Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Distances between zeroes and critical points for random polynomials with
  i.i.d. zeroes   Consider a random polynomial $Q_n$ of degree $n+1$ whose zeroes are i.i.d.
random variables $\xi_0,\xi_1,\ldots,\xi_n$ in the complex plane. We study the
pairing between the zeroes of $Q_n$ and its critical points, i.e. the zeroes of
its derivative $Q_n'$. In the asymptotic regime when $n\to\infty$, with high
probability there is a critical point of $Q_n$ which is very close to $\xi_0$.
We localize the position of this critical point by proving that the difference
between $\xi_0$ and the critical point has approximately complex Gaussian
distribution with mean $1/(nf(\xi_0))$ and variance of order $\log n \cdot
n^{-3}$. Here, $f(z)= \mathbb E[1/(z-\xi_k)]$ is the Cauchy-Stieltjes transform
of the $\xi_k$'s. We also state some conjectures on critical points of
polynomials with dependent zeroes, for example the Weyl polynomials and
characteristic polynomials of random matrices.

28
The categories of this paper are: alg-geom
34
The categories of this paper are: chem-ph
35
The categories of this paper are: mtrl-th
37
The categories of this paper are: supr-con
25
The categories of this paper are: patt-sol
30
The categories of this paper are: q-alg
16
The categories of this paper are: stat
33
The categories of this paper are: plasm-ph
21
The categories of this paper are: adap-org
18
The categories of this paper are: econ
7
The categories of this paper are: hep-th
14
The categories of this paper are: math-ph
26
The categories of this paper are: bayes-an
20
The categories of this paper are: acc-phys
10
The categories of this paper are: q-bio
3
The categories of this paper are: physics