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

• URL: http://arxiv.org/abs/2502.14436v2
• Date: Fri, 09 May 2025 11:08:00 GMT
• Title: Multiplicative character sums over two classes of subsets of quadratic extensions of finite fields
• Authors: Kaimin Cheng, Arne Winterhof,
• Abstract summary: We improve estimates for character sums $sumlimits_g inmathcalGchi(f(g))$, where $mathcalG$ is either a subset of $mathbbF_qr$ of sparse elements.<n>These estimates can be used to prove the existence of primitive elements in $mathcalG$ in the standard way.
• Score: 6.5990719141691825
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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