Computing monodromy via parallel homotopy continuation

Numerical homotopy continuation gives a powerful tool for the applied scientist who seeks solutions to a system of polynomial equations. Techniques from numerical homotopy continuation can also be useful in pure mathematical research. We discuss applications of a particular homotopy continuation idea that leads to probabilistic numerical algorithms for construction of monodromy groups. One such application is used to analyze positive-dimensional solutions of polynomial systems. It is called the monodromy breakup method and partitions a witness set representing a positive-dimensional solution into irreducible components. The first author in collaboration with Jan Verschelde has implemented two parallel versions of this algorithm which show good speedup.