SOS approximations of nonnegative polynomials via simple high degree perturbations Article

First Online: 08 November 2006 Received: 10 October 2005 Accepted: 09 May 2006 DOI :
10.1007/s00209-006-0061-8

Cite this article as: Lasserre, J.B. & Netzer, T. Math. Z. (2007) 256: 99. doi:10.1007/s00209-006-0061-8
Abstract We show that every real polynomial f nonnegative on [−1,1]^{n} can be approximated in the l _{1} -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 K _{S} 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

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