We find new characterizations for the points in the symmetrized polydisc 𝔾n, a family of domains associated with the spectral interpolation, defined by 𝔾n := ∑1≤i≤nzi,∑1≤i<j≤nzizj,…,∏i=1nz i : |zi| < 1,i = 1,…,n. We introduce a new family of domains which we call the extended symmetrized polydisc 𝔾̃n, and define in the following way: 𝔾̃n:=(y1,…,yn−1,q) ∈ ℂn : q ∈ 𝔻,y j = βj + β̄n−jq,βj ∈ ℂand |βj|+|βn−j| < n j,forj = 1,…,n−1. We show that 𝔾n = 𝔾̃n for n = 1, 2 and that 𝔾n ⊊ 𝔾̃n for n ≥ 3. We first obtain a variety of characterizations for the points in 𝔾̃n and we apply these necessary and sufficient conditions to produce an analogous set of characterizations for the points in 𝔾n. Also, we obtain similar characterizations for the points in Γn ∖ 𝔾n, where Γn = 𝔾n¯. A set of n − 1 fractional linear transformations plays central role in the entire program. We also show that for n ≥ 2, 𝔾̃n is nonconvex but polynomially convex and is starlike about the origin but not circled.

中文翻译：

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通过另一个域族表征对称多圆盘

我们为图中的点找到了新的特征对称多圆盘 𝔾n，与光谱插值相关的一系列域，由下式定义 𝔾n ：= ∑1≤一世≤nz一世,∑1≤一世<j≤nz一世zj,…,∏一世=1nz 一世 ： |z一世| < 1,一世 = 1,…,n. 我们引入了一个新的域家族，我们称之为扩展的对称多圆盘 𝔾̃n, 并按以下方式定义： 𝔾̃n：=(是的1,…,是的n-1,q) ∈ ℂn ： q ∈ 𝔻,是的 j = βj + β̄n-jq,βj ∈ ℂ和 |βj|+|βn-j| < n j,为了j = 1,…,n-1. 我们表明𝔾n = 𝔾̃n为了n = 1, 2然后𝔾n ⊊ 𝔾̃n为了n ≥ 3. 我们首先获得了对中点的各种表征𝔾̃n我们应用这些必要和充分条件来产生一组类似的特征𝔾n. 此外，我们对中的点获得了类似的表征Γn ∖ 𝔾n， 在哪里Γn = 𝔾n¯. 一套n - 1分数线性变换在整个程序中起着核心作用。我们还表明，对于n ≥ 2,𝔾̃n是非凸的，但多项式凸的，并且关于原点是星状的，但不是圆形的。