Janet-Like Monomial Division

  • Vladimir P. Gerdt
  • Yuri A. Blinkov
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3718)

Abstract

In this paper we introduce a new type of monomial division called Janet-like, since its properties are similar to those of Janet division. We show that the former division improves the latter one. This means that a Janet divisor is always a Janet-like divisor but the converse is generally not true. Though Janet-like division is not involutive, it preserves all algorithmic merits of Janet division, including Noetherianity, continuity and constructivity. Due to superiority of Janet-like division over Janet division, the algorithm for constructing Gröbner bases based on the new division is more efficient than its Janet division counterpart.

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References

  1. 1.
    Gerdt, V.P., Blinkov, Y.A.: Involutive Bases of Polynomial Ideals. Mathematics and Computers in Simulation 45, 519–542 (1998), http://arXiv.org/math.AC/9912027 Minimal Involutive Bases. Ibid., 543–560, http://arXiv.org/math.AC/9912029
  2. 2.
    Apel, J.: Theory of Involutive Divisions and an Application to Hilbert Function Computations. Journal of Symbolic Computation 25, 683–704 (1998)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Janet, M.: Leçons sur les Systèmes d’Equations aux Dérivées Partielles, Cahiers Scientifiques, IV. Gauthier-Villars, Paris (1929)Google Scholar
  4. 4.
    Gerdt, V.P., Blinkov, Y.A., Yanovich, D.A.: Construction of Janet Bases. I. Monomial Bases. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) Computer Algebra in Scientific Computing / CASC 2001, pp. 233–247. Springer, Berlin (2001); II. Polynomial bases, ibid., 249–263Google Scholar
  5. 5.
    Berth, M., Gerdt, V.: Computation of Involutive Bases with Mathematica. In: Proceedings of the Third International Workshop on Mathematica System in Teaching and Research, Institute of Mathematics & Physics, University of Podlasie, pp. 29–34 (2001)Google Scholar
  6. 6.
    Hausdorf, M., Seiler, W.M.: Involutive Bases in MuPAD – Part I: Involutive Divisions. mathPAD 11, 51–56 (2002)Google Scholar
  7. 7.
    Blinkov, Y.A., Cid, C.F., Gerdt, V.P., Plesken, W., Robertz, D.: The Maple Package “Janet”: I. Polynomial Systems. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) Computer Algebra in Scientific Computing / CASC 2003. Institute of Informatics, pp. 31–40. Technical University of Munich, Garching (2003)Google Scholar
  8. 8.
    Hemmecke, R.: Involutive Bases for Polynomial Ideals. PhD Thesis, RISC Linz (2003)Google Scholar
  9. 9.
    Gerdt, V.P.: Computational Commutative and Non-Commutative algebraic geometry. In: Cojocaru, S., Pfister, G., Ufnarovski, V. (eds.) Computational Commutative and Non-Commutative algebraic geometry. NATO Science Series, pp. 199–225. IOS Press, Amsterdam (2005), http://arXiv.org/math.AC/0501111 Google Scholar
  10. 10.
    Buchberger, B.: Gröbner Bases: an Algorithmic Method in Polynomial Ideal Theory. In: Bose, N.K. (ed.) Recent Trends in Multidimensional System Theory, Reidel, Dordrecht, pp. 184–232 (1985)Google Scholar
  11. 11.
    Gerdt, V.P., Blinkov, Y.A.: Janet Bases of Toric Ideals. In: Kredel, H., Seiler, W.K. (eds.) Proceedings of the 8th Rhine Workshop on Computer Algebra, pp. 125–135. University of Mannheim (2002), http://arXiv.org/math.AC/0501180
  12. 12.
    Gerdt, V.P., Blinkov, Y.A.: Janet-like Gröbner Bases. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2005. LNCS, vol. 3718, pp. 184–195. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  13. 13.
    Giovinni, A., Mora, T., Niesi, G., Robbiano, L., Traverso, C.: One sugar cube, please, or selection strategies in the Buchberger algorithm. In: Proceedings of ISSAC 1991, pp. 49–54. ACM Press, New York (1991)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Vladimir P. Gerdt
    • 1
  • Yuri A. Blinkov
    • 2
  1. 1.Laboratory of Information TechnologiesJoint Institute for Nuclear ResearchDubnaRussia
  2. 2.Department of Mathematics and MechanicsSaratov UniversitySaratovRussia

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