Permutation polynomials over $\mathbb{F}_{q^2}$ from rational functions

TL;DR

This paper introduces a method to construct permutation polynomials over finite fields using rational functions that induce bijections on specific subsets, generalizing previous results by Zieve.

Contribution

It presents a new approach to generate permutation polynomials over _{q^2} using rational functions of any degree, extending prior work by Zieve.

Findings

Constructed permutation polynomials over _{q^2} using rational functions.

Generalized Zieve's results to broader classes of rational functions.

Demonstrated bijections on _{q^2} subsets with new polynomial constructions.

Abstract

Let $\mu_{q+1}$ denote the set of $(q+1)$-th roots of unity in $\mathbb{F}_{q^2 }$. We construct permutation polynomials over $\mathbb{F}_{q^2}$ by using rational functions of any degree that induce bijections either on $\mu_{q+1}$ or between $\mu_{q+1}$ and $\mathbb{F}_q \cup \{\infty\}$. In particular, we generalize results from Zieve.