An evolution equation modeling roots of polynomials under differentiation
Speaker: Stefan STEINERBERGER, Yale
Location: Warren Weaver Hall 1302
Date: Thursday, April 16, 2020, 11 a.m.
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 remarkably simple nonlinear,
nonlocal transport equation on the line governed by the Hilbert
transform. This equation has at least two very nice closed-form
solutions: a shrinking semi-circle 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 are being discussed.