Mathematische Zeitschrift

, Volume 256, Issue 1, pp 99–112

SOS approximations of nonnegative polynomials via simple high degree perturbations

Article

Abstract

We show that every real polynomial f nonnegative on [−1,1]n can be approximated in the l1-norm of coefficients, by a sequence of polynomials \({\{f_{\epsilon r}\}}\) that are sums of squares (s.o.s). This complements the existence of s.o.s. approximations in the denseness result of Berg, Christensen and Ressel, as we provide a very simple and explicit approximation sequence. Then we show that if the moment problem holds for a basic closed semi-algebraic set \({K_{S} \subset \mathbb{R}^n}\) with nonempty interior, then every polynomial nonnegative on KS can be approximated in a similar fashion by elements from the corresponding preordering. Finally, we show that the degree of the perturbation in the approximating sequence depends on \({\epsilon}\) as well as the degree and the size of coefficients of the nonnegative polynomial f, but not on the specific values of its coefficients.

Keywords

Real algebraic geometry Positive polynomials Sum of squares Semidefinite programming Moment problem 

Mathematics Subject Classification (1991)

12E05 12Y05 90C22 44A60 

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Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.LAAS-CNRS and Institute of MathematicsLAASToulouse cedex 4France
  2. 2.Universität Konstanz, Fachbereich Mathematik und StatistikKonstanzGermany

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