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

TL;DR

This paper estimates the count of primitive points on affine hypersurfaces over finite fields, providing existence results, proofs of conjectures, and explicit formulas in special cases, advancing understanding of algebraic points over finite fields.

Contribution

It introduces new bounds and formulas for primitive points on hypersurfaces, including existence results for Dwork-regular and Fermat-type polynomials, and proves a recent conjecture.

Findings

Existence of primitive points for Dwork-regular and Fermat-type hypersurfaces.

Explicit formulas for primitive points when the field size is a Fermat prime.

Proof of a recently posed conjecture regarding primitive points.

Abstract

In this paper, we estimate the number of $\mathbb{F}_q$-primitive points on the affine hypersurface defined by the equation $f(x_1,\ldots,x_s)=0$, 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.