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.
 1.
Christian
Berg, The multidimensional moment problem and semigroups,
Moments in mathematics (San Antonio, Tex., 1987) Proc. Sympos. Appl.
Math., vol. 37, Amer. Math. Soc., Providence, RI, 1987,
pp. 110–124. MR 921086
(89k:44013), http://dx.doi.org/10.1090/psapm/037/921086
 2.
C.
Berg, J.
P. R. Christensen, and C.
U. Jensen, A remark on the multidimensional moment problem,
Math. Ann. 243 (1979), no. 2, 163–169. MR 543726
(81e:44008), http://dx.doi.org/10.1007/BF01420423
 3.
Christian
Berg, Jens
Peter Reus Christensen, and Paul
Ressel, Harmonic analysis on semigroups, Graduate Texts in
Mathematics, vol. 100, SpringerVerlag, New York, 1984. Theory of
positive definite and related functions. MR 747302
(86b:43001)
 4.
Christian
Berg and P.
H. Maserick, Polynomially positive definite sequences, Math.
Ann. 259 (1982), no. 4, 487–495. MR 660043
(84k:44012), http://dx.doi.org/10.1007/BF01466054
 5.
E. K. Haviland, On the momentum problem for distribution functions in more than one dimension, Amer. J. Math. 57 (1935), 562572.
 6.
E. K. Haviland, On the momentum problem for distribution functions in more than one dimension II, Amer. J. Math. 58 (1936), 164168.
 7.
Thomas
Jacobi and Alexander
Prestel, Distinguished representations of strictly positive
polynomials, J. Reine Angew. Math. 532 (2001),
223–235. MR 1817508
(2001m:14080), http://dx.doi.org/10.1515/crll.2001.023
 8.
Morris
Marden, The Geometry of the Zeros of a Polynomial in a Complex
Variable, Mathematical Surveys, No. 3, American Mathematical Society,
New York, N. Y., 1949. MR 0031114
(11,101i)
 9.
M.
Marshall, Extending the Archimedean Positivstellensatz to the
noncompact case, Canad. Math. Bull. 44 (2001),
no. 2, 223–230. MR 1827856
(2002b:14073), http://dx.doi.org/10.4153/CMB20010222
 10.
M. Marshall, Positive polynomials and sums of squares, Dottorato de Ricerca in Matematica, Dept. di Mat., Univ. Pisa, 2000.
 11.
Victoria
Powers and Claus
Scheiderer, The moment problem for noncompact semialgebraic
sets, Adv. Geom. 1 (2001), no. 1, 71–88.
MR
1823953 (2002c:14086), http://dx.doi.org/10.1515/advg.2001.005
 12.
V. Powers, C. Scheiderer, Correction to the paper ``The moment problem for noncompact semialgebraic sets'', to appear, Advances in Geometry.
 13.
Claus
Scheiderer, Sums of squares of regular functions
on real algebraic varieties, Trans. Amer. Math.
Soc. 352 (2000), no. 3, 1039–1069. MR 1675230
(2000j:14090), http://dx.doi.org/10.1090/S0002994799025222
 14.
C. Scheiderer, Sums of squares in coordinate rings of compact real varieties, work in progress.
 15.
Konrad
Schmüdgen, The 𝐾moment problem for compact
semialgebraic sets, Math. Ann. 289 (1991),
no. 2, 203–206. MR 1092173
(92b:44011), http://dx.doi.org/10.1007/BF01446568
 16.
N. Schwartz, Positive polynomials in the plane, work in progress, September, 2001.
 17.
T. Wörmann, Short algebraic proofs of theorems of Schmüdgen and Pólya, preprint.
 1.
 C. Berg, The multidimensional moment problem and semigroups, Proc. of Symposia in Applied Math. 37 (1987), 110124. MR 89k:44013
 2.
 C. Berg, J. Christensen, C. Jensen, A remark on the multidimensional moment problem, Math. Ann. 243 (1979), 163169. MR 81e:44008
 3.
 C. Berg, J. Christensen, P. Ressel, Harmonic analysis on semigroups: theory of positive definite and related functions, SpringerVerlag, 1984. MR 86b:43001
 4.
 C. Berg, P. H. Maserick, Polynomially positive definite sequences, Math. Ann. 259 (1982), 487495. MR 84k:44012
 5.
 E. K. Haviland, On the momentum problem for distribution functions in more than one dimension, Amer. J. Math. 57 (1935), 562572.
 6.
 E. K. Haviland, On the momentum problem for distribution functions in more than one dimension II, Amer. J. Math. 58 (1936), 164168.
 7.
 T. Jacobi, A. Prestel, Distinguished representations of strictly positive polynomials, J. reine angew. Math. 532 (2001), 223235. MR 2001m:14080
 8.
 M. Marden, The geometry of the zeros of a polynomial of a complex variable, Math. Surveys 3, Amer. Math. Soc., Providence, RI, 1949. MR 11:101i
 9.
 M. Marshall, Extending the archimedean Positivstellensatz to the noncompact case, Canad. Math. Bull. 44 (2001), 223230. MR 2002b:14073
 10.
 M. Marshall, Positive polynomials and sums of squares, Dottorato de Ricerca in Matematica, Dept. di Mat., Univ. Pisa, 2000.
 11.
 V. Powers, C. Scheiderer, The moment problem for noncompact semialgebraic sets, Advances in Geometry 1 (2001), 7188. MR 2002c:14086
 12.
 V. Powers, C. Scheiderer, Correction to the paper ``The moment problem for noncompact semialgebraic sets'', to appear, Advances in Geometry.
 13.
 C. Scheiderer, Sums of squares of regular functions on real algebraic varieties, Transactions Amer. Math. Soc. 352 (1999), 10301069. MR 2000j:14090
 14.
 C. Scheiderer, Sums of squares in coordinate rings of compact real varieties, work in progress.
 15.
 K. Schmüdgen, The moment problem for compact semialgebraic sets, Math. Ann. 289 (1991), 203206. MR 92b:44011
 16.
 N. Schwartz, Positive polynomials in the plane, work in progress, September, 2001.
 17.
 T. Wörmann, Short algebraic proofs of theorems of Schmüdgen and Pólya, preprint.
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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
