Midwest Algebra, Geometry and their Interactions Conference MAGIC05 University of Notre Dame, Notre Dame October 7-11, 2005 Solving polynomial systems by the polyhedral homotopy continuation method by T.Y. LI, Michigan State University Abstract: Numerically solving isolated zeros of polynomial systems in affine space has become increasingly important in applications. In this talk, a numerical approach, developed in the last two decades, by using the homotopy continuation method will be surveyed. The method involves first solving a trivial system, and then deforming these solutions along smooth paths to the solutions of the target system. The method has been successfully implemented in solving many polynomial systems, and the amount of computation required to find all solutions can be made roughly proportional to the number of solutions.