Merging solutions of non-linear algebraic systems

Please always quote using this URN: urn:nbn:de:0297-zib-11769
  • In solving large polynomial algebraic systems that are too big for standard Gröbner basis techniques one way to make progress is to introduce case distinctions. This divide and conquer technique can be beneficial if the algorithms and computer programs know how to take advantage of inequalities. A further hurdle is the form of the resulting general solutions which often have unnecessarily many branches. In this paper we discuss a procedure to merge solutions by dropping inequalities which are associated with them and, if necessary, by re-parametrizing solutions. In the appendix the usefulness of the procedure is demonstrated in the classification of quadratic Hamiltonians with a Lie-Poisson bracket $e(3)$. This application required the solution of algebraic systems with over 200 unknowns, 450 equations and between 5000 and 9000 terms.

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Metadaten
Author:Thomas Wolf
Document Type:ZIB-Report
Tag:Lie-Poisson brackets; computer algebra; polynomial systems; quadratic Hamiltonians
MSC-Classification:37-XX DYNAMICAL SYSTEMS AND ERGODIC THEORY [See also 26A18, 28Dxx, 34Cxx, 34Dxx, 35Bxx, 46Lxx, 58Jxx, 70-XX] / 37Jxx Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems [See also 53Dxx, 70Fxx, 70Hxx] / 37J35 Completely integrable systems, topological structure of phase space, integration methods
PACS-Classification:00.00.00 GENERAL / 02.00.00 Mathematical methods in physics / 02.70.-c Computational techniques; simulations (for quantum computation, see 03.67.Lx; for computational techniques extensively used in subdivisions of physics, see the appropriate section; for example, see 47.11.-j Computational methods in fluid dynamics) / 02.70.Wz Symbolic computation (computer algebra)
Contributing Corporation:Department of Mathematics, Brock University St. Catharines Ontario, Canada and ZIB Fellow
Date of first Publication:2010/07/05
Series (Serial Number):ZIB-Report (10-13)
ISSN:1438-0064