Homotopies to Compute Intersections of Solution
Components of Polynomial Systems
Abstract:
We (Andrew Sommese, Jan Verschelde, and Charles Wampler) show how
to use numerical continuation to compute the intersection C
of two irreducible algebraic sets A and B,
where A, B, and C are numerically
represented by witness sets. We show this by first showing how
to find the irreducible decomposition of the solution set 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 AxB.
This diagonal homotopy also allows us to find the
intersection of two components of two polynomial systems (possibly
the same system), which is not possible with any previous numerical
continuation approach. A major offshoot, of this new approach, which
will be discussed in the talk of Charles Wampler, is that one can
solve a large system of equations by finding the solution components
of its subsystems and then intersecting these.