Using Monodromy to decompose Solution Sets of Polynomial Systems into Irreducible Components

Abstract:

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.