Skip to main content
SpringerLink
Log in

On solving large systems of polynomial equations appearing in discrete differential geometry

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

Abstract

The paper describes methods for solving very large overdetermined algebraic polynomial systems on an example that appears from a classification of all integrable 3-dimensional scalar discrete quasilinear equations Q 3=0 on an elementary cubic cell of the lattice ℤ3. The overdetermined polynomial algebraic system that has to be solved is far too large to be formulated. A “probing” technique, which replaces independent variables by random integers or zero, allows to formulate subsets of this system.

An automatic alteration of equation formulating steps and equation solving steps leads to an iteration process that solves the computational problem.

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. Sokolov, V.V. and Wolf, T., Integrable Quadratic Hamiltonians on so(4) and so(3, 1), J. Phys. A: Math. Gen., 2006, vol. 39, pp. 1915–1926. Also preprint at arXiv:nlin.SI/0405066.

    Article  MATH  MathSciNet  Google Scholar 

  2. Tsuchida, T. and Wolf, T., Classification of Polynomial Integrable Systems of Mixed Scalar and Vector Evolution Equations. I, J. Phys. A: Math. Gen., 2005, vol. 38, pp. 7691–7733. Also preprint at arXiv:nlin.SI/0412003.

    Article  MATH  MathSciNet  Google Scholar 

  3. Anco, S. and Wolf, T., Some Symmetry Classifications of Hyperbolic Vector Evolution Equations, JNMP, 2005, vol. 12(supplement 1), pp. 13–31. Also preprint at arXi:nlin.SI/0412015.

    Article  MathSciNet  Google Scholar 

  4. Kiselev, A. and Wolf, T., On Weakly Non-Local, Nilpotent, and Super-Recursion Operators for N = 1 Homogeneous Superequations, Proc. of Dubna Int. Workshop “Supersymmetries and Quantum Symmetries” (SQS’05), JINR, 2005, pp. 231–237. Online: http://theor.jinr.ru/:_sqs05/SQS05.pdf. Also preprint at arXiv:math-ph/051107.

  5. Kiselev, A. and Wolf, T., Supersymmetric Representations and Integrable Super-Extensions of the Burgers and Bussinesq Equations, SIGMA, 2006, no. 2, pp. 19, Paper 030. Also preprint at arXiv:math-ph/0511071.

  6. Kiselev, A. and Wolf, T., Classification of Integrable Super-systems Using the SsTools Environment, Comp. Phys. Commun., 2007, vol. 177, no. 3, pp. 315–328. Also preprint at arXiv:nlin.SI/0609065.

    Article  MATH  MathSciNet  Google Scholar 

  7. Tsarev, S.P. and Wolf, T., Classification of 3-dimensional Integrable Scalar Discrete Equations, 2007. Preprint at arXiv:0706.2464.

  8. Vermaseren, J.A.M., New Features of Form, 2000. Also preprint at arXiv:math-ph/0010025 (a complete distribution can be downloaded from http://www.nikhef.nl/:_form/.

  9. Wolf, T., Applications of Crack in the Classification of Integrable Systems, CRM Proc. and Lecture Notes, Montreal, Centre de Recherches Mathematiques, 2004, vol. 37, pp. 283–300. Also preprint at arXiv: nlin.SI/0301032.

    Google Scholar 

  10. Wolf, T., An Online Tutorial for the Package CRACK, 2004, http://lie.math.brocku.ca/crack/demo.

  11. Adler, V.E., Bobenko, A.I., and Suris, Yu.B., Classification of Integrable Equations on Quad-graphs. The Consistency Approach, Comm. Math. Phys., 2003, vol. 233, pp. 513–543.

    MATH  MathSciNet  Google Scholar 

  12. Adler, V.E., Bobenko, A.I., and Suris, Yu.B., Discrete Nonlinear Hyperbolic Equations. Classification of Integrable Cases, 2007. Preprint at arXiv:nlin.SI/0705.1663.

  13. Konopelchenko, B.G. and Schief, W.K., Reciprocal Figures, Graphical Statics and Inversive Geometry of the Schwarzian BKP Hierarchy, Stud. Appl. Math., 2002, vol. 109, pp. 89–124. Also preprint at arXiv:nlin.SI/0107001.

    Article  MATH  MathSciNet  Google Scholar 

  14. Nimmo, J.J.C. and Schief, W.K., An Integrable Discretization of a 2+1-Dimensional Sine-Gordon Equation, Stud. Appl. Math., 1998, vol. 100, pp. 295–309.

    Article  MATH  MathSciNet  Google Scholar 

  15. Wolf, T., Neun, W., and Tsarev, S.P., About Parallelizing the Search for 3-dim. Scalar Discrete Integrable Equations, Talk given at PASCO, London, Ontario, 2007. http://lie.math.brocku.ca/twolf/papers/paratalk.pdf.

  16. Wolf, T., Merging Solutions of Polynomial Algebraic Systems, 2003, Preprint http://lie.math.brocku.ca/twolf/papers/merge-sig.ps.

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Brock University, St. Catharines, Ontario, Canada, L2S 3A1

    T. Wolf

Authors
  1. T. Wolf
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to T. Wolf.

Additional information

The text was submitted by the author in English.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wolf, T. On solving large systems of polynomial equations appearing in discrete differential geometry. Program Comput Soft 34, 75–83 (2008). https://doi.org/10.1134/S0361768808020047

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0361768808020047

Keywords

Search

Navigation