Advertisement

Journal of Systems Science and Complexity

, Volume 28, Issue 5, pp 1194–1211 | Cite as

Generic regular decompositions for parametric polynomial systems

  • Zhenghong ChenEmail author
  • Xiaoxian Tang
  • Bican Xia
Article

Abstract

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.

Keywords

Generic regular decomposition parametric polynomial system regular-decompositionunstable variety 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    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
  2. [2]
    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
  3. [3]
    Nabeshima K, A speed-up of the algorithm for computing comprehensive Gröbner systems, Proc. ISSAC 2007, 2007, 299–306.Google Scholar
  4. [4]
    Suzuki A and Sato Y, An alternative approach to comprehensive Gröbner bases, Proc. ISSAC 2002, 2002, 255–261.Google Scholar
  5. [5]
    Suzuki A and Sato Y, A simple algorithm to compute comprehensive Gröbner bases, Proc. ISSAC 2006, 2006, 326–331.Google Scholar
  6. [6]
    Weispfenning V, Comprehensive Gröbner bases, J. Symb. Comp., 1992, 14: 1–29.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Aubry P, Lazard D, and Maza M, On the theories of triangular sets, J. Symb. Comp.,, 1999, 28: 105–124.CrossRefzbMATHGoogle Scholar
  8. [8]
    Chen C, Golubitsky O, Lemaire F, Maza M, and Pan W, Comprehensive Triangular Decomposition. Proc. CASC 2007, LNCS 4770, 2007, 73–101.Google Scholar
  9. [9]
    Gao X and Chou S, Solving parametric algebraic systems, Proc. ISSAC 1992, 1992, 335–341.Google Scholar
  10. [10]
    Kalkbrener M, A generalized Euclidean algorithm for computing for computing triangular representationa of algebraic varieties, J. Symb. Comput., 1993, 15: 143–167.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    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
  12. [12]
    Wang D, Zero decomposition algorithms for system of polynomial equations, Computer Mathematics, Proc. ASCM 2000, 2000, 67–70.CrossRefGoogle Scholar
  13. [13]
    Wang M, Computing triangular systems and regular systems, J. Symb. Comput., 2000, 30: 221–236.CrossRefzbMATHGoogle Scholar
  14. [14]
    Wu W, Basic principles of mechanical theorem proving in elementary geometries, J. Syst. Sci. Math. Sci., 1984, 4: 207–235.Google Scholar
  15. [15]
    Yang L, Hou X, and Xia B, A complete algorithm for automated discovering of a class of inequality-type theorems, Science in China, Series F, 2001, 44(6): 33–49.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    Yang L and Xia B, Automated Proving and Discovering Inequalities, Science Press Beijing, 2008 (in Chinese).Google Scholar
  17. [17]
    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
  18. [18]
    Wang M, Elimination Methods, Springer New York, 2001.CrossRefzbMATHGoogle Scholar
  19. [19]
    Wang M, Elimination Practice: Software Tools and Applications, Imperial College Press London, 2004.CrossRefGoogle Scholar
  20. [20]
    Tang X, Chen Z, and Xia B, Generic regular decompositions for generic zero-dimensional systems, Science China Information Sciences, 2014, 57(9): 1–14.MathSciNetCrossRefGoogle Scholar
  21. [21]
    Xia B, DISCOVERER: A tool for solving semi-algebraic systems, ACM Commun. Comput. Algebra., 2007, 41(3): 102–103.CrossRefGoogle Scholar
  22. [22]
    Chou S, Mechanical Geometry Theorem Proving. D. Reidel Publishing Company, 1987.CrossRefGoogle Scholar
  23. [23]
    Chen C and Maza M, Algorithms for computing triangular decomposition of polynomial systems, J. Symb. Comp., 2012, 47: 610–642.CrossRefzbMATHGoogle Scholar
  24. [24]
    Cox D, Little J, and O’Shea D, Using Algebraic Geometry, Springer, New York, 1998.CrossRefzbMATHGoogle Scholar

Copyright information

© Institute of Systems Science, Academy of Mathematics and Systems Science, CAS and Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.School of Mathematical SciencesPeking UniversityBeijingChina

Personalised recommendations