Skip to main content
SpringerLink
Log in

Semidefinite Characterization and Computation of Zero-Dimensional Real Radical Ideals

  • Published:
Foundations of Computational Mathematics Aims and scope Submit manuscript

Abstract

For an ideal I⊆ℝ[x] given by a set of generators, a new semidefinite characterization of its real radical I(V ℝ(I)) is presented, provided it is zero-dimensional (even if I is not). Moreover, we propose an algorithm using numerical linear algebra and semidefinite optimization techniques, to compute all (finitely many) points of the real variety V ℝ(I) as well as a set of generators of the real radical ideal. The latter is obtained in the form of a border or Gröbner basis. The algorithm is based on moment relaxations and, in contrast to other existing methods, it exploits the real algebraic nature of the problem right from the beginning and avoids the computation of complex components.

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. E. Becker and R. Neuhaus, Computation of real radicals of polynomial ideals, in Computational Algebraic Geometry (F. Eyssette and A. Galligo, eds.), Progress in Mathematics, Vol. 109, pp. 1–20, Birkhäuser, Boston, 1993.

    Google Scholar 

  2. E. Becker and J. Schmid, On the real Nullstellensatz, in Algorithmic Algebra and Number Theory (B. H. Matzat, G.-M. Greuel, G. Hiss, eds.), pp. 173–185, Springer, New York, 1997.

    Google Scholar 

  3. E. Becker and T. Wörmann, Radical computations of zero-dimensional ideals and real root counting, Math. Comput. Simul., 42 (1996), 561–569.

    Article  MATH  Google Scholar 

  4. F. Bihan, J. M. Rojas, and C. E. Stella, First steps in algorithmic fewnomial theory, 2004. Available from http://www.arxiv.org/abs/math/0411107.

  5. F. Bihan and F. Sottile, New fewnomial upper bounds from Gale dual polynomial systems, Moscow Math. J., 7(3) (2007).

  6. D. Bini and B. Mourrain, Polynomial test suite, 1996. Available from http://www-sop.inria.fr/saga/POL.

  7. J. Bochnak, M. Coste, and M.-F. Roy, Géométrie Algébrique Réelle, Springer, New York, 1987.

    MATH  Google Scholar 

  8. M. Caboara, P. Conti, and C. Traverso, Yet another ideal decomposition algorithm, in Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, Lecture Notes in Computer Science, Vol. 1255, pp. 39–54, Springer, Berlin, 1997.

    Google Scholar 

  9. P. Conti and C. Traverso, Algorithms for the real radical, Preprint, 1998.

  10. D. Cox, J. Little, and D. O’Shea, Ideals, Varieties and Algorithms, Springer, New York, 1997.

    Google Scholar 

  11. D. Cox, J. Little, and D. O’Shea, Using Algebraic Geometry, Springer, New York, 1998.

    MATH  Google Scholar 

  12. R. Curto and L. Fialkow, Solution of the truncated complex moment problem for flat data, Mem. Amer. Math. Soc. 119(568) (1996).

  13. R. Curto and L. Fialkow, The truncated complex K-moment problem, Trans. Amer. Math. Soc. 352 (2000), 2825–2855.

    Article  MATH  MathSciNet  Google Scholar 

  14. R. Curto and L. Fialkow, Solution of the singular quartic moment problem, J. Oper. Theory 48 (2002), 315–354.

    MATH  MathSciNet  Google Scholar 

  15. E. de Klerk, Aspects of Semidefinite Programming—Interior Point Algorithms and Selected Applications, Kluwer Academic, Amsterdam, 2002.

    MATH  Google Scholar 

  16. A. Dickenstein and I. Z. Emiris (eds.), Solving Polynomial Equations: Foundations, Algorithms, and Applications, Algorithms and Computation in Mathematics, Vol. 14, Springer, Berlin, 2005.

    MATH  Google Scholar 

  17. D. Eisenbud, C. Huneke, and W. Vasconcelos, Direct methods for primary decompositions, Invent. Math. 110 (1992), 207–235.

    Article  MATH  MathSciNet  Google Scholar 

  18. G.-M. Greuel, G. Pfister, and H. Schönemann, Singular 3.0. A computer algebra system for polynomial computations, Centre for Computer Algebra, University of Kaiserslautern, 2005. Available from http://www.singular.uni-kl.de.

  19. P. Gianni, B. Trager, and G. Zacharias, Gröbner bases and primary decomposition of polynomial ideals, J. Symb. Comput. 6 (1988), 149–167.

    Article  MATH  MathSciNet  Google Scholar 

  20. D. Goldfarb and K. Scheinberg, Interior point trajectories in semidefinite programming, SIAM J. Optim. 8 (1998), 871–886.

    Article  MATH  MathSciNet  Google Scholar 

  21. D. Henrion and J. B. Lasserre, Detecting global optimality and extracting solutions in gloptiPoly, in Positive Polynomials in Control (D. Henrion and A. Garulli, eds.), Lectures Notes in Control and Information Sciences, Vol. 312, pp. 293–310, Springer, New York, 2005.

    Google Scholar 

  22. A. G. Khovanski, Fewnomials, Am. Math. Soc., Providence, 1991.

    Google Scholar 

  23. T. Krick and A. Logar, An algorithm for the computation of the radical of an ideal in the ring of polynomials, in Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (New Orleans, LA, 1991), Lecture Notes in Computer Science, Vol. 539, pp. 195–205, Springer, Berlin, 1991.

    Google Scholar 

  24. Y. N. Lakshman and D. Lazard, On the complexity of zero-dimensional algebraic systems, in Effective Methods in Algebraic Geometry (T. Mora and C. Traverso, eds.), Progress in Mathematics, Vol. 94, pp. 217–226, Birkhäuser, Boston, 1991.

    Google Scholar 

  25. J. B. Lasserre, Global optimization with polynomials and the problem of moments, SIAM J. Optim. 11 (2001), 796–817.

    Article  MATH  MathSciNet  Google Scholar 

  26. J. B. Lasserre, A moment approach to analyze zeros of triangular polynomial sets, Trans. Amer. Math. Soc. 358 (2006), 1403–1420.

    Article  MATH  MathSciNet  Google Scholar 

  27. M. Laurent, Revisiting two theorems of Curto and Fialkow, Proc. Amer. Math. Soc. 133(10) (2005), 2965–2976.

    Article  MATH  MathSciNet  Google Scholar 

  28. M. Laurent, Semidefinite representations for finite varieties, Math. Program. 109 (2007), 1–26.

    Article  MATH  MathSciNet  Google Scholar 

  29. M. Laurent, Moment matrices and optimization over polynomials—A survey on selected topics, Preprint, 2005. Available from http://homepages.cwi.nl/~monique/.

  30. J. Löfberg, YALMIP: A toolbox for modeling and optimization in MATLAB, in Proceedings of CACSD, Taipei, Taiwan, 2004. Available from http://control.ee.ethz.ch/~joloef/yalmip.php.

  31. B. Mourrain, A new criterion for normal form algorithms, in Proc. Conf. AAECC-13, Honolulu, 1999 (M. Fossorier et al., eds.), Lecture Notes in Computer Science, Vol. 1719, pp. 431–443, Springer, Berlin, 1999.

    Google Scholar 

  32. P. Pedersen, M.-F. Roy, and A. Szpirglas, Counting real zeros in the multivariate case, in Computational Algebraic Geometry (F. Eyssette, A. Galligo, eds.), Progress in Mathematics, Vol. 109, pp. 203–224, Birkhäuser, Boston, 1993.

    Google Scholar 

  33. G. Reid and L. Zhi, Solving nonlinear polynomial system via symbolic-numeric elimination method, in Proceedings of the International Conference on Polynomial System Solving, 2004.

  34. N. Revol and F. Rouillier, Motivations for an arbitrary precision interval arithmetic and the MPFI library, Reliable Comput. 11 (2005), 1–16.

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  36. A. Seidenberg, Constructions in algebra, Trans. Amer. Math. Soc. 197 (1974), 273–313.

    Article  MATH  MathSciNet  Google Scholar 

  37. A. J. Sommese and C. W. Wampler, The Numerical Solution of Systems of Polynomials Arising in Engineering and Science, World Scientific, Singapore, 2005.

    MATH  Google Scholar 

  38. G. Stengle, A Nullstellensatz and a Positivstellensatz in semialgebraic geometry, Math. Ann. 207 (1974), 87–97.

    Article  MathSciNet  Google Scholar 

  39. H.J. Stetter, Numerical Polynomial Algebra, SIAM, Philadelphia, 2004.

    MATH  Google Scholar 

  40. J. F. Sturm, Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones, Optim. Methods Softw. 11/12 (1999), 625–653. Special issue on Interior Point Methods (CD supplement with software).

    Article  MathSciNet  Google Scholar 

  41. J. F. Sturm, Implementation of interior point methods for mixed semidefinite and second order cone optimization problems, Optim. Methods Softw. 17(6) (2002), 1105–1154.

    Article  MATH  MathSciNet  Google Scholar 

  42. L. Vandenberghe and S. Boyd, Semidefinite programming, SIAM Rev. 38(1) (1996), 49–95.

    Article  MATH  MathSciNet  Google Scholar 

  43. J. Verschelde, Algorithm 795: PHCpack: A general-purpose solver for polynomial systems by homotopy continuation, ACM Trans. Math. Softw. 25(2) (1999), 251–276.

    Article  MATH  Google Scholar 

  44. J. Verschelde and K. Gatermann, Symmetric Newton polytopes for solving sparse polynomial systems, Adv. Appl. Math. 16(1) (1995), 95–127.

    Article  MATH  MathSciNet  Google Scholar 

  45. H. Wolkowicz, R. Saigal, and L. Vandenberghe (eds.), Handbook of Semidefinite Programming, Kluwer Academic, Boston, 2000.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. LAAS-CNRS, Toulouse, France

    Jean Bernard Lasserre

  2. Institute of Mathematics, Toulouse, France

    Jean Bernard Lasserre

  3. Centrum voor Wiskunde en Informatica, Kruislaan 413, 1098 SJ, Amsterdam, The Netherlands

    Monique Laurent

  4. Automatic Control Laboratory, ETH Zurich, Physikstrasse 3, 8092, Zurich, Switzerland

    Philipp Rostalski

Authors
  1. Jean Bernard Lasserre
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Monique Laurent
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Philipp Rostalski
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jean Bernard Lasserre.

Additional information

Communicated by Marie-Francoise Roy.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lasserre, J.B., Laurent, M. & Rostalski, P. Semidefinite Characterization and Computation of Zero-Dimensional Real Radical Ideals. Found Comput Math 8, 607–647 (2008). https://doi.org/10.1007/s10208-007-9004-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10208-007-9004-y

Keywords

Mathematics Subject Classification (2000)

Search

Navigation