×

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.)

Citations:

Zbl 1012.14019
PDF BibTeX XML Cite
Full Text: DOI Link