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

DOI:
https://doi.org/10.1090/S0002-9947-02-03075-1

Published electronically:
July 8, 2002

MathSciNet review:
1926876

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be the basic closed semi-algebraic set in defined by some finite set of polynomials and , the preordering generated by . For compact, a polynomial in variables nonnegative on and real , we have that [15]. In particular, the -Moment Problem has a positive solution. In the present paper, we study the problem when is not compact. For , we show that the -Moment Problem has a positive solution if and only if is the natural description of (see Section 1). For , we show that the -Moment Problem fails if contains a cone of dimension 2. On the other hand, we show that if is a cylinder with compact base, then the following property holds:

This property is strictly weaker than the one given in [15], but in turn it implies a positive solution to the -Moment Problem. Using results of [9], we provide many (noncompact) examples in hypersurfaces for which () 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:
https://doi.org/10.1090/S0002-9947-02-03075-1

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