Homotopies for intersecting solution components of polynomial systems

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


We show how to use numerical continuation to compute the intersection C=A cap B of two algebraic sets A and B, where A, B, and C are numerically represented by witness sets. Enroute to this result, we first show how to find the irreducible decomposition of a system of polynomials restricted to an algebraic set. The intersection of components A and B then follows by considering the decomposition of the diagonal system of equations u-v=0 restricted to {u,v} in A x B. One offshoot of this new approach is that one can solve a large system of equations by finding the solution components of its subsystems and then intersecting these. It also allows one to find the intersection of two components of the two polynomial systems, which is not possible with any previous numerical continuation approach.

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

Key words and phrases. Components of solutions, embedding, generic points, homotopy continuation, irreducible components, numerical algebraic geometry, polynomial system.

SIAM J. Numerical Anal. 42(4):1552-1571, 2004.