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