Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Orthogonal Polynomials and $S$-curves   This paper is devoted to a study of $S$-curves, that is systems of curves in
the complex plane whose equilibrium potential in a harmonic external field
satisfies a special symmetry property ($S$-property). Such curves have many
applications. In particular, they play a fundamental role in the theory of
complex (non-hermitian) orthogonal polynomials. One of the main theorems on
zero distribution of such polynomials asserts that the limit zero distribution
is presented by an equilibrium measure of an $S$-curve associated with the
problem if such a curve exists. These curves are also the starting point of the
matrix Riemann-Hilbert approach to srtong asymptotics. Other approaches to the
problem of strong asymptotics (differential equations, Riemann surfaces) are
also related to $S$-curves or may be interpreted this way. Existence problem
$S$-curve in a given class of curves in presence of a nontrivial external field
presents certain challenge. We formulate and prove a version of existence
theorem for the case when both the set of singularities of the external field
and the set of fixed points of a class of curves are small (in main case --
finite). We also discuss various applications and connections of the theorem.

12
The categories of this paper are: nucl-th
10
The categories of this paper are: q-bio
8
The categories of this paper are: hep-ex
18
The categories of this paper are: econ
6
The categories of this paper are: astro-ph
24
The categories of this paper are: dg-ga
7
The categories of this paper are: hep-th
23
The categories of this paper are: solv-int
13
The categories of this paper are: quant-ph
5
The categories of this paper are: gr-qc
17
The categories of this paper are: q-fin
21
The categories of this paper are: adap-org
37
The categories of this paper are: supr-con
35
The categories of this paper are: mtrl-th
9
The categories of this paper are: nlin
16
The categories of this paper are: stat