Positivity, sums of squares and the multi-dimensional moment problem II

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.