2 problems

Let $\mathbb{F}_q$ be a finite field, and let $N_{v,u}(q)$ denote the number of permutation rational functions $V(x)/U(x)$ over $\mathbb{F}_q$ with numerator degree $v$ and denomin…

Let $k$ be a positive integer, set $q=3^k$, and define … the set of $(q+1)$-th roots of unity in ${\mathbb F}_{q^2}$. A rational function permutes $\mu_{q+1}$ if it is defined on t…