An estimate for incomplete mixed character sums and applications

TL;DR

This paper establishes a new bound for incomplete mixed character sums over finite fields using function field techniques, and applies it to demonstrate the weak line property for primitive normal elements in certain field extensions.

Contribution

It introduces a novel bound for mixed character sums in finite fields and applies this to prove the weak line property for primitive normal elements, strengthening previous results.

Findings

Established a nontrivial bound for mixed character sums.

Proved the weak line property for primitive normal elements under certain conditions.

Strengthened existing results on finite field properties.

Abstract

Let $q$ be a prime power and $m>1$ be any integer. Let $\mathbb F_{q^m}$ be the finite field of order $q^m$ and $\theta\in\mathbb F_{q^m}$ be such that $\mathbb F_{q^m} = \mathbb F(\theta)$. We obtain a nontrivial bound for the mixed character sum $\sum_{x \in\mathbb F}\chi(\theta+x)\psi(x)$, where $\chi$ and $\psi$ are multiplicative and additive characters of $\mathbb F_{q^m}$ and $\mathbb F$, respectively, using function field methods. As an application of our main result, we prove that for fixed $m$ and sufficiently large prime powers $q$, that satisfy certain conditions, $\mathbb F_{q^m}/\mathbb F$ possesses the weak line property for primitive normal elements. In particular, our result is a strengthening of existing results.