Improved bounds on Gauss sums in arbitrary finite fields

Let q be a power of a prime and let Fq be the finite field consisting of q elements. We establish new explicit estimates on Gauss sums of the form S-n(a) = Sigma F-x is an element of(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 vertical bar S-n(a)vertical bar 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.