**Homotopies for intersecting solution components
of polynomial systems**

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

#### Abstract:

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.