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Positivity sums of squares and the multidimensional moment problem

By S. Kuhlmann and M. Marshall


Abstract. Let K be the basic closed semi-algebraic set in Rn defined by some finite set of polynomials S and T, the preordering generated by S. For K compact, f apolynomialinnvariables nonnegative on K and real ɛ>0, we have that f + ɛ ∈ T. In particular, the K-Moment Problem has a positive solution. In the present paper, we study the problem when K is not compact. For n = 1, we show that the K-Moment Problem has a positive solution if and only if S is the natural description of K (see Section 1). For n ≥ 2, we show that the K-Moment Problem fails if K contains a cone of dimension 2. On the other hand, we show that if K is a cylinder with compact base, then the following property holds: (‡) ∀f ∈ R[X],f ≥ 0onK ⇒∃q ∈ T such that ∀ real ɛ>0,f + ɛq ∈ T. This property is strictly weaker than the one given in Schmüdgen (1991), but in turn it implies a positive solution to the K-Moment Problem. Using results of Marshall (2001), we provide many (noncompact) examples in hypersurfaces for which (‡) holds. Finally, we provide a list of 8 open problems

Year: 2005
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