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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 | import OAI.NumberTheory.PiExponent.Jets.NormalFrameBasis
import OAI.NumberTheory.PiExponent.Polynomials.PolynomialFrame
namespace OAI
noncomputable section
namespace PiExponent
open scoped BigOperators
open PiExponentApprox NormalBasisRigidity
theorem residue_polynomialFrame_eq_frameBasis_dot {E : Type*} [Field E]
(m : ℕ) (phi : FramePolynomial m →+* E)
(hy : phi (MvPolynomial.X (0 : Fin (m + 1))) ≠ 0)
(j : Fin (m + 1)) (p : FramePolynomial m) :
phi (polynomialFrame m j p) =
∑ i, frameBasis m (phi (MvPolynomial.X 0)) hy j i *
phi (MvPolynomial.pderiv i p) := by
refine Fin.cases ?_ (fun l => ?_) j
· rw [polynomialFrame_zero, logarithmicDerivation_apply, map_add, map_mul,
map_sum, Fin.sum_univ_succ]
simp only [frameBasis_zero_zero, frameBasis_zero_succ, one_mul]
· rw [polynomialFrame_pos m l.succ (Fin.succ_ne_zero l), frameBasis_succ]
simp [Pi.basisFun_apply, Pi.single_apply]
end PiExponent
end
end OAI
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