Homotopy continuation methods are numerically stable algorithms to
compute all isolated solutions of a polynomial system. A recent application
of these methods finds generic points on each positive dimensional solution
component of the system. Using monodromy we can predict a classification of
those generic points along the breakup of components into irreducible ones.
Besides an increased efficiency, the new methods improve the numerical
conditioning of the multivariate interpolation problem to generate equations
for the components. The performance of our new algorithms will be
illustrated on some well-known examples in science and engineering: systems
of adjacent minors of a general 2-by-n matrix, the cyclic 8- and 9-roots
problems, and the Stewart-Gough platform in mechanical systems design.
This is a joint work with Andrew Sommese and Charles Wampler.
AMS Special Session on Computational Algebraic Geometry and Its Applications 28 April 2001.