Foundations of Computational Mathematics

, Volume 8, Issue 5, pp 607–647

Semidefinite Characterization and Computation of Zero-Dimensional Real Radical Ideals

  • Jean Bernard Lasserre
  • Monique Laurent
  • Philipp Rostalski
Article

Abstract

For an ideal I⊆ℝ[x] given by a set of generators, a new semidefinite characterization of its real radical I(V(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(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.

Keywords

Algebraic geometry Zero-dimensional ideal (Real) radical ideal Semidefinite programming 

Mathematics Subject Classification (2000)

14P05 13P10 12E12 12D10 90C22 

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References

  1. 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. 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. 3.
    E. Becker and T. Wörmann, Radical computations of zero-dimensional ideals and real root counting, Math. Comput. Simul., 42 (1996), 561–569. MATHCrossRefGoogle Scholar
  4. 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. 5.
    F. Bihan and F. Sottile, New fewnomial upper bounds from Gale dual polynomial systems, Moscow Math. J., 7(3) (2007). Google Scholar
  6. 6.
    D. Bini and B. Mourrain, Polynomial test suite, 1996. Available from http://www-sop.inria.fr/saga/POL.
  7. 7.
    J. Bochnak, M. Coste, and M.-F. Roy, Géométrie Algébrique Réelle, Springer, New York, 1987. MATHGoogle Scholar
  8. 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. 9.
    P. Conti and C. Traverso, Algorithms for the real radical, Preprint, 1998. Google Scholar
  10. 10.
    D. Cox, J. Little, and D. O’Shea, Ideals, Varieties and Algorithms, Springer, New York, 1997. Google Scholar
  11. 11.
    D. Cox, J. Little, and D. O’Shea, Using Algebraic Geometry, Springer, New York, 1998. MATHGoogle Scholar
  12. 12.
    R. Curto and L. Fialkow, Solution of the truncated complex moment problem for flat data, Mem. Amer. Math. Soc. 119(568) (1996). Google Scholar
  13. 13.
    R. Curto and L. Fialkow, The truncated complex K-moment problem, Trans. Amer. Math. Soc. 352 (2000), 2825–2855. MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    R. Curto and L. Fialkow, Solution of the singular quartic moment problem, J. Oper. Theory 48 (2002), 315–354. MATHMathSciNetGoogle Scholar
  15. 15.
    E. de Klerk, Aspects of Semidefinite Programming—Interior Point Algorithms and Selected Applications, Kluwer Academic, Amsterdam, 2002. MATHGoogle Scholar
  16. 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. MATHGoogle Scholar
  17. 17.
    D. Eisenbud, C. Huneke, and W. Vasconcelos, Direct methods for primary decompositions, Invent. Math. 110 (1992), 207–235. MATHCrossRefMathSciNetGoogle Scholar
  18. 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. 19.
    P. Gianni, B. Trager, and G. Zacharias, Gröbner bases and primary decomposition of polynomial ideals, J. Symb. Comput. 6 (1988), 149–167. MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    D. Goldfarb and K. Scheinberg, Interior point trajectories in semidefinite programming, SIAM J. Optim. 8 (1998), 871–886. MATHCrossRefMathSciNetGoogle Scholar
  21. 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. 22.
    A. G. Khovanski, Fewnomials, Am. Math. Soc., Providence, 1991. Google Scholar
  23. 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. 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. 25.
    J. B. Lasserre, Global optimization with polynomials and the problem of moments, SIAM J. Optim. 11 (2001), 796–817. MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    J. B. Lasserre, A moment approach to analyze zeros of triangular polynomial sets, Trans. Amer. Math. Soc. 358 (2006), 1403–1420. MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    M. Laurent, Revisiting two theorems of Curto and Fialkow, Proc. Amer. Math. Soc. 133(10) (2005), 2965–2976. MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    M. Laurent, Semidefinite representations for finite varieties, Math. Program. 109 (2007), 1–26. MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    M. Laurent, Moment matrices and optimization over polynomials—A survey on selected topics, Preprint, 2005. Available from http://homepages.cwi.nl/~monique/.
  30. 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. 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. 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. 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. Google Scholar
  34. 34.
    N. Revol and F. Rouillier, Motivations for an arbitrary precision interval arithmetic and the MPFI library, Reliable Comput. 11 (2005), 1–16. CrossRefMathSciNetGoogle Scholar
  35. 35.
    F. Rouillier, Solving zero-dimensional systems through the rational univariate representation, J. Appl. Algebra Eng. Commun. Comput. 9 (1999), 433–461. MATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    A. Seidenberg, Constructions in algebra, Trans. Amer. Math. Soc. 197 (1974), 273–313. MATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    A. J. Sommese and C. W. Wampler, The Numerical Solution of Systems of Polynomials Arising in Engineering and Science, World Scientific, Singapore, 2005. MATHGoogle Scholar
  38. 38.
    G. Stengle, A Nullstellensatz and a Positivstellensatz in semialgebraic geometry, Math. Ann. 207 (1974), 87–97. CrossRefMathSciNetGoogle Scholar
  39. 39.
    H.J. Stetter, Numerical Polynomial Algebra, SIAM, Philadelphia, 2004. MATHGoogle Scholar
  40. 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). CrossRefMathSciNetGoogle Scholar
  41. 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. MATHCrossRefMathSciNetGoogle Scholar
  42. 42.
    L. Vandenberghe and S. Boyd, Semidefinite programming, SIAM Rev. 38(1) (1996), 49–95. MATHCrossRefMathSciNetGoogle Scholar
  43. 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. MATHCrossRefGoogle Scholar
  44. 44.
    J. Verschelde and K. Gatermann, Symmetric Newton polytopes for solving sparse polynomial systems, Adv. Appl. Math. 16(1) (1995), 95–127. MATHCrossRefMathSciNetGoogle Scholar
  45. 45.
    H. Wolkowicz, R. Saigal, and L. Vandenberghe (eds.), Handbook of Semidefinite Programming, Kluwer Academic, Boston, 2000. Google Scholar

Copyright information

© SFoCM 2007

Authors and Affiliations

  • Jean Bernard Lasserre
    • 1
    • 2
  • Monique Laurent
    • 3
  • Philipp Rostalski
    • 4
  1. 1.LAAS-CNRSToulouseFrance
  2. 2.Institute of MathematicsToulouseFrance
  3. 3.Centrum voor Wiskunde en InformaticaAmsterdamThe Netherlands
  4. 4.Automatic Control LaboratoryETH ZurichZurichSwitzerland

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