Improved bounds on Gauss sums in arbitrary finite fields

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 $\mathrm{\mid }{S}_n(a)\mathrm{\mid }$ 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.