Foundations of Computational Mathematics

, Volume 8, Issue 5, pp 607–647 | Cite as

Semidefinite Characterization and Computation of Zero-Dimensional Real Radical Ideals

  • Jean Bernard LasserreEmail author
  • Monique Laurent
  • Philipp Rostalski


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.


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

Mathematics Subject Classification (2000)

14P05 13P10 12E12 12D10 90C22 


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Copyright information

© SFoCM 2007

Authors and Affiliations

  • Jean Bernard Lasserre
    • 1
    • 2
    Email author
  • 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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