Heuristic methods for computing the minimal multi-homogeneous B ezout number Ting Li, Zhenjiang Lin, Fengshan Bai * ,1 Department of Mathematical Sciences, Tsinghua University, Beijing 100084, PR China Abstract The multi-homogeneous B ezout number of a polynomial system is the number of paths that one has to follow in computing all its isolated solutions with continuation method. Each partition of variables corresponds to a multi-homogeneous B ezout number. It is a challenging problem to find a partition with minimal multi-homogeneous B ezout number. Two heuristic numerical methods for computing the minimal multi- homogeneous B ezout number are presented in this paper. Some analysis of computa- tional complexity are given. Numerical examples show the efficiency of these two methods. Ó 2002 Elsevier Inc. All rights reserved. Keywords: Multi-homogeneous B ezout number; Continuation method; System of polynomial equations 1. Introduction For a system of polynomial equations one may wish to determine the number of isolated solutions and then compute all of them. Garcia and Zangwill [3] and Drexler [1] suggest that homotopy continuation could be used to find the full set of isolated solutions of polynomial systems. During the last * Corresponding author. E-mail address: fbai@math.tsinghua.edu.cn (F. Bai). 1 Supported by National Science Foundation of China G19871047 and National Key Basic Research Special Fund G1998020306. 0096-3003/$ - see front matter Ó 2002 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(02)00540-4 Applied Mathematics and Computation 146 (2003) 237–256 www.elsevier.com/locate/amc