Of finding all isolated roots of polynomial systems in complex n-space via stable mixed volume
Tien-Yien Li, Tangan Gao · 1999
To find all the isolated zeroes of a polynomial system P( x) in Cn (as opposed to in C* n ) via the polyhedral homotopy method of Huber and Sturmfels [7], one first finds all stable mixed cells in a stable mixed subdivision and establishes a fine mixed subdivision for each stable mixed cell. One then solves a collection of polynomial subsystems corresponding to the stable mixed cells, and uses their solutions as starting points for the homotopy paths of a set of nonlinear homotopies which lead to all the isolated zeros of P(x) in Cn . This method offers a dramatic computational improvement over earlier homotopy algorithms at the cost of many costly recursive liftings at the preprocessing step of finding the stable mixed cells and their fine mixed subdivisions. The main goal of this dissertation is to present a new strategy which can quickly (and simultaneously) find the stable mixed subdivision, the fine mixed subdivisions of the stable mixed cells, and the necessary subsystems by means of a single lifting.