Independence of ordinary multivariate Taylor coefficient functionals

Proved

OAI.PiExponent.AdmissibleFormalInterpolation.ordinaryTaylor_coefficient_functionals_linearIndependent

by Eyal1990 · Oct 15, 2026 · Mathlib 0df444a (Lean v4.33.1)

number-theoryopenai-pi-exponent-017

Ordinary multivariate Taylor coefficient functionals on polynomials are linearly independent whenever their center–exponent pairs are distinct.

Preamble

import Definitions.Def_OAI_PiExponent_AdmissibleFormalInterpolation
import Mathlib.LinearAlgebra.LinearIndependent.Basic

open OAI.PiExponent.AdmissibleFormalInterpolation

Formal statement

theorem OAI.PiExponent.AdmissibleFormalInterpolation.ordinaryTaylor_coefficient_functionals_linearIndependent
    {m : ℕ} {ι : Type} [Fintype ι] [DecidableEq ι]
    (center : ι → Fin (m + 1) → ℂ)
    (exponent : ι → (Fin (m + 1) →₀ ℕ))
    (hindex : Function.Injective (fun x => (center x, exponent x))) :
    LinearIndependent ℂ
      (fun x => (MvPowerSeries.coeff (exponent x)).comp
        (MvPolynomial.aeval (fun i =>
          MvPowerSeries.C (center x i) + MvPowerSeries.X i)).toLinearMap) := by sorry

Source

https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/NumberTheory/PiExponent/Approximation/AdmissibleMatrixGeometry.lean#L41-L44

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