Mathematics in Computer Science

, Volume 5, Issue 2, pp 179–194 | Cite as

Pivoting in Extended Rings for Computing Approximate Gröbner Bases



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 F5 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.


Approximate Gröbner basis Artificial discontinuity Monomial basis F5 algorithm 

Mathematics Subject Classification (2000)

Primary I.1.2 Secondary F.2.1 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abbott J., Fassino C., Torrente M.-L.: Stable border bases for ideals of points. J. Symb. Comput. 43(12), 883–894 (2008)MATHMathSciNetCrossRefGoogle Scholar
  2. 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)Google Scholar
  3. 3.
    Becker T., Kredel H., Weispfenning V.: Gröbner Bases: A Computational Approach to Commutative Algebra. Springer, London (1993)MATHCrossRefGoogle Scholar
  4. 4.
    Buchberger, B.: Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal. PhD thesis, Innsbruck (1965)Google Scholar
  5. 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)Google Scholar
  6. 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)Google Scholar
  7. 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)Google Scholar
  8. 8.
    Corless R.M.: Groebner Bases and Matrix Eigenproblems. SIGSAM Bull. (Commun. Comput. Algebra) 30(4), 26–32 (1996)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Cox D., Little J., O’Shea D.: Using Algebraic Geometry, 2nd edn. Springer, Reading (2005)MATHGoogle Scholar
  10. 10.
    Faugère J.-C.: A new efficient algorithm for computing Gröbner basis (F4). J. Pure Appl. Algebra 139(1–3), 61–88 (1999)MATHMathSciNetCrossRefGoogle Scholar
  11. 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)Google Scholar
  12. 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)MATHCrossRefGoogle Scholar
  13. 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. (2007)
  14. 14.
    Faugère J.-C., Liang Y.: Artificial discontinuities of single-parametric Gröbner bases. J. Symb. Comput. 46(4), 459–466 (2011)MATHCrossRefGoogle Scholar
  15. 15.
    Kehrein A., Kreuzer M.: Characterizations of border bases. J. Pure Appl. Algebra 196, 251–270 (2005)MATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Kehrein A., Kreuzer M.: Computing border bases. J. Pure Appl. Algebra 205(2), 279–295 (2006)MATHMathSciNetCrossRefGoogle Scholar
  17. 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. 18.
    Kondratyev, A.: Numerical Computation of Gröbner Bases. Technical report, University of Linz, Austria, March 2004. RISC Report SeriesGoogle Scholar
  19. 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)Google Scholar
  20. 20.
    Möller, H., Buchberger, B.: The construction of multivariate polynomials with preassigned zeros. In: EUROCAM, pp. 24–31 (1982)Google Scholar
  21. 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)Google Scholar
  22. 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)Google Scholar
  23. 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)Google Scholar
  24. 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)Google Scholar
  25. 25.
    Rouillier F.: Solving zero-dimensional systems through the rational univariate representation. J. Appl. Algebra Eng. Commun. Comput. 9, 433–461 (1999)MATHMathSciNetCrossRefGoogle Scholar
  26. 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)MATHMathSciNetCrossRefGoogle Scholar
  27. 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)Google Scholar
  28. 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)Google Scholar
  29. 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)Google Scholar
  30. 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)Google Scholar
  31. 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)Google Scholar
  32. 32.
    Shirayanagi K., Sweelder M.: Remarks on automatic algorithm stabilization. J. Symb. Comput. 26(6), 761–765 (1998)MATHCrossRefGoogle Scholar
  33. 33.
    Stetter, H.: Stabilization of polynomial systems solving with Groebner bases. In: International Symposium on Symbolic and Algebraic Computation, pp. 117–124. ACM (1997)Google Scholar
  34. 34.
    Stetter H.: Numerical Polynomial Algebra. Society for Industrial and Applied Mathematics, Philadelphia (2004)MATHCrossRefGoogle Scholar
  35. 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)Google Scholar
  36. 36.
    Trébuchet, P.: Generalized normal forms for positive dimensional ideals. In: International Conference on Polynomial System Solving (2004)Google Scholar
  37. 37.
    Weispfenning, V.: Gröbner bases for inexact input data. In: Proc. of CASC2003, pp. 403–411, Passau, Germany (2003)Google Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.INRIA, Paris-Rocquencourt Center, SALSA Project, UPMC, Univ Paris 06, CNRS, UMR 7606ParisFrance
  2. 2.LMIB, School of Mathematics and Systems SciencesBeihang UniversityBeijingChina
  3. 3.KLMM, Institute of Systems Science, Academy of Mathematics and System ScienceChinese Academy of SciencesBeijingChina

Personalised recommendations