Positivity, sums of squares and the multi-dimensional moment problem II ⁄
Positivity, sums of squares and the multi-dimensional moment problem II ⁄
复制标题
正性、平方和和多维矩问题 II ⁄
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发表时间：
2005
期刊：
影响因子：
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通讯作者：
Niels Schwartz
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文献类型：
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作者：
S. Kuhlmann;M. Marshall;Niels Schwartz
这篇论文是[K-M]中前两位作者开始的工作的继续。第一节是引言。在第二节中，我们证明了一个基本引理，引理2.1，并利用它在紧情形下给出了Scheider在[S1][S2]中的关键技术结果的新证明；见推论2.3，2.4和2.5。引理2.1也被用在第三节中，我们继续研究在[K-M]中开始的情形n=1，集中在紧情形上。在第四节中，我们证明了在n=1的情况下表示的某些一致度界，然后我们在第五节中用它来证明(‡)对具有紧截面的柱面的基本闭半代数子集成立，只要生成元满足一定的条件；见定理5.3和推论5.5。定理5.3部分回答了Schudgen在[SC2]中提出的一个问题。我们还证明了，对于具有紧截面的柱面的基本闭半代数子集，[SC2]中给出的(SMP)的充分条件也是必要条件，见推论5.2(B)。在第6节中，我们证明了[SC2]中结果的模变量，其精神与紧致情形下[SC1]中结果的Putina变量[Pu]相同；见定理6.1。我们将其应用于具有紧截面的圆柱体的基本封闭半代数子集；见推论6.4。在第7节中，我们应用第5节的结果来解决[K-M]中列出的两个公开问题；见推论7.1和推论7.4。在第8节中，我们考虑了平面中的一些例子。在第9节中，我们列出了一些未解决的问题。
The paper is a continuation of work initiated by the first two authors in [K– M]. 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 [S1] [S2] 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 [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 Schmudgen in [Sc2]. We also show that, for basic closed semi-algebraic subsets of cylinders with compact cross-section, the sufficient conditions for (SMP) given in [Sc2] are also necessary; 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 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 7.4. In Section 8 we consider a number of examples in the plane. In Section 9 we list some open problems.