Fq-primitive points on varieties over finite fields

Abstract

Let r be a positive divisor of $q-1$ and $f(x,y)$ a rational function of degree sum d over $F_q$ with some restrictions, where the degree sum of a rational function $f(x,y)=f_1(x,y)/f_2(x,y)$ is the sum of the degrees of $f_1(x,y)$ and $f_2(x,y)$. In this article, we discuss the existence of triples $(\alpha ,\beta ,f(\alpha ,\beta \left)\right)$ over $F_q$, where $\alpha ,\beta$ are primitive and $f(\alpha ,\beta )$ is an r-primitive element of $F_q$. In particular, this implies the existence of $F_q$-primitive points on the surfaces of the form $z^r=f(x,y)$. As an example, we apply our results on the unit sphere over $F_q$.