The distribution of zeros of the derivative of a random polynomial

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.