Primitive values of rational functions at primitive elements of a finite field

Abstract

Given a prime power $q$ and an integer $n\geq2$, we establish a sufficient condition for the existence of a primitive pair $(\alpha,f(\alpha))$ where $\alpha \in \mathbb{F}_q$ and $f(x) \in \mathbb{F}_q(x)$ is a rational function of degree $n$. (Here $f=f_1/f_2$, where $f_1, f_2$ are coprime polynomials of degree $n_1,n_2$, respectively, and $n_1+n_2=n$.) For any $n$, such a pair is guaranteed to exist for sufficiently large $q$. Indeed, when $n=2$, such a pair definitely does {\em not} exist only for 28 values of $q$ and possibly (but unlikely) only for at most $3911$ other values of $q$.