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

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 $\sum _{g\in \mathcal{G}}\chi (f(g))$, 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.

12pages, 1 table, 1 figure