Characterizations of Symmetrized Polydisc

TL;DR

This paper characterizes elements of the symmetrized polydisc in complex n-space, providing conditions that relate these elements to lower-dimensional symmetrized polydiscs and the unit disk.

Contribution

It offers a new characterization of points in the symmetrized polydisc using recursive relations involving lower-dimensional symmetrized polydiscs and the unit disk.

Findings

Characterization of elements in mma_n via recursive relations.

Conditions involving mma_{n-1} and the unit disk.

Explicit formulas for elements in mma_n.

Abstract

Let $\Gamma_n$, $n \geq 2$, denote the symmetrized polydisc in $\mathbb{C}^n$, and $\Gamma_1$ be the closed unit disc in $\mathbb{C}$. We provide some characterizations of elements in $\Gamma_n$. In particular, an element $(s_1, \ldots, s_{n-1}, p) \in \mathbb{C}^n$ is in $\Gamma_n$ if and only if $s_j = \beta_j + \overline{\beta_{n-j}} p$, $j = 1, \ldots, n-1$, for some $(\beta_1, \ldots, \beta_{n-1}) \in \Gamma_{n-1}$, and $|p| \leq 1$.