Positivity, sums of squares and the multi-dimensional moment problem
Positivity, sums of squares and the multi-dimensional moment problem
复制标题
正性、平方和和多维矩问题
DOI：
--
复制
发表时间：
2002
期刊：
影响因子：
--
通讯作者：
M. Marshall
中科院分区：
文献类型：
--
作者：
S. Kuhlmann;M. Marshall
设K是Rn中由S和T组成的有限多项式集所定义的基本闭半代数集，是S生成的预序集，f是K上的n元非负多项式且实e>；0，我们有f+∈E，特别地，K矩问题有正解.本文研究了当K不紧时的问题。对于n=1，我们证明了K-矩问题有正解的充要条件是S是K的自然描述(见第1节)。对于n>；2，我们证明了当K包含2维锥时K矩问题是失败的。另一方面，我们证明了如果K是具有紧基的圆柱体，则下列性质成立：(+)∀f∈R[X]，f≥0 on K⇒∃Q∈T使得∀∈>；0，f+eq∈T。这个性质严格弱于Schudgen(1991年)给出的性质，但反过来它又意味着K矩问题的正解。利用马歇尔(2001)的结果，我们在()成立的超曲面上提供了许多(非紧的)例子。最后，我们列出了8个有待解决的问题。
Let K be the basic closed semi-algebraic set in R n defined by some finite set of polynomials S and T, the preordering generated by S. For K compact, f a polynomial in n variables nonnegative on K and real e > 0, we have that f + ∈ E T. In particular, the K-Moment Problem has a positive solution. In the present paper, we study the problem when K is not compact. For n = 1, we show that the K-Moment Problem has a positive solution if and only if S is the natural description of K (see Section 1). For n > 2, we show that the K-Moment Problem fails if K contains a cone of dimension 2. On the other hand, we show that if K is a cylinder with compact base, then the following property holds: (+) ∀ f ∈ R[X], f ≥ 0 on K ⇒ ∃q ∈ T such that ∀ real ∈ > 0, f + eq ∈ T. This property is strictly weaker than the one given in Schmudgen (1991), but in turn it implies a positive solution to the K-Moment Problem. Using results of Marshall (2001), we provide many (noncompact) examples in hypersurfaces for which () holds. Finally, we provide a list of 8 open problems.