Improved bounds on Gauss sums in arbitrary finite fields

TL;DR

This paper provides improved explicit bounds on Gauss sums over finite fields, extending the range of n for which nontrivial bounds are known, thus advancing understanding of exponential sums in finite field theory.

Contribution

The paper introduces new explicit estimates on Gauss sums that extend the known bounds to larger values of n, surpassing previous results by Zhelezov.

Findings

Nontrivial upper bounds on |S_n(a)| for n up to q^{1/2 + 1/68}

Extension of bounds range beyond previous results

Improved understanding of exponential sums in finite fields

Abstract

Let $q$ be a power of a prime and let $\mathbb{F}_q$ be the finite field consisting of $q$ elements. We establish new explicit estimates on Gauss sums of the form $S_n(a) = \sum_{x\in \mathbb{F}_q}\psi_a(x^n)$, where $\psi_a$ is a nontrivial additive character. In particular, we show that one has a nontrivial upper bound on $|S_n(a)|$ for certain values of $n$ of order up to $q^{1/2 + 1/68}$. Our results improve on the previous best known bound, due to Zhelezov.