Improved bounds on Gauss sums in arbitrary 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.

12 pages. This new version fixes some silly errors and adds some new results

Cited by in corpus (1)

• Bounds of Trilinear and Trinomial Exponential Sums