Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Lifting maps from the symmetrized polydisk in small dimensions   The spectral unit ball $\Omega_n$ is the set of all $n\times n$ matrices with
spectral radius less than $1$. Let $\pi(M) \in \mathbb C^n$ stand for the
coefficients of its characteristic polynomial of $M$ (up to signs), i.e. the
elementary symmetric functions of its eigenvalues. The symmetrized polydisk is
$\mathbb G_n:=\pi(\Omega_n)$. When investigating Nevanlinna-Pick problems for
maps from the disk to the spectral ball, it is often useful to project the map
to the symmetrized polydisk (for instance to obtain continuity results for the
Lempert function): if $\psi \in \mathcal O(\mathbb D, \Omega_n)$, then $\pi
\circ \psi \in \mathcal O(\mathbb D, \mathbb G_n)$. Given a map $\varphi \in
\mathcal O(\mathbb D, \mathbb G_n)$, we are looking for necessary and
sufficient conditions for this map to "lift through given matrices", i.e. find
$\psi$ as above so that $\pi \circ \psi = \varphi$ and $\psi (\alpha_j) = M_j$,
$1\le j \le N$. A natural necessary condition is $\varphi(\alpha_j)=\pi(M_j)$,
$1\le j \le N$. When the matrices $M_j$ are derogatory (i.e. do not admit a
cyclic vector) new necessary conditions appear, involving derivatives of
$\varphi$ at the points $\alpha_j$. Those conditions are necessary and
sufficient for a local lift. We give a scheme which shows that the necessary
conditions are also sufficient for a global lift in small dimensions (up to
$5$), and a counter-example to show that the scheme fails in dimension $6$ (and
above).

5
The categories of this paper are: gr-qc
22
The categories of this paper are: chao-dyn
18
The categories of this paper are: econ
17
The categories of this paper are: q-fin
19
The categories of this paper are: eess
25
The categories of this paper are: patt-sol
20
The categories of this paper are: acc-phys
9
The categories of this paper are: nlin
16
The categories of this paper are: stat
30
The categories of this paper are: q-alg
12
The categories of this paper are: nucl-th
24
The categories of this paper are: dg-ga
37
The categories of this paper are: supr-con
27
The categories of this paper are: comp-gas
21
The categories of this paper are: adap-org
13
The categories of this paper are: quant-ph