Root counting in product homotopy method

Abstract

Product Homotopy method is used to solve dense multivariate polynomial systems for finding all isolated solutions (real or complex). There are two stages in the computation of Homotopy method which are root counting and root finding. This study focuses on root counting which involves the computation of multi-homogeneous Bézout number (MHBN). This value determines the number of solution path in the second stage. Homogenization of partition each gives its own MHBN. Therefore, it is crucial to have minimum MHBN. The computation of minimum MHBN using local search method, fission and assembly method and genetic algorithm had become intractable when the system size gets larger. Hence, this study applied recent heuristic method, Tabu Search. Other than that, the computation of estimating MHBN is of exponential time. For large size system, the usage of row expansion with memory becomes impossible, hence, this study focus on implementing General Random Path algorithm (GRPA). This study implements Tabu search method and GRPA into several systems of different sizes. Tabu search is effective since the global minimum is obtained instead of the local minimum. Other than that, the number of visited partition is much smaller compared with the previous method. Although GRPA gives estimated value, it helps for large size system. We implement two accuracy level in the computation and in the result, the N=1000 gives more accurate result. Hence, GRPA is important when it comes to solve estimated MHBN for large size system.