The distribution of zeros of the derivative of a random polynomial

TL;DR

This paper studies the distribution of critical points of large degree random polynomials with roots from a fixed distribution, proving convergence results under certain conditions and exploring special cases like the uniform measure on the circle.

Contribution

It introduces a probabilistic framework for analyzing the zeros of derivatives of large random polynomials and proves convergence of their distribution under specific conditions.

Findings

Zeros of f' converge to mu for measures with finite energy

Special case: uniform measure on the circle leads to convergence to a Gaussian power series

Provides new insights into the critical points of random polynomials

Abstract

In this note we initiate the probabilistic study of the critical points of polynomials of large degree with a given distribution of roots. Namely, let f be a polynomial of degree n whose zeros are chosen IID from a probability measure mu on the complex numbers. We conjecture that the zero set of f' always converges in distribution to mu as n goes to infinity. We prove this for measures with finite one-dimensional energy. When mu is uniform on the unit circle this condition fails. In this special case the zero set of f' converges in distribution to that the IID Gaussian random power series, a well known determinantal point process.