A hybrid algorithm for multi-homogeneous Bézout number

The multi-homogenous homotopy continuation method can solve all isolated solutions of polynomial systems. Different variable partition yields different multi-homogenous Bézout number, which gives the upper bound of the number of isolated solutions. However, the computation of the multi-homogenous Bézout number is hard. In this paper, the permanent formulation of the multi-homogenous Bézout number is considered. The intensive and systemic computations are made for the method of row expansion with memory, the precise and the approximate permanent methods. Each of these methods has its own advantage. Hence a hybrid algorithm is naturally presented. This method works for n about 30 contrasting with 15 before, where n is the number of the variables of the polynomial system.