Computing zeta functions of sparse nondegenerate hypersurfaces

TL;DR

This paper presents a deterministic algorithm based on Dwork cohomology to compute zeta functions of sparse, nondegenerate hypersurfaces over finite fields, applicable to various types of hypersurfaces and exponential sums.

Contribution

It introduces a novel, efficient method for computing zeta functions of nondegenerate hypersurfaces using Dwork cohomology, especially for polynomials with few monomials in small characteristic.

Findings

Algorithm effectively computes zeta functions for various hypersurfaces.

Method is efficient for polynomials with few monomials in small characteristic.

Applicable to toric, affine, and projective hypersurfaces.

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.