Multiplicative character sums over two classes of subsets of quadratic extensions of finite fields
Kaimin Cheng, Arne Winterhof
TL;DR
This work studies multiplicative character sums $S(\mathcal{G},\chi,f)=\sum_{\gamma\in\mathcal{G}}\chi(f(\gamma))$ over two structured subset classes of the even-degree extension ${\mathbb F}_{q^r}$, by reducing the sums to the subfield ${\mathbb F}_{q^{r/2}}$ using a Wan-type bound. It defines ${\mathcal{G}}_{\mathcal{A}}$ (restricted coordinates) and ${\mathcal{G}}_s$ (sparse elements) with bases containing ${\mathbb F}_{q^{r/2}}$, and proves a general bound for ${\mathcal{G}}_{\mathcal{A}}$ and a refined bound for ${\mathcal{G}}_s$ (with ρ-parameter) that improves earlier results by reducing to subfield sums. The main contributions are Theorem 3.1, giving a nontrivial bound for ${\mathcal{G}}_{\mathcal{A}}$, and Theorem 4.2, providing a sharper bound for ${\mathcal{G}}_s$ with $q=2$, both enabling stronger statements about the existence of primitive elements in these subsets for large even $r$ and small $q$. By sharpening these character-sum estimates, the paper enhances the toolbox for constructing primitive elements in restricted subsets of finite fields and highlights the advantage of subfield reductions in character-sum analysis.
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 $χ$ 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 $\sum\limits_{g \in\mathcal{G}}χ(f(g))$, 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.