Generic regular decompositions for parametric polynomial systems
- 55 Downloads
This paper presents a generalization of the authors’ earlier work. In this paper, the two concepts, generic regular decomposition (GRD) and regular-decomposition-unstable (RDU) variety introduced in the authors’ previous work for generic zero-dimensional systems, are extended to the case where the parametric systems are not necessarily zero-dimensional. An algorithm is provided to compute GRDs and the associated RDU varieties of parametric systems simultaneously on the basis of the algorithm for generic zero-dimensional systems proposed in the authors’ previous work. Then the solutions of any parametric system can be represented by the solutions of finitely many regular systems and the decomposition is stable at any parameter value in the complement of the associated RDU variety of the parameter space. The related definitions and the results presented in the authors’ previous work are also generalized and a further discussion on RDU varieties is given from an experimental point of view. The new algorithm has been implemented on the basis of DISCOVERER with Maple 16 and experimented with a number of benchmarks from the literature.
KeywordsGeneric regular decomposition parametric polynomial system regular-decompositionunstable variety
Unable to display preview. Download preview PDF.
- Kapur D, Sun Y, and Wang D, A new algorithm for computing comprehensive Gröbner systems, Proc. ISSAC 2010, ACM Press, 2010, 25–28.Google Scholar
- Montes A and Recio T, Automatic discovery of geometry theorems using minimal canonical comprehensive Gröbner systems, Eds. by Botana F and Recio T, ADG 2006, Lecture Notes in Artificial Intelligence 4869, Berlin Heidelberg: Springer-Verlag, Berlin Heidelberg, 2007, 113–138.Google Scholar
- Nabeshima K, A speed-up of the algorithm for computing comprehensive Gröbner systems, Proc. ISSAC 2007, 2007, 299–306.Google Scholar
- Suzuki A and Sato Y, An alternative approach to comprehensive Gröbner bases, Proc. ISSAC 2002, 2002, 255–261.Google Scholar
- Suzuki A and Sato Y, A simple algorithm to compute comprehensive Gröbner bases, Proc. ISSAC 2006, 2006, 326–331.Google Scholar
- Chen C, Golubitsky O, Lemaire F, Maza M, and Pan W, Comprehensive Triangular Decomposition. Proc. CASC 2007, LNCS 4770, 2007, 73–101.Google Scholar
- Gao X and Chou S, Solving parametric algebraic systems, Proc. ISSAC 1992, 1992, 335–341.Google Scholar
- Maza M, On triangular decompositions of algebraic varieties, Technical Report TR 4/99, NAG Ltd, Oxford, UK, 1999, Presented at the MEGA-2000 Conference, Bath, England.Google Scholar
- Wu W, Basic principles of mechanical theorem proving in elementary geometries, J. Syst. Sci. Math. Sci., 1984, 4: 207–235.Google Scholar
- Yang L and Xia B, Automated Proving and Discovering Inequalities, Science Press Beijing, 2008 (in Chinese).Google Scholar
- Yang L and Zhang J, Searching dependency between algebraic equations: An algorithm applied to automated reasoning, Technical Report ICTP/91/6, International Center for Theoretical Physics, 1991, 1–12.Google Scholar