NDLI: Robust Algorithms For Generalized Pham Systems

Content Provider	Springer Nature Link
Author	Dratman, Ezequiel Matera, Guillermo Waissbein, Ariel
Copyright Year	2009
Abstract	We discuss the complexity of robust symbolic algorithms solving a significant class of zero–dimensional square polynomial systems with rational coefficients over the complex numbers, called generalized Pham systems, which represent the class of zero–dimensional homogeneous complete–intersection systems with “no points at infinity”. Our notion of robustness models the behavior of all known universal methods for solving (parametric) polynomial systems avoiding unnecessary branchings and allowing the solution of certain limit problems. We first show that any robust algorithm solving generalized Pham systems has complexity at least polynomial in the Bézout number of the system under consideration. Then we exhibit a robust probabilistic algorithm which solves generalized Pham systems with quadratic complexity in the Bézout number of the input system. The algorithm consists in a series of homotopies deforming the input system into a system which is “easy to solve”, together with a “projection algorithm” which allows to move the solutions of the known instance to the solutions of an arbitrary instance along the parameter space.
Starting Page	105
Ending Page	154
Page Count	50
File Format	PDF
ISSN	10163328
Journal	Computational complexity
Volume Number	18
Issue Number	1
e-ISSN	14208954
Language	English
Publisher	Birkhäuser-Verlag
Publisher Date	2009-04-24
Publisher Place	Basel
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Polynomial system solving robust algorithms geometric solutions Newton–Hensel lifting probabilistic algorithms complexity Algorithm Analysis and Problem Complexity Computational Mathematics and Numerical Analysis
Content Type	Text
Resource Type	Article
Subject	Theoretical Computer Science Computational Theory and Mathematics Computational Mathematics

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