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. Leykin and Jan Verschelde have implemented two parallel versions of this algorithm which show good speedups. We use numerical homotopy continuation to compute Galois groups of certain enumerative geometric problems coming from Schubert calculus. The basic idea is similar: given a parametric family of 0-dimensional polynomial systems, we construct loops in the parameter...

• Type: Conference
• Year: 2007
• Where: ISSAC
• Authors: Anton Leykin, Frank Sottile
• Keywords: Homotopy Continuation, Homotopy Continuation Idea, ISSAC 2007, Numerical Homotopy Continuation