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Exact $ p$-divisibility of exponential sums via the covering method


Authors: Francis Castro and Ivelisse M. Rubio
Journal: Proc. Amer. Math. Soc. 143 (2015), 1043-1056
MSC (2010): Primary 11L03; Secondary 11A07
DOI: https://doi.org/10.1090/S0002-9939-2014-12315-X
Published electronically: October 29, 2014
MathSciNet review: 3293721
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Abstract: In general, the methods to estimate the $ p$-divisibility of exponential sums or the number of solutions of systems of polynomial equations over finite fields are non-elementary. In this paper we present the covering method, an elementary combinatorial method that can be used to compute the exact $ p$-divisibility of exponential sums over a prime field. The results here allow us to compute the exact $ p$-divisibility of exponential sums of new families of polynomials, to unify and improve previously known results, and to construct families of systems of polynomial equations over finite fields that are solvable.


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Additional Information

Francis Castro
Affiliation: Department of Mathematics, University of Puerto Rico, Río Piedras, PO Box 70377, San Juan, Puerto Rico 00936
Email: franciscastr@gmail.com

Ivelisse M. Rubio
Affiliation: Department of Computer Science, University of Puerto Rico, Río Piedras, PO Box 70377, San Juan, Puerto Rico 00936
Email: iverubio@gmail.com

DOI: https://doi.org/10.1090/S0002-9939-2014-12315-X
Received by editor(s): May 16, 2013
Received by editor(s) in revised form: July 22, 2013
Published electronically: October 29, 2014
Communicated by: Ken Ono
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.