A Method to Determine if Two Parametric Polynomial Systems Are Equal

  • Jie Zhou
  • Dingkang Wang
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8592)

Abstract

The comprehensive Gröbner systems of parametric polynomial ideal were first introduced by Volker Weispfenning. Since then, many improvements have been made to improve these algorithms to make them useful for different applications. In contract to reduced Groebner bases, which is uniquely determined by the polynomial ideal and the term ordering, however, comprehensive Groebner systems do not have such a good property. Different algorithm may give different results even for a same parametric polynomial ideal. In order to treat this issue, we give a decision method to determine whether two comprehensive Groebner systems are equal. The polynomial ideal membership problem has been solved for the non-parametric case by the classical Groebner bases method, but there is little progress on this problem for the parametric case until now. An algorithm is given for solving this problem through computing comprehensive Groebner systems. What’s more, for two parametric polynomial ideals and a constraint over the parameters defined by a constructible set, an algorithm will be given to decide whether one ideal contains the other under the constraint.

Keywords

Constructible Set Quasi-algebraic set Gröbner Bases Comprehensive Gröbner System 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Cox, D., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms, 3rd edn. Springer, New York (2007)CrossRefMATHGoogle Scholar
  2. 2.
    Gonzalez–Vega, L., Traverso, C., Zanoni, A.: Hilbert stratification and parametric Gröbner bases. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2005. LNCS, vol. 3718, pp. 220–235. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  3. 3.
    Kapur, D., Sun, Y., Wang, D.K.: A new algorithm for computing comprehensive Gröbner systems. In: Proc. ISSAC 2010, pp. 29–36. ACM Press, New York (2010)Google Scholar
  4. 4.
    Manubens, M., Montes, A.: Minimal canonical comprehensive Gröbner systems. Journal of Symbolic Computation 44(5), 463–478 (2009)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Montes, A.: A new algorithm for discussing Gröbner bases with parameters. Journal of Symbolic Computation 33(2), 183–208 (2002)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Montes, A., Wibmer, M.: Gröbner bases for polynomial systems with parameters. Journal of Symbolic Computation 45(12), 1391–1425 (2010)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Nabeshima, K.: A speed-up of the algorithm for computing comprehensive Gröbner systems. In: Proc. ISSAC 2010, pp. 299–306. ACM Press, New York (2007)Google Scholar
  8. 8.
    Sato, Y., Suzuki, A.: Computation of inverses in residue class rings of parametric polynomial ideal. In: Proc. ISSAC 2009, pp. 311–316. ACM Press, New York (2009)Google Scholar
  9. 9.
    Suzuki, A., Sato, Y.: An alternative approach to comprehensive Gröbner bases. Journal of Symbolic Computation 36(3), 649–667 (2003)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Suzuki, A., Sato, Y.: A simple algorithm to compute comprehensive Gröbner bases using Gröbner bases. In: Proc. ISSAC 2006, pp. 326–331. ACM Press, New York (2006)Google Scholar
  11. 11.
    Sit, W.: Computations on Quasi-Algebraic Sets (2001)Google Scholar
  12. 12.
    Weispfenning, V.: Comprehensive Gröbner bases. Journal of Symbolic Computation 14(1), 1–29 (1992)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Weispfenning, V.: Canonical comprehensive Gröbner bases. Journal of Symbolic Computation 36(3), 669–683 (2003)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Wibmer, M.: Gröbner bases for families of affine or projective schemes. Journal of Symbolic Computation 42(8), 803–834 (2007)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Jie Zhou
    • 1
  • Dingkang Wang
    • 1
  1. 1.KLMM, Academy of Mathematics and Systems ScienceCASBeijingChina

Personalised recommendations