Skip to main content
SpringerLink
Log in

Pivoting in Extended Rings for Computing Approximate Gröbner Bases

  • Published:
Mathematics in Computer Science Aims and scope Submit manuscript

Abstract

It is well known that in the computation of Gröbner bases arbitrarily small perturbations in the coefficients of polynomials may lead to a completely different staircase, even if the solutions of the polynomial system change continuously. This phenomenon is called artificial discontinuity in Kondratyev’s Ph.D. thesis. We show how such phenomenon may be detected and even “repaired” by using a new variable to rename the leading term each time we detect a “problem”. We call such strategy the TSV (Term Substitutions with Variables) strategy. For a zero-dimensional polynomial ideal, any monomial basis (containing 1) of the quotient ring can be found with the TSV strategy. Hence we can use TSV strategy to relax term order while keeping the framework of Gröbner basis method so that we can use existing efficient algorithms (for instance the F 5 algorithm) to compute an approximate Gröbner basis. Our main algorithms, named TSVn and TSVh, can be used to repair artificial \({\epsilon}\)-discontinuities. Experiments show that these algorithms are effective for some nontrivial problems.

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. Abbott J., Fassino C., Torrente M.-L.: Stable border bases for ideals of points. J. Symb. Comput. 43(12), 883–894 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  2. Auzinger, W., Stetter, H.: An elimination algorithm for the computation of all zeros of a system of multivariate polynomial equations. In: Conference in Numerical Analysis, pp. 11–30. Birkhäuser-Verlag (1988)

  3. Becker T., Kredel H., Weispfenning V.: Gröbner Bases: A Computational Approach to Commutative Algebra. Springer, London (1993)

    Book  MATH  Google Scholar 

  4. Buchberger, B.: Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal. PhD thesis, Innsbruck (1965)

  5. Buchberger, B.: Gröbner-bases: an algorithmic method in polynomial ideal theory. In: Multidimensional Systems Theory—Progress Directions and Open Problems in Multidimensional Systems, pp. 184–232. Reidel Publishing Company, Dordrecht (1985)

  6. Buchberger, B.: An algorithm for finding the basis elements in the residue class ring modulo a zero dimensional polynomial ideal. J. Symb. Comput. 41(3–4) (2006)

  7. Chen, Y., Meng, X.: Border bases of positive dimensional polynomial ideals. In: SNC ’07: Proceedings of the 2007 International Workshop on Symbolic-Numeric Computation, pp. 65–71, New York, NY, USA. ACM (2007)

  8. Corless R.M.: Groebner Bases and Matrix Eigenproblems. SIGSAM Bull. (Commun. Comput. Algebra) 30(4), 26–32 (1996)

    Article  MathSciNet  Google Scholar 

  9. Cox D., Little J., O’Shea D.: Using Algebraic Geometry, 2nd edn. Springer, Reading (2005)

    MATH  Google Scholar 

  10. Faugère J.-C.: A new efficient algorithm for computing Gröbner basis (F4). J. Pure Appl. Algebra 139(1–3), 61–88 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  11. Faugère, J.-C.: A new efficient algorithm for computing Gröbner bases without reduction to zero (F5). In: Proceedings of the 2002 International Symposium on Symbolic and Algebraic Computation ISSAC, pp. 75–83, New York, NY, USA. ACM (2002)

  12. Faugère J.-C., Gianni P., Lazard D., Mora T.: Efficient computation of zero-dimensional Gröbner basis by change of ordering. J. Symb. Comput. 16(4), 329–344 (1993)

    Article  MATH  Google Scholar 

  13. Faugère, J.-C., Liang, Y.: Numerical computation of Gröbner bases for zero-dimensional polynomial ideals. In: Electronic Proceedings of MACIS 2007, Paris, December 2007. http://www-spiral.lip6.fr/MACIS2007/Papers/FL_MACIS2007final.pdf (2007)

  14. Faugère J.-C., Liang Y.: Artificial discontinuities of single-parametric Gröbner bases. J. Symb. Comput. 46(4), 459–466 (2011)

    Article  MATH  Google Scholar 

  15. Kehrein A., Kreuzer M.: Characterizations of border bases. J. Pure Appl. Algebra 196, 251–270 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  16. Kehrein A., Kreuzer M.: Computing border bases. J. Pure Appl. Algebra 205(2), 279–295 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  17. Kehrein A., Kreuzer M., Robbiano L.: An algebraist’s view on border bases. In: Dickenstein, A., Emiris, I. (eds) Solving Polynomial Equations: Foundations, Algorithms, and Applications, Algorithms and Computation in Mathematics, pp. 160–202. Springer, Heidelberg (2005)

    Google Scholar 

  18. Kondratyev, A.: Numerical Computation of Gröbner Bases. Technical report, University of Linz, Austria, March 2004. RISC Report Series

  19. Möller, H.: Systems of algebraic equations solved by means of endomorphisms. In: AAECC-10: Proceedings of the 10th International Symposium on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, pp. 43–56. Springer, Berlin (1993)

  20. Möller, H., Buchberger, B.: The construction of multivariate polynomials with preassigned zeros. In: EUROCAM, pp. 24–31 (1982)

  21. Mourrain, B.: A new criterion for normal form algorithms. In: AAECC-13: Proceedings of the 13th International Symposium on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, pp. 430–443. Springer, Berlin (1999)

  22. Mourrain, B., Trébuchet, P.: Generalized normal forms and polynomial system solving. In: International Symposium on Symbolic and Algebraic Computation, pp. 253–260. ACM, New York (2005)

  23. Reid, G., Tang, J., Yu, J., Zhi, L.: Hybrid method for solving new pose estimation equation system. In: Proceedings of the 2004 International Workshop on Computer and Geometric Algebra with Applications, pp. 46–57. Springer, Berlin (2005)

  24. Reid, G., Tang, J., Zhi, L.: A complete symoblic-numeric linear method for camera pose determination. In: Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation, pp. 215–223, Philadelphia, Pennsylvania, USA. ACM (2003)

  25. Rouillier F.: Solving zero-dimensional systems through the rational univariate representation. J. Appl. Algebra Eng. Commun. Comput. 9, 433–461 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  26. Rouillier F., Roy M.-F., Din M.S.E.: Finding at least one point in each connected component of a real algebraic set defined by a single equation. J. Complexity 16(4), 716–750 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  27. Sasaki, T., Kako, F.: Computing floating-point Gröbner bases stably. In: SNC ’07: Proceedings of the 2007 International Workshop on Symbolic-Numeric Computation, pp. 180–189, New York, NY, USA. ACM (2007)

  28. Sasaki, T., Kako, F.: Floating-point Gröbner basis computation with ill-conditionedness estimation. In: Proc. of ASCM2007 (LNAI 5081), pp. 278–292. Springer, Berlin (2008)

  29. Sasaki, T., Kako, F.: A practical method for floating-point Gröbner basis computation. In: Proc. of Joint Conf. of ASCM2009 and MACIS2009, pp. 167–176 (2009)

  30. Sasaki, T., Kako, F.: Term cancellations in computing floating-point Gröbner bases. In: Proc. of CASC2010 (LNAI 6244), pp. 220–231. Springer, Berlin (2010)

  31. Shirayanagi, K.: Floating point Gröbner bases. In: Selected papers presented at the International IMACS Symposium on Symbolic Computation, New Trends and Developments, pp. 509–528, Amsterdam, Netherlands. Elsevier (1996)

  32. Shirayanagi K., Sweelder M.: Remarks on automatic algorithm stabilization. J. Symb. Comput. 26(6), 761–765 (1998)

    Article  MATH  Google Scholar 

  33. Stetter, H.: Stabilization of polynomial systems solving with Groebner bases. In: International Symposium on Symbolic and Algebraic Computation, pp. 117–124. ACM (1997)

  34. Stetter H.: Numerical Polynomial Algebra. Society for Industrial and Applied Mathematics, Philadelphia (2004)

    Book  MATH  Google Scholar 

  35. Traverso, C., Zanoni, A.: Numerical stability and stabilization of Groebner basis computation. In: International Conference on Symbolic and Algebraic Computation, pp. 262–269, New York, NY, USA. ACM (2002)

  36. Trébuchet, P.: Generalized normal forms for positive dimensional ideals. In: International Conference on Polynomial System Solving (2004)

  37. Weispfenning, V.: Gröbner bases for inexact input data. In: Proc. of CASC2003, pp. 403–411, Passau, Germany (2003)

Download references

Author information

Authors and Affiliations

  1. INRIA, Paris-Rocquencourt Center, SALSA Project, UPMC, Univ Paris 06, CNRS, UMR 7606, UFR Ingénierie 919, LIP6, Case 169, 4, Place Jussieu, 75252, Paris, France

    Jean-Charles Faugère & Ye Liang

  2. LMIB, School of Mathematics and Systems Sciences, Beihang University, 37 Xueyuan Road, Haidian District, Beijing, 100191, China

    Ye Liang

  3. KLMM, Institute of Systems Science, Academy of Mathematics and System Science, Chinese Academy of Sciences, 55 Zhongguancun East Road, Haidian District, Beijing, 100190, China

    Ye Liang

Authors
  1. Jean-Charles Faugère
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ye Liang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ye Liang.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Faugère, JC., Liang, Y. Pivoting in Extended Rings for Computing Approximate Gröbner Bases. Math.Comput.Sci. 5, 179–194 (2011). https://doi.org/10.1007/s11786-011-0089-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11786-011-0089-y

Keywords

Mathematics Subject Classification (2000)

Search

Navigation