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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Authors: S. Kuhlmann and M. Marshall
Journal: Trans. Amer. Math. Soc. 354 (2002), 4285-4301
MSC (2000): Primary 14P10, 44A60
Published electronically: July 8, 2002
MathSciNet review: 1926876
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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:

\begin{displaymath}(\ddagger)\quad\quad\forall f\in \mathbb{R}[X], f\ge 0 \text{... ...hat }\forall \text{ real } \epsilon>0, f+\epsilon q\in T.\quad \end{displaymath}

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.


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Additional Information

S. Kuhlmann
Affiliation: Mathematical Sciences Group, Department of Computer Science, University of Saskatchewan, Saskatoon, SK Canada, S7N 5E6
Email: skuhlman@math.usask.ca

M. Marshall
Affiliation: Mathematical Sciences Group, Department of Computer Science, University of Saskatchewan, Saskatoon, SK Canada, S7N 5E6
Email: marshall@math.usask.ca

DOI: http://dx.doi.org/10.1090/S0002-9947-02-03075-1
PII: S 0002-9947(02)03075-1
Received by editor(s): October 3, 2000
Received by editor(s) in revised form: March 21, 2002
Published electronically: July 8, 2002
Additional Notes: This research was supported in part by NSERC of Canada
Article copyright: © Copyright 2002 American Mathematical Society