Flow of the zeros of polynomials under iterated differentiation

Abstract

For a monic polynomial $Q_n$ of degree $n$, let $Q_{n,k}$ be its $k$-th derivative normalized to be monic. Under the only assumption that the sequence $\{Q_n\}$ has a weak* limiting zero distribution (an empirical distribution of zeros) represented by a probability measure ${\mu }_0$ with compact support in the complex plane, we show that as $n,k\to \infty$ such that $k/n\to t\in (0,1)$, the Cauchy transform of the normalized zero-counting measure of the polynomials $Q_{n,k}$ converges in a neighborhood of infinity to an analytic function, uniquely determined by ${\mu }_0$ and $t$, that can be written as the Cauchy transform of a measure ${\mu }_t$, not necessarily uniquely determined unless ${\mu }_0$ is supported on the real line. The family of these Cauchy transforms and, when well defined, the corresponding measures ${\mu }_t$, $t\in (0,1)$, whose dependence on the parameter $t$ can be interpreted as a flow of the zeros under iterated differentiation, has several interesting connections with the inviscid Burgers equation, the fractional free convolution of ${\mu }_0$, or a nonlocal diffusion equation governing the density of ${\mu }_t$ on $\mathbb{R}$. We provide an elementary and unified approach that not only recovers, but also explains various phenomena observed in prior works - from Burgers-type PDEs to free probability limits.

28 pages; to appear in Analysis & PDE