Heuristic methods for computing the minimal multi-homogeneous Bézout number

摘要

The multi-homogeneous Bézout 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 Bézout number. It is a challenging problem to find a partition with minimal multi-homogeneous Bézout number. Two heuristic numerical methods for computing the minimal multi-homogeneous Bézout number are presented in this paper. Some analysis of computational complexity are given. Numerical examples show the efficiency of these two methods.