Heuristic methods for computing the minimal multi-homogeneous Bézout number

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 multihomogeneous B ezout number are presented in this paper. Some analysis of computational complexity are given. Numerical examples show the efficiency of these two methods. 2002 Elsevier Inc. All rights reserved.