Stefan Steinerberger
Mathematics Department, Yale University

Title: Polynomials with many roots -- the mean field limit of differentiation

Abstract: Let p_n be a polynomial of very large degree n such that all its roots lie on the real line. Suppose the roots are roughly distributed like random variables coming from, say, a Gaussian. What can you say about the roots of, say, the (n/2)-th derivative of the polynomial? We propose the underlying dynamical system might indeed have a mean field limit and identify a nonlinear and nonlocal partial differential equation. This equation has at least two very nice closed-form solutions: a shrinking semicircle and a family of distributions evolving in the Marchenko-Pastur family of probability distributions; we also show that these solutions satisfy an infinite number of conservation laws. Many open problems, including a connection to random matrices, are being discussed.