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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Bounds of Gauss sums in finite fields


Author: Igor E. Shparlinski
Journal: Proc. Amer. Math. Soc. 132 (2004), 2817-2824
MSC (2000): Primary 11L05, 11T24; Secondary 11B37
Published electronically: June 2, 2004
MathSciNet review: 2063098
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Abstract: We consider Gauss sums of the form

\begin{displaymath}G_n(a) = \sum_{x \in \mathbb{F} _{p^m}} \chi(x^n) \end{displaymath}

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 $p^{m/2 + 1/2}$.


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Additional Information

Igor E. Shparlinski
Affiliation: Department of Computing, Macquarie University, Sydney, New South Wales 2109, Australia
Email: igor@ics.mq.edu.au

DOI: http://dx.doi.org/10.1090/S0002-9939-04-07133-3
PII: S 0002-9939(04)07133-3
Keywords: Gauss sums, finite fields, linear recurrence sequences
Received by editor(s): February 1, 2002
Received by editor(s) in revised form: June 7, 2002
Published electronically: June 2, 2004
Communicated by: Wen-Ching Winnie Li
Article copyright: © Copyright 2004 American Mathematical Society