Finding all isolated zeros of polynomial systems in \(\mathbb{C}^n\) via stable mixed volumes
In 1997, \textit{B. Huber} and \textit{B. Sturmfels} [Discrete Comput Geom. 17, No. 2, 137-141 (1997; Zbl 0891.65055)] developed a method, based on the polyhedral homotopy, for finding all of the isolated roots in \(\mathbb{C}^n\) of a polynomial system. The method requires one to find all stable mixed cells in the stable mixed subdivision of a polyhedron, and then to create a fine mixed subdivision for each such cell. While this greatly reduces the number of extraneous (not leading to a root) paths that must be followed in the ensuing homotopy continuation, the steps described above may require several recursive liftings. This paper presents an algorithm which simultaneously builds the stable mixed subdivision and the fine mixed subdivisions of the stable mixed cells using a single lifting. The algorithm has been implemented, and results (without timings) are presented for 7 well-known examples.
scientific article; zbMATH DE number 1369544