Bounds of Gauss sums in finite fields

Shparlinski, Igor

doi:10.1090/S0002-9939-04-07133-3

Bounds of Gauss sums in finite fields
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by Igor E. Shparlinski PDF

Proc. Amer. Math. Soc. 132 (2004), 2817-2824 Request permission

Abstract:

We consider Gauss sums of the form \[ G_n(a) = \sum _{x \in \mathbb {F}_{p^m}} \chi (x^n) \] 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 $|G_n(a)| \le (n-1) p^{m/2}$ 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 $|G_n(a)|$ 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 $p^{m/2 + 1/2}$.

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