Positivity, sums of squares and the multi-dimensional moment problem

Abstract:

Let $K$ be the basic closed semi-algebraic set in $\mathbb {R}^n$ defined by some finite set of polynomials $S$ and $T$, the preordering generated by $S$. For $K$ compact, $f$ a polynomial in $n$ variables nonnegative on $K$ and real $\epsilon >0$, we have that $f+\epsilon \in T$ [15]. 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 $n\ge 2$, 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: \[ (\ddagger )\quad \quad \forall f\in \mathbb {R}[X], f\ge 0 \text { on } K\Rightarrow \exists q\in T \text { such that }\forall \text { real } \epsilon >0, f+\epsilon q\in T.\quad \] This property is strictly weaker than the one given in [15], but in turn it implies a positive solution to the $K$-Moment Problem. Using results of [9], we provide many (noncompact) examples in hypersurfaces for which ($\ddagger$) holds. Finally, we provide a list of 8 open problems.