## Positivity, sums of squares and the multi-dimensional moment problem. II.(English)Zbl 1095.14055

This article, as the authors indicate in its abstract, is a continuation of work initiated by the first two ones in [Trans. Am. Math. Soc. 354, No. 11, 4285–4301 (2002; Zbl 1012.14019)]. More precisely, the main aim of this paper is to improve the results about certain conditions concerning the moment problem. Let us describe more precisely such conditions. Let $${\mathbb R}[X]$$ denote the polynomial ring in $$n$$ variables $$X=(x_1,\dots ,x_n)$$ over the field $$\mathbb R$$ of real numbers. For a finite subset $$S=\{g_1,\dots,g_s\}$$ of $${\mathbb R}[X]$$, write $$K_S$$ for the closed semi-algebraic subset $$\{\alpha\in \mathbb R^n| \;g_i(\alpha)\geq0,\,i=1,\,\dots,\,s\}$$, and $$T_S$$ for the preordering of $${\mathbb R}[X]$$ generated by $$S$$, i.e., the set of all the polynomials of the form $$\sum_{i=1}^n h_i^2g_1^{e_{i1}}\cdots g_s^{e_{is}}$$, where $$n$$ is a positive integer, $$h_i\in \mathbb R[X]$$ and $$e_{ij}=0$$ or $$1$$. Also they consider the two subsets of $$\mathbb R[X]$$ defined as follows: $T_S^{\text{alg}}:=\{f\in {\mathbb R}[X]| \;f\geq 0\text{ on }K_S\}\qquad\text{and}$
$T_S^{\text{lin}}:=\{f\in\mathbb R[X]\mid L(f)\geq 0\quad \forall\;L:{\mathbb R}[X]\to{\mathbb R}\text{ linear functional: }L(T_S)\geq0\}.$ The authors explore the following conditions for several types of semialgebraic sets:
(SMP) $$T_S^{\text{alg}}=T_S^{\text{lin}}$$.
($$\dagger$$) For every $$f\in\mathbb R[X]$$ with $$f\geq0$$ on $$K_S$$, $$f+\epsilon\in T_S$$ for all positive numbers $$\epsilon$$.
(MP) $$T_\emptyset^{\text{alg}}=T_S^{\text{lin}}$$.
$$(\ddagger)$$ For every $$f\in \mathbb R[X]$$ with $$f\geq 0$$ on $$K_S$$, there exists a $$q\in\mathbb R[X]$$ such that $$f+\epsilon q\in T_S$$ for all positive numbers $$\epsilon$$.
More precisely, they mainly focus their attention in the study of the (SMP), (MP), $$(\dagger)$$ and $$(\ddagger)$$ conditions for basic closed semi–algebraic subsets of cylinders with compact cross-sections. The authors also solve some open problems listed in [loc. cit.] and propose a list of six new open ones.

### MSC:

 14P10 Semialgebraic sets and related spaces 44A60 Moment problems 12D15 Fields related with sums of squares (formally real fields, Pythagorean fields, etc.)

### Keywords:

positivity; sums of squares; moment problem; semialgebraic set

Zbl 1012.14019
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