Permutation Rational Functions over Quadratic Extensions of Finite Fields

TL;DR

This paper introduces a new class of permutation rational functions over quadratic extensions of finite fields, focusing on $q$-quadratic polynomials and their zero distributions to determine permutation properties.

Contribution

It characterizes permutation rational functions over $\

Findings

Determined the exact number of zeros of specific $q$-quadratic polynomials in $\

Developed criteria to identify permutation rational functions over $\

Connected character sums and quadratic forms to permutation properties in finite fields.

Abstract

Permutation rational functions over finite fields have attracted much attention in recent years. In this paper, we introduce a class of permutation rational functions over $\mathbb F_{q^2}$, whose numerators are so-called $q$-quadratic polynomials. To this end, we will first determine the exact number of zeros of a special $q$-quadratic polynomial in $\mathbb F_{q^2}$, by calculating character sums related to quadratic forms of $\mathbb F_{q^2}/\mathbb F_q$. Then given some rational function, we can demonstrate whether it induces a permutation of $\mathbb F_{q^2}$.