论文标题
关于$ 6 \ times 6 $共同锥的结构
On the structure of the $6 \times 6$ copositive cone
论文作者
论文摘要
在这项工作中，我们补充了$ 6 \ times 6 $共同锥与某些拓扑结构的极端光线的描述。在上一篇论文中，我们将该锥体的极端元素分解为不同尺寸的代数品种的不相交的结合。在本文中，我们将此分类与最近引入的组合特征联系起来，称为扩展最小零支持集。我们确定那些必不可少的组件，即未嵌入其他组件边界中的组件。这允许在研究$ 6 \ times 6 $共同锥的不同特性时，大大减少人们必须考虑的案例数。作为一个应用程序，我们构建了一个共同呈阳性$ 6 \ times 6 $矩阵的示例，该示例与单位对角线，该矩阵不属于parrilo squares lesse hallows lesse $ {\ cal k}^{（1）} _ 6 $。
In this work we complement the description of the extreme rays of the $6 \times 6$ copositive cone with some topological structure. In a previous paper we decomposed the set of extreme elements of this cone into a disjoint union of pieces of algebraic varieties of different dimension. In this paper we link this classification to the recently introduced combinatorial characteristic called extended minimal zero support set. We determine those components which are essential, i.e., which are not embedded in the boundary of other components. This allows to drastically decrease the number of cases one has to consider when investigating different properties of the $6 \times 6$ copositive cone. As an application, we construct an example of a copositive $6 \times 6$ matrix with unit diagonal which does not belong to the Parrilo inner sum of squares relaxation ${\cal K}^{(1)}_6$.
