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

TL;DR

This paper investigates the existence of primitive points on algebraic surfaces over finite fields, establishing conditions under which points with primitive coordinates exist, with applications to surfaces like the unit sphere.

Contribution

It provides new criteria for the existence of primitive points on varieties over finite fields, extending previous results to rational functions and specific surfaces.

Findings

Existence of primitive points on surfaces $z^r = f(x,y)$ over $F_q$

Conditions for triples $( mi, mii,f( mi, mii))$ with primitive elements

Application to primitive points on the unit sphere over $F_q$

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 $\mathbb{F}_q$.