SOS approximations of nonnegative polynomials via simple high degree perturbation

Details SOS approximations of nonnegative polynomials via simple high degree perturbation SOS approximations of nonnegative polynomials via simple high degree perturbation

2005-10-21

arxiv.org/abs/math/0510456v1

Mathematics

Algebraic Geometry

12 pages

Scientific paper

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 $\{f_{\ep r}\}$ that are sums of squares. 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 \textit{explicit} approximation sequence. Then we show that if the Moment Problem holds for a basic closed semi-algebraic set $K_S\subset\R^n$ 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.

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