Homotopy methods for equations over finite fields

Abstract

This paper describes an application of some ideas from homotopy theory to the problem of computing the number of solutions to a multivariate polynomial equation over a finite field. The benefit of the homotopy approach over more direct methods is that the runningtime is far less dependent on the number of variables. The method was introduced by the author in another paper, where specific complexity estimates were obtained for certain special cases. Some consequences of these estimates are stated in the present paper. © Springer-Verlag Berlin Heidelberg 2003.