Dynamics of zeroes under repeated differentiation

Abstract

Consider a random polynomial $P_n$ of degree $n$ whose roots are independent random variables sampled according to some probability distribution $\mu_0$ on the complex plane $\mathbb C$. It is natural to conjecture that, for a fixed $t\in [0,1)$ and as $n\to\infty$, the zeroes of the $[tn]$-th derivative of $P_n$ are distributed according to some measure $\mu_t$ on $\mathbb C$. Assuming either that $\mu_0$ is concentrated on the real line or that it is rotationally invariant, Steinerberger [Proc. AMS, 2019] and O'Rourke and Steinerberger [arXiv:1910.12161] derived nonlocal transport equations for the density of roots. We introduce a different method to treat such problems. In the rotationally invariant case, we obtain a closed formula for $\psi(x,t)$, the asymptotic density of the radial parts of the roots of the $[tn]$-th derivative of $P_n$. Although its derivation is non-rigorous, we provide numerical evidence for its correctness and prove that it solves the PDE of O'Rourke and Steinerberger. Moreover, we present several examples in which the solution is fully explicit (including the special case in which the initial condition $\psi(x,0)$ is an arbitrary convex combination of delta functions) and analyze some properties of the solutions such as the behavior of void annuli and circles of zeroes. As an additional support for the correctness of the method, we show that a similar method, applied to the case when $\mu_0$ is concentrated on the real line, gives a correct result which is known to have an interpretation in terms of free probability.