Skip to main content
Log in

Computation of Involutive and Gröbner Bases Using the Tableau Representation of Polynomials

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

Abstract

For the work with polynomials such data representations as lists of terms, geobuckets, and heaps are usually used. In this paper, an attempt to give a new look on the representation of polynomials for computing involutive and Gröbner bases of systems of nonlinear polynomial equations is made. The proposed approach makes it possible to execute a part of this computational task on the GPU, which opens prospects for solving more complicated problems.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

REFERENCES

  1. B. Buchberger, B., Gröbner-bases: An algorithmic method in polynomial ideal theory, Multidimensional Systems Theory— Progress, Directions and Open Problems in Multidimensional Systems, Dodrecht: Reidel, 1985, pp. 184–232.

    Google Scholar 

  2. Cox, D., Little, J., and O’Shea, D., Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, New York: Springer, 1997.

    Book  Google Scholar 

  3. Gerdt, V.P. and Blinkov, Yu.A., Involutive bases of polynomial ideals, Math. Comp. Simul., 1998, vol. 45, pp. 519–542.

    Article  MathSciNet  Google Scholar 

  4. Gerdt, V.P. and Blinkov, Yu.A., Involutive divisions of monomials, Program. Comput. Software, 1998, vol. 24, no. 6, pp. 283–285.

    MathSciNet  MATH  Google Scholar 

  5. Yanovich, D.A., Parallel modular computation of Gröbner and involutive bases, Program. Comput. Software, 2013, vol. 39, no. 2, pp. 110–115.

    Article  MathSciNet  Google Scholar 

  6. Johnson, S.C., Sparse polynomial arithmetic, ACM SIGSAM Bull., 1974, vol. 8, no. 3, pp. 63–71.

    Article  Google Scholar 

  7. Yan, T., The geobucket data structure for polynomials, J. Symb. Comput., 1998, vol. 25, pp. 285–293.

    Article  MathSciNet  Google Scholar 

  8. Cid, C., Murphy, S., and Robshaw, M.J.B., Small scale variants of the AES, 12th Int. Workshop, FSE 2005, Paris, 2005, pp. 145–162.

  9. Faugère, J.-C. and Joux, A., Algebraic cryptanalysis of hidden field equation (HFE) cryptosystems using Gröobner bases, Proc. of the 23rd Annual Int. Cryptology Conf., Santa Barbara, Calif., 2003, pp. 44–60.

  10. Yanovich, D.A., Compact representation of polynomials for algorithms for computing Gröbner and involutive bases, Program. Comput. Software, 2015, vol. 41, no. 2, pp. 126–131.

    Article  MathSciNet  Google Scholar 

  11. Zimmermann, P., Casamayou, A., Cohen, N., Connan, G., et al. Computational mathematics with SageMath, SIAM, 2018, pp. 203–205.

    Book  Google Scholar 

  12. Verschelde, J., The Database with Test Examples. http://www.math.uic.edu/ jan/demo.html

  13. Monagan, M., Sparse polynomial division using a heap, J. Symb. Comput., 2011, vol. 46, pp. 807–822.

    Article  MathSciNet  Google Scholar 

  14. http://www.valgrind.org

  15. https://github.com/sparsehash/sparsehash

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to D. A. Yanovich.

Additional information

Translated by A. Klimontovich

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yanovich, D.A. Computation of Involutive and Gröbner Bases Using the Tableau Representation of Polynomials. Program Comput Soft 46, 162–166 (2020). https://doi.org/10.1134/S0361768820020115

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0361768820020115

Navigation