Positivity, sums of squares and the multidimensional moment problem
Authors:
S. Kuhlmann and M. Marshall
Journal:
Trans. Amer. Math. Soc. 354 (2002), 42854301
MSC (2000):
Primary 14P10, 44A60
Published electronically:
July 8, 2002
MathSciNet review:
1926876
Fulltext PDF Free Access
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Abstract: Let be the basic closed semialgebraic 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:
http://dx.doi.org/10.1090/S0002994702030751
PII:
S 00029947(02)030751
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
