The paper is a continuation of work initiated by the first two authors in [K–M]. Section 1 is introductory. In Section 2 we give new proofs of results of Scheiderer in [S1] [S2] in the compact case; see Corollaries 2.3, 2.4 and 2.5. The main tool in Section 2, Lemma 2.1, is also used in Section 3 where we continue the examination of the case n = 1 initiated in [K–M], 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 [Sc2]. We also show that for basic closed semi-algebraic subsets of cylinders with compact cross-section the necessary conditions for (SMP) given in [Sc2] are also sufficient; see Corollary 5.2(b). In Section 6 we prove a module variant of the result in [Sc2] in the same spirit as Putinar’s variant [Pu] of the result in [Sc1] in the compact case; see Theorem 6.1. We then 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 [K–M]; see Corollary 7.1 and Corollary 7.5. In Section 8 we consider a number of examples in the plane. In Section 9 we list some open problems.
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