Let f(z) = 0 denote a system of n complex nonlinear equations in n unknowns. Homotopy continuation is a method to find geometrically isolated solutions as follows. Embed f in a system of n equations in n + 1 unknowns, where this new system includes the variables of f and a new variable, called the homotopy parameter. For one value of this parameter, the new system can be satisfactorily solved; for another value it is identically equal to f. The continuation process attempts to solve f(z) = 0 by evolving or continuing the full set of known solutions into the full set of solutions of f(z) = 0. If the homotopy system is h(z,t) = 0, where h(z,1) = f for all z, we assume we know the solutions to h(z,1) = 0; the homotopy parameter t is viewed as varying between 0 and 1.

Implementation of this method involves defining the homotopy h(z,t) and finding a numerical method for tracking the paths defined by h(z,t)- = 0. HOMPACK is a collection of FORTRAN subroutines that can be applied to this general problem. In this paper the authors consider the more specialized but important case when f(z) represents a system of polynomial equations. They discuss the theoretical basis for the method and some of the numerical considerations that are essential to its implementation. Several algorithms that are not available in HOMPACK are also considered and compared. The authors analyze the results of some numerical tests and suggest ways that HOMPACK’s performance can be improved. Because the authors have a great deal of experience and insight, the paper is important for all those who are interested in either the theoretical or practical issues of solving polynomial systems.