# Pairing between zeros and critical points of random polynomials with
independent roots

Research paper by **Sean O'Rourke, Noah Williams**

Indexed on: **19 Oct '16**Published on: **19 Oct '16**Published in: **arXiv - Mathematics - Probability**

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#### Abstract

Let $p_n$ be a random, degree $n$ polynomial whose roots are chosen
independently according to the probability measure $\mu$ on the complex plane.
For a deterministic point $\xi$ lying outside the support of $\mu$, we show
that almost surely the polynomial $q_n(z):=p_n(z)(z - \xi)$ has a critical
point at distance $O(1/n)$ from $\xi$. In other words, conditioning the random
polynomials $p_n$ to have a root at $\xi$, almost surely forces a critical
point near $\xi$. More generally, we prove an analogous result for the critical
points of $q_n(z):=p_n(z)(z - \xi_1)\cdots (z - \xi_k)$, where $\xi_1, \ldots,
\xi_k$ are deterministic. In addition, when $k=o(n)$, we show that the
empirical distribution constructed from the critical points of $q_n$ converges
to $\mu$ in probability as the degree tends to infinity, extending a recent
result of Kabluchko.