Determination of a class of permutation quadrinomials

TL;DR

This paper classifies a specific class of permutation polynomials over finite fields, resolves multiple open problems, and introduces novel geometric techniques to analyze high-degree rational functions in small finite fields.

Contribution

It provides a complete classification of certain permutation quadrinomials over finite fields and applies geometric methods to problems previously considered intractable.

Findings

Classified all permutation quadrinomials of a specific form over F_{q^2}.

Resolved eight open conjectures and problems in the literature.

Connected the classification to 58 recent results.

Abstract

We determine all permutation polynomials over F_{q^2} of the form X^r A(X^{q-1}) where, for some Q which is a power of the characteristic of F_q, the integer r is congruent to Q+1 (mod q+1) and all terms of A(X) have degrees in {0, 1, Q, Q+1}. We then use this classification to resolve eight conjectures and open problems from the literature, and we show that the simplest special cases of our result imply 58 recent results from the literature. Our proof makes a novel use of geometric techniques in a situation where they previously did not seem applicable, namely to understand the arithmetic of high-degree rational functions over small finite fields, despite the fact that in this situation the Weil bounds do not provide useful information.