On primitive elements of finite fields avoiding affine hyperplanes

TL;DR

This paper investigates the existence and distribution of primitive elements in finite fields that avoid certain affine hyperplanes, providing both asymptotic and concrete results related to digit representations.

Contribution

It introduces new results on primitive elements avoiding affine hyperplanes in finite fields, extending previous work on digit distributions.

Findings

Existence of primitive elements avoiding given hyperplanes established.

Asymptotic formulas for distribution of such elements derived.

Concrete bounds and examples provided.

Abstract

Let $n\ge 2$ be an integer and let $\mathbb F_q$ be the finite field with $q$ elements, where $q$ is a prime power. Given $\mathbb F_q$-affine hyperplanes $\mathcal A_1, \ldots, \mathcal A_n$ of $\mathbb F_{q^n}$ in general position, we study the existence and distribution of primitive elements of $\mathbb F_{q^n}$, avoiding each $\mathcal A_i$. We obtain both asymptotic and concrete results, relating to past works on digits over finite fields.