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

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.