Permutation rational functions over quadratic extensions of 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 $F_{q^2}$, 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 $F_{q^2}$, by calculating character sums related to quadratic forms of $F_{q^2}/F_q$. Then given some rational function, we can demonstrate whether it induces a permutation of $F_{q^2}$.