$\mathbb{F}_q$-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 $\mathbb{F}_q$ 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 $(\alpha, \beta, f(\alpha, \beta))$ over $\mathbb{F}_q$, where $\alpha, \beta$ are primitive and $f(\alpha, \beta)$ is an $r$-primitive element of $\mathbb{F}_q$. In particular, this implies the existence of $\mathbb{F}_q$-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 $\mathbb{F}_q$.