Multiplicative character sums over two classes of subsets of quadratic extensions of finite fields

TL;DR

This paper improves bounds on multiplicative character sums over special subsets of quadratic extensions of finite fields, enabling better results on the existence of primitive elements within these subsets.

Contribution

The authors develop sharper estimates for character sums over subsets of finite field extensions by reducing the problem to sums over subfields, enhancing previous bounds.

Findings

Sharper bounds for character sums over quadratic extension subsets

Reduction of sums to subfield sums for improved estimates

Application to prove existence of primitive elements in subsets

Abstract

Let $q$ be a prime power and $r$ a positive even integer. Let $\mathbb{F}_{q}$ be the finite field with $q$ elements and $\mathbb{F}_{q^r}$ be its extension field of degree $r$. Let $\chi$ be a nontrivial multiplicative character of $\mathbb{F}_{q^r}$ and $f(X)$ a polynomial over $\mathbb{F}_{q^r}$ with a simple root in $\mathbb{F}_{q^r}$. In this paper, we improve estimates for character sums $\sum\limits_{g \in\mathcal{G}}\chi(f(g))$, where $\mathcal{G}$ is either a subset of $\mathbb{F}_{q^r}$ of sparse elements, with respect to some fixed basis of $\mathbb{F}_{q^r}$ which contains a basis of $\mathbb{F}_{q^{r/2}}$, or a subset avoiding affine hyperplanes in general position. While such sums have been previously studied, our approach yields sharper bounds by reducing them to sums over the subfield $\mathbb{F}_{q^{r/2}}$ rather than sums over general linear spaces. These estimates can be used to prove the existence of primitive elements in $\mathcal{G}$ in the standard way.