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

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 ${\xi }_1,\dots ,{\xi }_k$ 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.

39 pages, 5 figures; incorporated comments and suggestions from Boris Hanin

References in corpus (1)

• Spectra of nearly Hermitian random matrices

Cited by in corpus (3)

• Zeros of random polynomials and its higher derivatives
• Conservation Laws for the Density of Roots of Polynomials under Differentiation
• Distances between zeroes and critical points for random polynomials with i.i.d. zeroes