Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Characterizations of the symmetrized polydisc via another family of
  domains   We find new characterizations for the points in the \textit{symmetrized
polydisc} $\mathbb G_n$, a family of domains associated with the spectral
interpolation, defined by \[ \mathbb G_n :=\left\{ \left(\sum_{1\leq i\leq n}
z_i,\sum_{1\leq i<j\leq n}z_iz_j \dots, \prod_{i=1}^n z_i \right): \,|z_i|<1,
i=1,\dots,n \right \}. \] We introduce a new family of domains which we call
\textit{the extended symmetrized polydisc} $\widetilde{\mathbb G}_n$, and
define in the following way: \begin{align*} \widetilde{\mathbb G}_n := \Bigg\{
(y_1,\dots,y_{n-1}, q)\in \mathbb C^n :\; q \in \mathbb D, \; y_j = \beta_j +
\bar{\beta}_{n-j} q, \; \beta_j \in \mathbb C &\text{ and }\\ |\beta_j|+
|\beta_{n-j}| < {n \choose j} &\text{ for } j=1,\dots, n-1 \Bigg\}.
\end{align*} We show that $\mathbb G_n=\widetilde{\mathbb G}_n$ for $n=1,2$ and
that ${\mathbb G}_n \subsetneq \widetilde{\mathbb G}_n$ for $n\geq 3$. We first
obtain a variety of characterizations for the points in $\widetilde{\mathbb
G}_n$ and we apply these necessary and sufficient conditions to produce an
analogous set of characterizations for the points in ${\mathbb G}_n$. Also we
obtain similar characterizations for the points in $\Gamma_n \setminus {\mathbb
G}_n$, where $\Gamma_n =\overline{{\mathbb G}_n}$. A set of $n-1$ fractional
linear transformations play central role in the entire program. We also show
that for $n\geq 2$, $\widetilde{\mathbb G}_n$ is non-convex but polynomially
convex and is starlike about the origin but not circled.

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The categories of this paper are: bayes-an
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The categories of this paper are: eess
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The categories of this paper are: astro-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: math-ph
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The categories of this paper are: q-bio
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The categories of this paper are: gr-qc
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The categories of this paper are: ao-sci
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The categories of this paper are: hep-ex
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The categories of this paper are: alg-geom
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The categories of this paper are: dg-ga
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The categories of this paper are: comp-gas
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The categories of this paper are: patt-sol
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The categories of this paper are: acc-phys
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The categories of this paper are: hep-ph
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The categories of this paper are: hep-lat