Solving Polynomial Systems by Polyhedral Homotopies

In the last two decades, the homotopy continuation method has been developed into a reliable and ecient numerical algorithm for solving all isolated zeros of polynomial systems. During the last few years, a major computational breakthrough has emerged in the area. Based on the Bernshtein theory on root count, the polyhedral homotopy is established to considerably reduce the number of homotopy paths that need to be traced to nd all the isolated roots, making the method much more powerful. This article reports the most recent development of this new method along with future considerations.