论文标题
自由四边形的阿尔维森边界由非交通性品种给出
The Arveson boundary of a Free Quadrilateral is given by a noncommutative variety
论文作者
论文摘要
令$ sm_n（\ mathbb {r}）^g $表示$ g $ - $ n \ times n $ n $真实对称矩阵，并设置$ sm（\ mathbb {r}）^g = \ cup_n sm_n（\ mathbb {r}）^g $。自由四边形是SM（\ Mathbb {r}）^2 $中的元组的集合，它们在定义经典四边形的线性方程式上具有正面的半芬特评估。这种集合在称为矩阵凸组合的丰富凸组合中关闭。 That is, given elements $X=(X_1, \dots, X_g) \in SM_{n_1}(\mathbb{R})^g$ and $Y=(Y_1, \dots, Y_g) \in SM_{n_2}(\mathbb{R})^g$ of a free quadrilateral $\mathcal{Q}$, one \ [v_1^t x v_1+v_1+v_2^t y v_2 \ in \ mathcal {q} \]对于任何收缩，$ v_1：\ mathbb {r}^n \ to \ mathbb {r} \ Mathbb {r}^{n_2} $满足$ v_1^t v_1+v_2^t v_2 = i_n $。这些矩阵凸组合是在无维度中的凸组合的天然类似物。 自由四边形的自然级别是自由的极端点：游离四边形的元素，该元素不能表示为自由四边形的元素的非平凡矩阵凸组合。这些自由的极端点是通过基质凸组合恢复自由四边形的最小集合。 在本文中，我们表明，自由四边形的一组自由极点集取决于非交换多项式集合的零集。更准确地说，给定一个免费的四边形$ \ MATHCAL {q} $，我们构造了非交易性多项式$ p_1，p_1，p_2，p_3，p_3，p_4 $，使得tuple $ x \ in Sm（\ m athbb {r}）^2 $是$ \ mathcal if $ h. \ Mathcal {q} $和$ p_i（x）= 0 $ for $ i = 1,2,3,4 $和$ x $是不可记论的。此外，我们为自由谱图和均匀的自由谱图的投影图建立了几个基本结果。
Let $SM_n(\mathbb{R})^g$ denote $g$-tuples of $n \times n$ real symmetric matrices and set $SM(\mathbb{R})^g = \cup_n SM_n(\mathbb{R})^g$. A free quadrilateral is the collection of tuples $X \in SM(\mathbb{R})^2$ which have positive semidefinite evaluation on the linear equations defining a classical quadrilateral. Such a set is closed under a rich class of convex combinations called matrix convex combination. That is, given elements $X=(X_1, \dots, X_g) \in SM_{n_1}(\mathbb{R})^g$ and $Y=(Y_1, \dots, Y_g) \in SM_{n_2}(\mathbb{R})^g$ of a free quadrilateral $\mathcal{Q}$, one has \[ V_1^T X V_1+V_2^T Y V_2 \in \mathcal{Q} \] for any contractions $V_1:\mathbb{R}^n \to \mathbb{R}^{n_1}$ and $V_2:\mathbb{R}^n \to \mathbb{R}^{n_2}$ satisfying $V_1^T V_1+V_2^T V_2=I_n$. These matrix convex combinations are a natural analogue of convex combinations in the dimension free setting. A natural class of extreme point for free quadrilaterals is free extreme points: elements of a free quadrilateral which cannot be expressed as a nontrivial matrix convex combination of elements of the free quadrilateral. These free extreme points serve as the minimal set which recovers a free quadrilateral through matrix convex combinations. In this article we show that the set of free extreme points of a free quadrilateral is determined by the zero set of a collection of noncommutative polynomials. More precisely, given a free quadrilateral $\mathcal{Q}$, we construct noncommutative polynomials $p_1,p_2,p_3,p_4$ such that a tuple $X \in SM (\mathbb{R})^2$ is a free extreme point of a $\mathcal{Q}$ if and only if $X \in \mathcal{Q}$ and $p_i(X) =0 $ for $i=1,2,3,4$ and $X$ is irreducible. In addition we establish several basic results for projective maps of free spectrahedra and for homogeneous free spectrahedra.
