In a recent joint work with Andrew J. Sommese (University of Notre Dame) and Charles W. Wampler (General Motors Research and Development) we have developed numerical homotopy methods to decompose positive dimensional solution sets of polynomial systems into irreducible components. The problem addressed in this talk is the intersection of two irreducible solution components of two possibly identical polynomial systems, a problem which could not be solved by any previous numerical homotopy. To develop new homotopies to solve this problem, we generalize our algorithms for a numerical irreducible decomposition to polynomial systems restricted to an algebraic set. Considering the diagonal system of equations u - v = 0 restricted to the product of the two components we wish to intersect leads to the "diagonal homotopy", providing a numerical representation of the intersection. Computational experiments illustrate the efficiency of this new diagonal homotopy.
UIC MSCS Mathematics and Applications Seminar, 28 January 2004.