Explicit determination of a class of permutation rational functions in any characteristic

TL;DR

This paper explicitly characterizes a broad class of permutation rational functions of small degree, focusing on their action on specific multiplicative subgroups, and applies these results to permutation quadrinomials over finite fields.

Contribution

It provides an explicit description of permutation rational functions of small degree and characterizes permutation quadrinomials over finite fields using geometric methods.

Findings

Explicit description of permutation rational functions permuting mu_q+1

Determination of permutation quadrinomials over F_q^2 induced by degree-3 functions

Unification and extension of existing results using geometric perspectives

Abstract

In this paper, we make use of the classification results of low-degree permutation rational functions together with their geometric properties to investigate rational functions that induce permutations on the multiplicative subgroup mu_q+1, where q is a prime power. By carefully analyzing the structural conditions under which such rational functions permute muq+1, we obtain an explicit description of a broad class of permutation rational functions of small degree. As a direct application of these findings, we explicitly determine many permutation quadrinomials over Fq2 that are induced by degree-3 rational functions permuting muq+1. Our approach not only unifies and extends several existing results in the literature but also provides a concrete geometric perspective for characterizing permutation polynomials over Fq2.