Real Algebraic and Analytic Geometry |

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

e-mail: , ,

Submission: 2003, August 9.

*Abstract:
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 (z) 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 suffcient; 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.*

Mathematics Subject Classification (2000): 14P10, 44A60.

**Full text**, 27p.:
dvi 116k,
ps.gz 200k,
pdf 256k.

Server Home Page