theorem liftSeries_C (m : ℕ) (r : R) :
liftSeries m (PowerSeries.C r) = C r := by
apply (finSuccEquiv R m).injective
simp [liftSeries]
theorem pderiv_zero_liftSeries (m : ℕ) (f : PowerSeries R) :
pderiv 0 (liftSeries m f) = liftSeries m (PowerSeries.derivative f) := by
ext d
rw [coeff_pderiv, coeff_liftSeries, coeff_liftSeries, PowerSeries.coeff_derivative]
have ht : (d + Finsupp.single (0 : Fin (m+1)) 1).tail = d.tail := by
ext i
simp [Finsupp.tail_apply]
simp only [ht, Finsupp.add_apply, Finsupp.single_eq_same]
split_ifs <;> simp_all
theorem pderiv_succ_liftSeries (m : ℕ) (i : Fin m) (f : PowerSeries R) :
pderiv i.succ (liftSeries m f) = 0 := by
ext d
rw [coeff_pderiv, coeff_liftSeries, coeff_zero]
have ht : (d + Finsupp.single i.succ 1).tail ≠ 0 := by
intro h
have hi := DFunLike.congr_fun h i
simp [Finsupp.tail_apply] at hi
simp [ht]
def formalLog (m : ℕ) : MvPowerSeries (Fin (m+1)) ℂ :=
liftSeries m (PowerSeries.log ℂ)
@[simp] theorem pderiv_succ_formalLog (m : ℕ) (i : Fin m) :
pderiv i.succ (formalLog m) = 0 := pderiv_succ_liftSeries m i _
theorem one_add_X_mul_pderiv_formalLog (m : ℕ) :
(1 + X 0) * pderiv 0 (formalLog m) = 1 := by
have h := congrArg (liftSeries m) (PowerSeries.derivative_log_mul_one_add_X (A := ℂ))
rw [map_mul, map_add, map_one, liftSeries_X] at h
rw [formalLog, pderiv_zero_liftSeries, mul_comm]
exact h
def formalJet {m : ℕ} (c : Fin m → ℂ) :
PiExponentApprox.FramePolynomial m →ₐ[ℂ] MvPowerSeries (Fin (m+1)) ℂ :=
MvPolynomial.aeval (Fin.cases (1 + X 0)
(fun i => C (c i) + X i.succ + formalLog m))
@[simp] theorem formalJet_Y {m : ℕ} (c : Fin m → ℂ) :
formalJet c (MvPolynomial.X (0 : Fin (m+1))) = 1 + X 0 := by
simp [formalJet]
@[simp] theorem formalJet_X {m : ℕ} (c : Fin m → ℂ) (i : Fin m) :
formalJet c (MvPolynomial.X i.succ) = C (c i) + X i.succ + formalLog m := by
simp [formalJet]
theorem derivation_map_of_X {σ S : Type*} [CommRing S] [Algebra R S]
(φ : MvPolynomial σ R →ₐ[R] S)
(D : Derivation R (MvPolynomial σ R) (MvPolynomial σ R))
(δ : Derivation R S S)
(hX : ∀ i, φ (D (MvPolynomial.X i)) = δ (φ (MvPolynomial.X i)))
(p : MvPolynomial σ R) : φ (D p) = δ (φ p) := by
induction p using MvPolynomial.induction_on with
| C a => simp
| add p q hp hq => simp only [map_add, hp, hq]
| mul_X p i hp =>
simp only [Derivation.leibniz, smul_eq_mul, map_add, map_mul, hp, hX]
theorem formalJet_frame_X {m : ℕ} (c : Fin m → ℂ) (i j : Fin (m+1)) :
formalJet c (PiExponentApprox.polynomialFrame m i (MvPolynomial.X j)) =
jetFrame m i (formalJet c (MvPolynomial.X j)) := by
classical
cases i using Fin.cases with
| zero =>
cases j using Fin.cases with
| zero =>
simp [PiExponentApprox.polynomialFrame_zero,
PiExponentApprox.logarithmicDerivation_apply, MvPolynomial.pderiv_X]
| succ j =>
simp [PiExponentApprox.polynomialFrame_zero,
PiExponentApprox.logarithmicDerivation_apply, MvPolynomial.pderiv_X,
Pi.single_apply, MvPowerSeries.pderiv_X_of_ne,
one_add_X_mul_pderiv_formalLog]
| succ i =>
rw [PiExponentApprox.polynomialFrame_pos m i.succ (Fin.succ_ne_zero i),
jetFrame_pos m i.succ (Fin.succ_ne_zero i)]
cases j using Fin.cases with
| zero => simp [MvPowerSeries.pderiv_X_of_ne (Ne.symm (Fin.succ_ne_zero i))]
| succ j =>
by_cases hij : i = j
· subst j; simp
· simp [MvPowerSeries.pderiv_X_of_ne, Ne.symm hij]
theorem formalJet_polynomialFrame {m : ℕ} (c : Fin m → ℂ)
(i : Fin (m+1)) (p : PiExponentApprox.FramePolynomial m) :
formalJet c (PiExponentApprox.polynomialFrame m i p) =
jetFrame m i (formalJet c p) :=
derivation_map_of_X (formalJet c) (PiExponentApprox.polynomialFrame m i)
(jetFrame m i) (formalJet_frame_X c i) p
theorem formalJet_polynomialFrameWord {m : ℕ} (c : Fin m → ℂ)
(word : List (Fin (m+1))) (p : PiExponentApprox.FramePolynomial m) :
formalJet c (PiExponentApprox.polynomialFrameWord m word p) =
jetFrameWord m word (formalJet c p) := by
induction word with
| nil => rfl
| cons i word ih =>
rw [PiExponentApprox.polynomialFrameWord_cons, formalJet_polynomialFrame, ih]
rfl
theorem formalJet_polynomialFrameWord_vanishing {m : ℕ} (c : Fin m → ℂ)
(v : Fin (m+1) → ℚ) (hv : ∀ i, 0 ≤ v i) (H : ℚ)
(p : PiExponentApprox.FramePolynomial m)
(hp : formalJet c p ∈ JetGeometry.rationalWeightedIdeal v hv H)
(word : List (Fin (m+1))) :
formalJet c (PiExponentApprox.polynomialFrameWord m word p) ∈
JetGeometry.rationalWeightedIdeal v hv (H - (word.map v).sum) := by
rw [formalJet_polynomialFrameWord]
exact jetFrameWord_mem_rationalWeightedIdeal v hv H (formalJet c p) hp word
end PiExponent.FormalLogJet
end
end OAI
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