Deformation Theory and the Computation of Zeta Functions

An attractive and challenging problem in computational number theory is to count in an e*cient manner the number of solutions to a multivariate polynomial equation over a -nite -eld. One desires an algorithm whose time complexity is a small polynomial function of some appropriate measure of the size of the polynomial. A natural measure of size is d logðqÞ for a polynomial of total degree d in n variables over the -eld of q elements. Despite intensive research over the last few decades, existing algorithms fall well short of this ideal. The case n 1⁄4 1, which is related to univariate polynomial factorisation, was solved essentially by Berlekamp and is comparatively straightforward; see [19, Chapter 14]. When n 1⁄4 2 one is counting points on curves, a topic of some practical importance [2]. Here one can achieve a complexity of logðqÞd where the exponent Cd depends exponentially on d (this can be improved to Cd a polynomial in d in some special cases; see [1]). The only algorithm which applies for general n has a complexity which is a polynomial function of ðpd logðqÞÞ [26] (see also the strategy in [23, x 2]). This is polynomial-time in what one might call the small characteristic input size of pd logðqÞ, but only for -xed dimension (that is, the exponent depends upon the dimension n). The purpose of this paper is to introduce a systematic new approach to counting solutions to equations over -nite -elds which aims to remove the exponential dependence on the dimension for small characteristic. That is, to obtain a single time complexity which is polynomial in pd logðqÞ uniformly over all n. A general method is sketched, and worked out for a particular, quite broad, family of polynomials. The new approach rests on two observations. First, the number of solutions to any equation de-ned by a diagonal polynomial can be computed easily within the required time bound. Second, any suitably generic polynomial can be deformed into a diagonal polynomial; more precisely, it lies in a one-parameter family of polynomials containing a diagonal form. Over a -nite -eld this deformation appears super-cially to be of little use. Remarkably though, one can associate a linear p-adic di<erential equation with the deformation, and by solving this one can recover the number of solutions to the original -nite -eld equation. Thus, in a sense, one reduces a high-dimensional solution counting problem to a one-dimensional deformation problem, whence the magical reduction in complexity. We note that homotopy methods are apparently well studied in the context of the numerical solution of systems of equations over