Numerical Irreducible Decomposition
using Projections from Points on the Components

Andrew J. Sommese, Jan Verschelde, and Charles W. Wampler


To classify positive dimensional solution components of a polynomial system, we construct polynomials interpolating points sampled from each component. In previous work, points on an i-dimensional component were linearly projected onto a generically chosen (i+1)-dimensional subspace. In this paper, we present two improvements. First, we reduce the dimensionality of the ambient space by determining the linear span of the component and restricting to it. Second, if the dimension of the linear span is greater than i+1, we use a less generic projection that leads to interpolating polynomials of lower degree, thus reducing the number of samples needed. While this more efficient approach still guarantees - with probability one - the correct determination of the degree of each component, the mere evaluation of an interpolating polynomial no longer certifies the membership of a point to that component. We present an additional numerical test that certifies membership in this new situation. We illustrate the performance of our new approach on some well-known test systems.

2000 AMS Subject Classification : Primary 65H10; Secondary 13P05, 14Q99, 68W30.

keywords : Components of solutions, central projection, embedding, generic points, homotopy continuation, irreducible components, numerical algebraic geometry, polynomial system, primary decomposition.

In Symbolic Computation: Solving Equations in Algebra, Geometry, and Engineering, edited by E.L. Green, S. Hosten, R. Laubenbacher, and V.A. Powers. Contemporary Mathematics, volume 286, AMS, 2001.