The problem of solving large systems of polynomial equations arises in many application areas and poses difficult mathematical questions and computational challenges. This paper critically reviews the literature of the past two decades, during which there has been rapid evolution of both theory and codes. Homotopy (continuation) methods are a principal tool for such problems. The code under consideration uses modules from the more general HOMPACK90 package and is compatible with it.

The middle game of following a homotopy path is well understood at this point. The paper focuses principally on the end and opening games. The authors treat mainly isolated, finite, regular solutions, but they also introduce suitable transformations for coping gracefully and efficiently with more awkward situations. They exploit partitioned linear product structure to provide convenient starting points and to prune the number of paths to be followed. The discussion relies heavily on the previous literature cited throughout, but the relevant results are clearly explained--except for minor mysteries.