Permutation Rational Functions over Quadratic Extensions of Finite Fields

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 ${\mathbb{F}}_{q^2}\mathrm{/}{\mathbb{F}}_q$. Then given some rational function, we can demonstrate whether it induces a permutation of ${\mathbb{F}}_{q^2}$.