${\mathbb{F}}_q$-primitive points on varieties over finite fields

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 $f(x,y)=f_1(x,y)\mathrm{/}{f}_2(x,y)$ 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 ${\mathbb{F}}_q$.