Character sums over affine spaces and applications

TL;DR

This paper introduces a new bound on character sums over affine spaces in finite fields and demonstrates its applications in establishing the existence of special primitive elements, improving previous results.

Contribution

The work provides a novel bound on character sums over affine spaces and applies it to primitive element existence problems, offering significant improvements over prior bounds.

Findings

New bound on character sums over affine spaces

Enhanced results on the existence of special primitive elements

Improved estimates compared to previous work

Abstract

Given a finite field $\mathbb F_q$, a positive integer $n$ and an $\mathbb F_q$-affine space $\mathcal A\subseteq \mathbb F_{q^n}$, we provide a new bound on the sum $\sum_{a\in \mathcal A}\chi(a)$, where $\chi$ a multiplicative character of $\mathbb F_{q^n}$. We focus on the applicability of our estimate to results regarding the existence of special primitive elements in $\mathbb F_{q^n}$. In particular, we obtain substantial improvements on previous works.