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arXiv · 2505.05733

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

José Alves Oliveira·Marcelo Oliveira Veloso
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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.

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José Alves Oliveira, Marcelo Oliveira Veloso. 2025-05-09. On $\mathbb{F}_q$-primitive points on hypersurfaces. https://arxiv.org/abs/2505.05733

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