A Tabu Search Method for Finding Minimal Multi-Homogeneous Bézout Number

Abstract

Problem statement: A homotopy method has proven to be reliable for computing all of the isolated solutions of a multivariate polynomial system. The multi-homogeneous Bézout number of a polynomial system is the number of paths that one has to trace in order to compute all of its isolated solutions. Each partition of the variables corresponds to a multi-homogeneous Bézout number. It is a crucial problem to find a partition with the minimum multi-homogeneous Bézout number since the size of the space of all the partitions increases exponentially. Approach: This study presented a new method by producing the Tabu Search Method (TSM) as a powerful technique for finding minimum multi-homogeneous Bézout number. Results: A comparison is made between the new method and some recent methods. It is shown that our algorithm is superior to the latter, besides being simple and efficient in the implementation. Conclusion: Furthermore the present study extended the applicability of the Tabu search method.