Computing zeta functions of nondegenerate hypersurfaces with few monomials

Steven Sperber; John Voight

doi:10.1112/S1461157012001179

Computing zeta functions of nondegenerate hypersurfaces with few monomials

Steven Sperber, John Voight

School of Mathematics

Research output: Contribution to journal › Article › peer-review

4 Scopus citations

Abstract

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.

Original language	English (US)
Pages (from-to)	9-44
Number of pages	36
Journal	LMS Journal of Computation and Mathematics
Volume	16
DOIs	https://doi.org/10.1112/S1461157012001179
State	Published - 2013

Bibliographical note

Funding Information:
The second author was partially supported by the National Security Agency under Grant Number H98230-09-1-0037.

Publisher Copyright:
© 2013 Author.

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10.1112/S1461157012001179

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abstract = "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.",

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AB - 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.

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Computing zeta functions of nondegenerate hypersurfaces with few monomials

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