论文标题
二次品种的奇特力矩问题
The Tracial Moment Problem on Quadratic Varieties
论文作者
论文摘要
截断的力矩问题要求表征实数的有限序列，这些序列是RN上阳性骨值的矩。通过整合对称矩阵的痕迹，可以获得其奇特的类似物，并且是本文的主要主题。 The solution of the bivariate quartic tracial moment problem with a nonsingular (7x7) moment matrix M(2) whose columns are indexed by words of degree 2 was established by Burgdorf and Klep, while in our previos work we completely solved all cases with M(2) of rank at most 5, split M(2) of rank 6 into four possible cases according to the column relation satisfied and solved two of them.本文中我们的第一个主要结果是满足第三列关系的m（2）解决方案，即y^2 = 1 + x^2。也就是说，代表度量的存在等同于某些线性基质不等式的可行性问题。第二个主要结果是对满足y^2 = 1的m（2）度量中原子的彻底分析，这是最苛刻的列关系。我们证明，在代表度量中不需要3个尺寸的原子，在所有其他情况下，事实证明是正确的。第三个主要结果将等级5的m（2）的解扩展到一般的m（n），n> 1，并具有两个二次柱关系。主要技术是将问题减少到经典的单变量截短力矩问题，这种方法也适用于经典的截短力矩问题。最后，我们的最后一个主要结果证明了这种方法，是简化了库尔托和Fialkow首先获得的退化截短双曲线矩问题的证明。
The truncated moment problem asks to characterize finite sequences of real numbers that are the moments of a positive Borel measure on Rn. Its tracial analog is obtained by integrating traces of symmetric matrices and is the main topic of this article. The solution of the bivariate quartic tracial moment problem with a nonsingular (7x7) moment matrix M(2) whose columns are indexed by words of degree 2 was established by Burgdorf and Klep, while in our previos work we completely solved all cases with M(2) of rank at most 5, split M(2) of rank 6 into four possible cases according to the column relation satisfied and solved two of them. Our first main result in this article is the solution for M(2) satisfying the third possible column relation, i.e., Y^2 = 1 + X^2. Namely, the existence of a representing measure is equivalent to the feasibility problem of certain linear matrix inequalities. The second main result is a thorough analysis of the atoms in the measure for M(2) satisfying Y^2 = 1, the most demanding column relation. We prove that size 3 atoms are not needed in the representing measure, a fact proved to be true in all other cases. The third main result extends the solution for M(2) of rank 5 to general M(n), n>1, with two quadratic column relations. The main technique is the reduction of the problem to the classical univariate truncated moment problem, an approach which applies also in the classical truncated moment problem. Finally, our last main result, which demonstrates this approach, is a simplification of the proof for the solution of the degenerate truncated hyperbolic moment problem first obtained by Curto and Fialkow.
