Mathematics in Computer Science

, Volume 6, Issue 4, pp 457–473 | Cite as

On Solving Parametric Polynomial Systems

  • Marc Moreno Maza
  • Bican Xia
  • Rong Xiao


Border polynomial and discriminant variety are two important notions related to parametric polynomial system solving, in particular, for partitioning the parameter space into regions where the solutions of the system depend continuously on the parameter values. In this paper, we study the relations between those notions in the case of parametric triangular systems. We also investigate the properties and computation of the non-properness locus of the canonical projection restricted at a parametric regular chain or at its saturated ideal.


Parametric polynomial system Border polynomial Discriminant variety Effective boundary Triangular decomposition Regular chain 

Mathematics Subject Classification (2010)

Primary 13P15 Secondary 68W30 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aubry P., Lazard D., Moreno Maza M.: On the theories of triangular sets. J. Symb. Comput. 28(1–2), 105–124 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Basu S., Pollack R., Roy M.-F.: Algorithms in Real Algebraic Geometry. Springer, Berlin (2006)zbMATHGoogle Scholar
  3. 3.
    Chen, C., Davenport, J.H., May, J., Moreno Maza, M., Xia, B., Xiao, R.: Triangular decomposition of semi-algebraic systems. In: Proceedings of ISSAC 2010, pp. 187–194. ACM, New York (2010)Google Scholar
  4. 4.
    Chen, C., Davenport, J.H., Moreno Maza, M., Xia, B., Xiao, R.: Computing with semi-algebraic sets represented by triangular decomposition. In: Proceedings of ISSAC 2011, pp. 75–82. ACM, New York (2011)Google Scholar
  5. 5.
    Chen, C., Golubitsky, O., Lemaire, F., Moreno Maza, M., Pan, W.: Comprehensive triangular decomposition. In: Proceedings of CASC’07, Lecture Notes in Computer Science, vol. 4770, pp. 73–101 (2007)Google Scholar
  6. 6.
    Chen C., Moreno Maza M.: Algorithms for computing triangular decomposition of polynomial systems. J. Symb. Comput. 47(6), 610–642 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Chou, S.C., Gao, X.S.: Solving parametric algebraic systems. In: Proceedings ISSAC’92, pp. 335–341, Berkeley (1992)Google Scholar
  8. 8.
    Collins G.E.: Quantifier elimination for real closed fields by cylindrical algebraic decomposition. Springer Lect. Notes Comput. Sci. 33, 515–532 (1975)Google Scholar
  9. 9.
    Collins G.E., Hong H.: Partial cylindrical algebraic decomposition. J. Symb. Comput. 12(3), 299–328 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Dolzmann, A., Sturm, T., Weispfenning, V.: Real quantifier elimination in practice. In: Algorithmic Algebra and Number Theory, pp. 221–247. Springer, New York (1998)Google Scholar
  11. 11.
    Jelonek Z.: Testing sets for properness of polynomial mappings. Math. Ann. 315, 1–35 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Kalkbrener M.: A generalized euclidean algorithm for computing triangular representations of algebraic varieties. J. Symb. Comput. 15, 143–167 (1933)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Kapur, D.: An approach for solving systems of parametric polynomial equations (1993)Google Scholar
  14. 14.
    Lazard D.: A new method for solving algebraic systems of positive dimension. Discret. Appl. Math. 33, 147–160 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Lazard D., Rouillier F.: Solving parametric polynomial systems. J. Symb. Comput. 42(6), 636–667 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Manubens M., Montes A.: Minimal canonical comprehensive Gröbner systems. J. Symb. Comput. 44, 463–478 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Moreno Maza, M., Xia, B., Xiao, R.: On solving parametric polynomial systems. In: Raschau, S. (ed.) Proceedings of the Fourth Internationa Conference on Mathematical Aspects of Computer Science and Information Sciences (MACIS 2011), pp. 205–215, Beijing (2011)Google Scholar
  18. 18.
    Montes A.: A new algorithm for discussing Gröbner bases with parameters. J. Symb. Comput. 33(2), 183–208 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Schost É.: Computing parametric geometric resolutions. Appl. Algebra Eng. Commun. Comput. 13, 349–393 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Stasica A.: An effective description of the Jelonek set. J. Pure Appl. Algebra 169(2–3), 321–326 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Wang D.M.: Elimination Methods. Springer, New York (2000)Google Scholar
  22. 22.
    Weispfenning V.: Comprehensive Gröbner bases. J. Symb. Comput. 14, 1–29 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Weispfenning, V.: Canonical comprehensive Gröbner bases. In: ISSAC 2002, pp. 270–276. ACM Press, New York (2002)Google Scholar
  24. 24.
    Wu W.T.: A zero structure theorem for polynomial equations solving. MM Res Preprints 1, 2–12 (1987)Google Scholar
  25. 25.
    Xiao, R.: Parametric polynomial system solving. PhD thesis, Peking University, Beijing (2009)Google Scholar
  26. 26.
    Yang, L., Hou, X., Xia, B.: Automated discovering and proving for geometric inequalities. In: Proceedings of ADG’98, pp. 30–46 (1999)Google Scholar
  27. 27.
    Yang, L., Xia, B.: Real solution classifications of a class of parametric semi-algebraic systems. In: Proceedings of the A3L’05, pp. 281–289 (2005)Google Scholar

Copyright information

© Springer Basel 2012

Authors and Affiliations

  1. 1.University of Western OntarioLondonCanada
  2. 2.Peking UniversityShenzhenChina

Personalised recommendations