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The Quarterly Journal of Mathematics Advance Access published online on June 5, 2008
The Quarterly Journal of Mathematics, doi:10.1093/qmath/han013
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© 2008. Published by Oxford University Press. All rights reserved. For permissions, please email: journals.permissions@oxfordjournals.org

DIVISIBILITY OF EXPONENTIAL SUMS AND SOLVABILITY OF CERTAIN EQUATIONS OVER FINITE FIELDS

Francis N. Castro{dagger}

Department of Mathematics, University of Puerto Rico, Box 23355, San Juan, PR 00931-3355, USA

Ivelisse Rubio {ddagger}

Department of Computer Science, University of Puerto Rico, Box 23355, San Juan, PR 00931-3355, USA

José M. Vega §

Department of Mathematics and Physics, University of Puerto Rico, Cayey, PR 00736, USA

{dagger} Corresponding author. E-mail: franciscastr{at}gmail.com

Received 25 January 2008; revised 26 March 2008
  Abstract

Carlitz [Solvability of certain equations in a finite field, Quart. J. Math. (Oxford) 7 (1956), 3–4] determined conditions under which infinite families of polynomials have solutions in a finite field. In this paper we extend some of Carlitz's results by computing the exact p-divisibility of certain exponential sums. As a by-product we obtain an upper bound for the Waring number for polynomials over extensions of finite fields.

{ddagger} E-mail: iverubio{at}uprrp.edu

§ E-mail: jmanueljmvg{at}yahoo.com


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