bibkey: mattareipizzato2017reciprocal
authors: Sandro Mattarei and Marco Pizzato
year: 2017
title: Generalizations of self-reciprocal polynomials
doi: 10.1016/j.ffa.2017.08.004
url: https://arxiv.org/abs/1609.07677v2
claim: "Section 5 identifies the polynomials invariant under translation by one in characteristic p as the polynomials in X^p-X."
strata_touched:
  - D5/S3/ArithUnits/SelfConjugateEvenPolynomialRefutation
license: arXiv non-exclusive distribution license; original summary only
triage: anchor

Translation-invariant polynomial algebra

Verified locator

DOI: https://doi.org/10.1016/j.ffa.2017.08.004

Source: https://arxiv.org/abs/1609.07677v2

The verified mathematical scope is Section 5 of arXiv:1609.07677v2: in characteristic p, translation-invariant polynomials have the form f(X^p-X). This is the existing invariant-algebra result credited below, not a prior settlement of the named sparse-power iff.

The source is https://arxiv.org/abs/1609.07677v2, Section 5, also published in Finite Fields and Their Applications 48, 271–288, DOI https://doi.org/10.1016/j.ffa.2017.08.004.

Over a field of characteristic p, the polynomials fixed by translation X maps to X+1 are exactly those of the form f(X^p-X). In characteristic two, X^4+X^2 = (X^2+X)^2 is an instance. This invariant-algebra mechanism is an existing result; the sparse-power refutation claims no new classifier or algebraic mechanism. Its direct proof only requires Frobenius and the coefficient obstruction for the family (X^4+X^2)^r.

This note cites the mathematical result and contains no transplanted source or proof code from the article.