Proposal of Singularization of Approximately Singular Polynomial Systems

By "approximately singular system" we mean a system of multivariate polynomials the dimension of whose variety is increased by small amounts of perturbations. By "singularization" we mean determining small perturbations such that the dimension of the variety of the perturbed system increases. We prove a theorem which says, roughly speaking, that the existence of approximately linear-dependent relation(s) among the input polynomials is a necessary condition for that the given system is approximately singular. We give a linear-algebraic lgorithm for computing approximately linear-dependent relations of restricted total-degrees. We then present a linear-algebraic algorithm of the singularization. The algorithm is simple but based on a detailed observation of the linear systems to be solved. The study of singularization is only at the beginning stage, and we point out various open problems.