Given a system of polynomial equations, we can compute with homotopy continuation methods much of the geometric information contained in the primary decomposition of the solution set. The basic data in a numerical irreducible decomposition are generic points, that certify the existence of irreducible components of the solution set, their dimensions, and their degrees. A decomposition then consists in the classification of those generic points according to the several irreducible components. Equations for the components are obtained by interpolation. A special case is the factorization of multivariate polynomials. Using monodromy increases efficiency and accuracy. This is a joint work with Andrew Sommese and Charles Wampler.
UIC Algebraic Geometry seminar, 7 March 2001.