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Ramification points arise from singularities along solution paths of a homotopy. This paper considers ramification points of homotopies, elucidating the total number of ramification points and providing general theory regarding the properties of the set of ramification points over the same branch point. The general approach utilized in this paper is to view homotopies as lines in the parameter spaces of families of polynomial systems on a projective manifold. With this approach, the number of singularities of systems parameterized by pencils is computed under broad conditions. General conditions are given for when the singularities of the systems parameterized by a line in a space of polynomial systems have multiplicity two. General conditions are also given for there to be at most one singularity in the solution set of any system parameterized by such a line. Several examples are included to demonstrate the theoretical results. 

Given a polynomial system f, this article provides a general construction for homotopies that yield at least one point of each con- nected 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 com- puting critical sets used in the decomposition of real algebraic sets. An example is presented which demonstrates the efficiency of this approach.