Critical points of polynomials with random zeros

Abstract

The study of zero distribution of random polynomials has a long history and is currently a very active research area. Traditionally, the randomness in these polynomials comes from the probability distribution followed by their coe cients. One can introduce randomness in the zeros (instead of the coe cients) of polynomials, and then investigate the locations of their critical points (relative to these zeros). Such a study was initiated by Rivin and the late Schramm in 2001, but only until 2011, Pemantle and Rivin proposed a precise probabilistic framework of it which will rst be explained in this talk. Following this framework, we will consider the problem of nding the zero distributions of the derivatives of random polynomials with i.i.d. zeros following a common distribution supported on a subset of the complex plane. Recently, Sean O'Rourke applied the same framework to study critical points of characteristic polynomial of a random matrix drawn from one of the compact classical matrix groups. This is a joint work with Pak-Leong Cheung, Jonathan Tsai and Phillip Yam and the work was supported by the RGC grant HKU 704611P and HKU 703313P.