manuscripta mathematica

, Volume 129, Issue 2, pp 251–271

Stability of quadratic modules



A finitely generated quadratic module or preordering in the real polynomial ring is called stable, if it admits a certain degree bound on the sums of squares in the representation of polynomials. Stability, first defined explicitly in Powers and Scheiderer (Adv Geom 1, 71–88, 2001), is a very useful property. It often implies that the quadratic module is closed; furthermore, it helps settling the Moment Problem, solves the Membership Problem for quadratic modules and allows applications of methods from optimization to represent nonnegative polynomials. We provide sufficient conditions for finitely generated quadratic modules in real polynomial rings of several variables to be stable. These conditions can be checked easily. For a certain class of semi-algebraic sets, we obtain that the nonexistence of bounded polynomials implies stability of every corresponding quadratic module. As stability often implies the non-solvability of the Moment Problem, this complements the result from Schmüdgen (J Reine Angew Math 558, 225–234, 2003), which uses bounded polynomials to check the solvability of the Moment Problem by dimensional induction. We also use stability to generalize a result on the Invariant Moment Problem from Cimpric et al. (Trans Am Math Soc, to appear).

Mathematics Subject Classification (2000)

12E05 12Y05 44A60 


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  1. 1.
    Adams E.W.: Elements of a theory of inexact measurements. Phil. Sci. 32, 205–228 (1965)CrossRefGoogle Scholar
  2. 2.
    Augustin, D.: The Membership Problem for Quadratic Modules with Focus on the One Dimensional Case. Doctoral Thesis, University of Regensburg (2008)Google Scholar
  3. 3.
    Cimpric J., Kuhlmann S., Scheiderer C.: Sums of squares and invariant moment problems in equivariant situations. Trans. Am. Math. Soc. 361, 735–765 (2009)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Kuhlmann S., Marshall M.: Positivity, sums of squares and the multi-dimensional moment problem. Trans. Am. Math. Soc. 354, 4285–4301 (2002)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Kuhlmann S., Marshall M., Schwartz N.: Positivity, sums of squares and the multi-dimensional moment problem II. Adv. Geom. 5, 583–606 (2005)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Lasserre J.B.: Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11, 796–817 (2001)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Marshall, M.: Positive polynomials and sums of squares. AMS Math. Surveys and Monographs 146, Providence (2008)Google Scholar
  8. 8.
    Netzer T.: An elementary proof of Schmüdgens theorem on the moment problem of closed semialgebrac sets. Proc. Am. Math. Soc. 136, 529–537 (2008)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Plaumann, D.: Stabilität von Quadratsummen auf Rellen Algebraischen Varietäten. Diplomarbeit, Universität Duisburg (2004)Google Scholar
  10. 10.
    Plaumann, D.: Bounded Polynomials, Sums of Squares and the Moment Problem. Doctoral Thesis, University of Konstanz (2008)Google Scholar
  11. 11.
    Powers V., Scheiderer C.: The moment problem for non-compact semialgebraic sets. Adv. Geom. 1, 71–88 (2001)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Prestel A., Delzell C.N.: Positive Polynomials. Springer, Berlin (2001)MATHGoogle Scholar
  13. 13.
    Scheiderer C.: Non-existence of degree bounds for weighted sums of squares representations. J. Complex. 21, 823–844 (2005)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Scheiderer C.: Sums of squares on real algebraic curves. Math. Z. 245, 725–760 (2003)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Schmüdgen K.: Unbounded Operator Algebras and Representation Theory. Birkhäuser, Basel (1990)Google Scholar
  16. 16.
    Schmüdgen K.: The K-moment problem for compact semi-algebraic sets. Math. Ann. 289, 203–206 (1991)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Schmüdgen K.: On the moment problem of closed semialgebraic sets. J. Reine Angew. Math. 558, 225–234 (2003)MATHMathSciNetGoogle Scholar
  18. 18.
    Schweighofer M.: On the complexity of Schmüdgen’s positivstellensatz. J. Complex. 20(4), 529–543 (2004)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Schweighofer M.: Optimization of polynomials on compact semialgebraic sets. SIAM J. Optim. 15, 805–825 (2005)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Vandenberghe L., Boyd S.: Semidefinite programming. SIAM Rev. 38, 49–95 (1996)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Fachbereich Mathematik und StatistikUniversität KonstanzKonstanzGermany

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