Homotopies for connected components of algebraic sets with application to computing critical sets

Abstract

Given a polynomial system f, this article provides a general construction for homotopies that yield at least one point of each connected component on the set of solutions of f = 0. This algorithmic approach is then used to compute a superset of the isolated points in the image of an algebraic set which arises in many applications, such as computing critical sets used in the decomposition of real algebraic sets. An example is presented which demonstrates the efficiency of this approach.