Limiting empirical distribution of zeros and critical points of random polynomials agree in general

TL;DR

This paper investigates the asymptotic distribution of zeros and critical points of certain random polynomials, showing they coincide under specific conditions, extending previous results to more general settings.

Contribution

It establishes that the limiting empirical measures of zeros and critical points agree for a broad class of random polynomials with deterministic roots, including triangular arrays.

Findings

Zeros and critical points share the same limiting distribution.

Results extend to generalized derivatives and rational functions.

Applicable to sequences with matching empirical measures.

Abstract

In this article, we study critical points (zeros of derivative) of random polynomials. Take two deterministic sequences $\{a_n\}_{n\geq1}$ and $\{b_n\}_{n\geq1}$ of complex numbers whose limiting empirical measures are same. By choosing $\xi_n = a_n$ or $b_n$ with equal probability, define the sequence of polynomials by $P_n(z)=(z-\xi_1)\dots(z-\xi_n)$. We show that the limiting measure of zeros and critical points agree for this sequence of random polynomials under some assumption. We also prove a similar result for triangular array of numbers. A similar result for zeros of generalized derivative (can be thought as random rational function) is also proved. Pemantle and Rivin initiated the study of critical points of random polynomials. Kabluchko proved the result considering the zeros to be i.i.d. random variables.