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Computing zeta functions of nondegenerate hypersurfaces with few monomials

Part of: Families, fibrations (Co)homology theory Zeta and $L$-functions: analytic theory Computational number theory

Published online by Cambridge University Press:  14 February 2013

Steven Sperber  and
John Voight
Steven Sperber
Affiliation:
School of Mathematics, University of Minnesota, 127 Vincent Hall Minneapolis 55455, USA (email: sperber@math.umn.edu)
John Voight
Affiliation:
Department of Mathematics and Statistics, University of Vermont, 16 Colchester Avenue Burlington VT 05401-1455, USA (email: jvoight@gmail.com)

Abstract

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Using the cohomology theory of Dwork, as developed by Adolphson and Sperber, we exhibit a deterministic algorithm to compute the zeta function of a nondegenerate hypersurface defined over a finite field. This algorithm is particularly well suited to work with polynomials in small characteristic that have few monomials (relative to their dimension). Our method covers toric, affine, and projective hypersurfaces, and also can be used to compute the L-function of an exponential sum.

MSC classification

Secondary: 11Y16: Algorithms; complexity 11M38: Zeta and $L$-functions in characteristic $p$ 14D10: Arithmetic ground fields (finite, local, global) 14F30: $p$-adic cohomology, crystalline cohomology
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 16 , October 2013 , pp. 9 - 44
Copyright
© The Author(s) 2013

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