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 $F_q$ be the finite field with q elements and $F_{q^r}$ be its extension field of degree r. Let χ be a nontrivial multiplicative character of $F_{q^r}$ and $f\left(X\right)$ a polynomial over $F_{q^r}$ with exactly one simple root in $F_{q^r}$. In this paper, we improve estimates for character sums $\sum _{g\in G}\chi \left(f\right(g\left)\right)$, where $G$ is either a subset of $F_{q^r}$ of sparse elements, with respect to some fixed basis of $F_{q^r}$ which contains a basis of $F_{q^{r/2}}$, 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 $F_{q^{r/2}}$ rather than sums over general linear spaces. These estimates can be used to prove the existence of primitive elements in $G$ in the standard way.