Skip to main content
SpringerLink
Log in

Involutive divisions and monomial orderings: Part II

  • Published:
Programming and Computer Software Aims and scope Submit manuscript

Abstract

This paper is a sequel to the studies on classification properties of involutive divisions reported in [1]. An example is given in which the minimal involutive basis of a particular monomial ideal for the “Janet antipode” <-division is neither an involutive Janet basis nor a minimal basis for a Janet-like division for any of n! orderings of variables. This example disproves the hypothesis that the minimal involutive basis for continuous and constructive divisions always coincides with the Janet basis for some ordering of variables.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Semenov, A. S. and Zyuzikov, P.A., Involutive Divisions and Monomial Orderings, Programmirovanie, 2007, no. 3, pp. 24–33 [Programming Comput. Software (Engl. Transl.), 2007, vol. 33, no. 3, pp. 139–146].

  2. Apel, J., The Theory of Involutive Divisions and an Application to Hilbert Function Computations, J. Symbolic Computation, 1998, vol. 25, no. 6, pp. 683–704.

    Article  MATH  MathSciNet  Google Scholar 

  3. Gerdt, V. P. and Blinkov, Yu. A., Involutive Bases of Polynomial Ideals, Math. Comput. Simulation, 1998, vol. 45, pp. 519–542.

    Article  MATH  MathSciNet  Google Scholar 

  4. Gerdt, V. P. and Blinkov, Yu. A., Minimal Involutive Bases, Math. Comput. Simulation, 1998, vol. 45, pp. 543–560.

    Article  MATH  MathSciNet  Google Scholar 

  5. Zharkov, A. Yu. and Blinkov, Yu. A., Involutive Approach to Investigating Polynomial Systems, Math. Comput. Simulation, 1996, vol. 42, pp. 323–332.

    Article  MathSciNet  Google Scholar 

  6. Zharkov, A. Yu. and Blinkov, Yu. A., Involutive Bases of Zero-Dimensional Ideals, Preprint of Joint Inst. for Nuclear Research, Dubna, 1994, no. E5-94-318.

  7. Gerdt, V. P., Involutive Division Technique: Some Generalizations and Optimizations, J. Math. Sci., 2002, vol. 108, no. 6, pp. 1034–1051.

    Article  MathSciNet  Google Scholar 

  8. Gerdt, V. P., Yanovich, D. A., and Blinkov, Yu. A., Fast Search for the Janet Divisor, Programmirovanie, 2001, vol. 27, no. 1, pp. 32–36 [Programming Comput. Software (Engl. Transl.), 2001, vol. 27, no. 1, pp. 22–24].

    MathSciNet  Google Scholar 

  9. Semenov, A. S., Pairwise Analysis of Involutive Divisions, Fundamental’naya Prikladnaya Mat., 2003, vol. 9,issue 3, pp. 199–212.

    MATH  Google Scholar 

  10. Semenov, A. S., On Connection between Constructive Involutive Divisions and Monomial Orderings, CASC-2006, Berlin: Springer, 2006.

    Google Scholar 

  11. Semenov, A. S. and Zyuzikov, P.A., Constructivity of Involutive Divisions: Facts and Examples, Proc. of Int. Workshop on Computer Algebra and Differential Equations (CADE-2007), Turku: Abo Akademi, 2007.

    Google Scholar 

  12. Gerdt, V. P., Blinkov, Yu. A., and Yanovich, D. A., Janet-like Monomial Division. Janet-like Gröbner Bases, Proc. of CASC-2005, Berlin: Springer, 2005, pp. 174–195.

    Google Scholar 

  13. Calmet, J., Hausdorf, M., and Seiler, W. M., A Constructive Introduction to Involution, Proc. Int. Symp. Applications of Computer Algebra—ISACA 2000, New Delhi, 2001, pp. 33–50.

  14. Semenov, A. S., On Constructivity of Involutive Divisions, Programmirovanie, 2006, no. 2, pp. 48–57 [Programming Comput. Software (Engl. Transl.), 2006, vol. 32, no. 2, pp. 96–102].

  15. Gerdt, V. P., Blinkov, Yu. A., and Yanovich, D. A., Construction of Janet Bases II: Polynomial Bases, CASC-2001, Berlin: Springer, 2001, pp. 249–263.

    Google Scholar 

  16. Hemmecke, R., Involutive Bases for Polynomial Ideals, Institut für Symbolisches Rechnen, Linz, 2003.

    Google Scholar 

  17. Mikhalev, A. V. and Pankrat’ev, E. V., Komp’yuternaya algebra. Vychisleniya v differentsial’noi i raznostnoi algebre (Computer Algebra: Computations in Differential and Difference Algebra), Moscow: Mosk. Gos. Univ., 1989.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanics and Mathematics, Moscow State University, Leninskie gory, Moscow, 119992, Russia

    A. S. Semenov & P. A. Zyuzikov

Authors
  1. A. S. Semenov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. P. A. Zyuzikov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. S. Semenov.

Additional information

Original Russian Text © A.S. Semenov, P.A. Zyuzikov, 2008, published in Programmirovanie, 2008, Vol. 34, No. 2.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Semenov, A.S., Zyuzikov, P.A. Involutive divisions and monomial orderings: Part II. Program Comput Soft 34, 107–111 (2008). https://doi.org/10.1134/S0361768808020084

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0361768808020084

Keywords

Search

Navigation