Bounds of Gauss sums in finite fields

We consider Gauss sums of the form

with a nontrivial additive character $\chi \ne \chi_0$of a finite field $\mathbb{F} _{p^m}$ of $p^m$ elements of characteristic $p$. The classical bound $\vert G_n(a)\vert \le (n-1) p^{m/2}$becomes trivial for $n \ge p^{m/2} + 1$. We show that, combining some recent bounds of Heath-Brown and Konyagin with several bounds due to Deligne, Katz, and Li, one can obtain the bound on $\vert G_n(a)\vert$ which is nontrivial for the values of $n$ of order up to $p^{m/2 + 1/6}$. We also show that for almost all primes one can obtain a bound which is nontrivial for the values of $n$ of order up to $p^{m/2 + 1/2}$.