Algorithmic theory of zeta functions over finite fields

Summary

We give an introductory account of the general algorithmic theory of the zeta function of an algebraic set defined over a finite field.

The zeta function contains important arithmetic and geometric information concerning X. It has been studied extensively in connection with the celebrated Weil conjectures [1949].

Both practical applications and theoretical investigations make a good understanding of the zeta function from an algorithmic point of view increasingly important. The aim of this paper is to present a brief introductory account of the various fundamental problems and results in the emerging algorithmic theory of zeta functions. We shall focus on general properties rather than on results that are restricted to special cases. In particular, in most of this paper we do not assume X to be smooth and projective, although in that case one can often say more.

The contents are organized as follows. In Section 2 we review general properties of zeta functions from an algorithmic point of view. A naive effective algorithm for computing the zeta function is given. If the characteristic p is small, one can use Dwork’s p-adic method to obtain a polynomial time algorithm for computing the zeta function in the case that the numbers of variables and defining equations for X are fixed.