On ${\mathbb{F}}_q$-primitive points on hypersurfaces

In this paper, we estimate the number of ${\mathbb{F}}_q$-primitive points on the affine hypersurface defined by the equation $f(x_1,\dots ,x_s)=0$, where $f\in {\mathbb{F}}_q[x_1,\dots ,x_s]$ is an appropriate polynomial. In particular, we provide existence results for the case when $f$ is Dwork-regular and when $f$ is of Fermat type. Additionally, we present a proof for a recently posed conjecture. Finally, in the case where $q$ is a Fermat prime, we provide an explicit formula for the number of ${\mathbb{F}}_q$-primitive points on hyperplanes.