Finding All Isolated Zeros of Polynomial Systems inCnvia Stable Mixed Volumes
Copyright policy )Finding All Isolated Zeros of Polynomial Systems inCnvia 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.
- Michigan State University United States
- University of Central Arkansas United States
Zero sets of holomorphic functions of several complex variables, algorithm, Algebra and Number Theory, polyhedral homotopy, Numerical computation of solutions to systems of equations, polynomial systems, isolated roots, Global methods, including homotopy approaches to the numerical solution of nonlinear equations, mixed subdivision, Computational Mathematics, stable mixed volumes, Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral), Computational aspects of field theory and polynomials, homotopy continuation
Zero sets of holomorphic functions of several complex variables, algorithm, Algebra and Number Theory, polyhedral homotopy, Numerical computation of solutions to systems of equations, polynomial systems, isolated roots, Global methods, including homotopy approaches to the numerical solution of nonlinear equations, mixed subdivision, Computational Mathematics, stable mixed volumes, Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral), Computational aspects of field theory and polynomials, homotopy continuation
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