Tuen-Wai Ng, HKU, Hong Kong

Critical points of random polynomials and random finite Blaschke products

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 coefficients. One can introduce randomness in the zeros (instead of the coefficients) 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 first be explained in this talk. Following this framework, we will consider the problem of finding the zero distributions of the derivatives of random polynomials and random finite Blaschke products with i.i.d. zeros following a common distribution supported on a subset of the complex plane. This is a joint work with Pak-Leong Cheung, Jonathan Tsai and Phillip Yam.