Excess intersections and numerical irreducible decompositions

Abstract

A fundamental problem in algebraic geometry is to decompose the solution set of a polynomial system. A numerical irreducible decomposition is a numerical description of this solution set. Standard algorithms to compute this use a sequence of several homotopies. Our new approach uses isosingular theory and a classical result to compute a smooth point on every irreducible component in every dimension with a single homotopy. Key words. Numerical irreducible decomposition, isosingular sets, numerical algebraic geometry, intersection theory.