## Abstract

The paper is a continuation of work initiated by the
first two authors in [S. Kuhlmann, M. Marshall, Positivity, sums of squares and
the multi-dimensional moment problem. *Trans. Amer. Math. Soc. *354 (2002), 4285–4301]. Section 1 is introductory. In Section 2
we prove a basic lemma, Lemma 2.1, and use it to give new proofs of key
technical results of Scheiderer in [C. Scheiderer, Sums of squares of regular
functions on real algebraic varieties. Trans. *Amer. Math. Soc. *352 (2000), 1039–1069] [C. Scheiderer, Sums of squares on real
algebraic curves. *Math. Z. *245 (2003), 725–760] in the compact case;
see Corollaries 2.3, 2.4 and 2.5. Lemma 2.1 is also used in Section 3 where we
continue the examination of the case *n* = 1 initiated in [S.
Kuhlmann, M. Marshall, Positivity, sums of squares and the multi-dimensional
moment problem. *Trans. Amer. Math. Soc. *354 (2002), 4285–4301],
concentrating on the compact case. In Section 4 we prove certain uniform degree
bounds for representations in the case *n* = 1, which we then use in
Section 5 to prove that (‡) holds for basic closed semi-algebraic subsets of
cylinders with compact cross-section, provided the generators satisfy certain
conditions; see Theorem 5.3 and Corollary 5.5. Theorem 5.3 provides a partial
answer to a question raised by Schmüdgen in [K. Schmüdgen, On the moment problem
of closed semi-algebraic sets. *J. Reine Angew. Math.* 558 (2003),
225–234]. We also show that, for basic closed semi-algebraic subsets of
cylinders with compact cross-section, the sufficient conditions for (SMP) given
in [K. Schmüdgen, On the moment problem of closed semi-algebraic sets. *J.
Reine Angew. Math.* 558 (2003), 225–234] are also necessary; see Corollary
5.2(b). In Section 6 we prove a module variant of the result in [K. Schmüdgen,
On the moment problem of closed semi-algebraic sets. *J. Reine Angew.
Math.* 558 (2003), 225–234], in the
same spirit as Putinar’s variant [M. Putinar, Positive polynomials on compact
semi-algebraic sets. *Indiana Univ. Math. J. *42 (1993), 969–984] of the
result in [K. Schmüdgen, The K-moment problem for compact semi-algebraic sets.
*Math. Ann. *289 (1991), 203–206] in the compact case; see Theorem 6.1.
We apply this to basic closed semi-algebraic subsets of cylinders with compact
cross-section; see Corollary 6.4. In Section 7 we apply the results from Section
5 to solve two of the open problems listed in [S. Kuhlmann, M. Marshall,
Positivity, sums of squares and the multi-dimensional moment problem. *Trans.
Amer. Math. Soc. *
354 (2002), 4285–4301]; see
Corollary 7.1 and Corollary 7.4. In Section 8 we consider a number of examples
in the plane. In Section 9 we list some open problems.

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