Applications of the theory of weakly nondegenerate conditions to zero decomposition for polynomial systems

This paper presents some applications of the theory of weakly nondegenerate conditions obtained by analytic methods to zero decomposition of polynomial systems and sets. Based on a known algorithm, a method is presented that can compute a strong regular series of any nonempty polynomial set. An algorithm is also devised that can decompose any polynomial system into two finite sets of strong regular sets with some good properties. In addition, we propose two alternative methods for decomposing any algebraic variety and quasi-algebraic variety into equidimensional components and removing redundant components respectively without computing Gröbner bases. Some examples are given to illustrate the performance and effectiveness of the applications.