Dynamics of rotationally invariant polynomial root sets under iterated differentiations

We associate to an $N$-sample of a given rotationally invariant probability measure $\mu_0$ with compact support in the complex plane, a polynomial $P_N$ with roots given by the sample. Then, for $t \in (0,1)$, we consider the empirical measure $\mu_t^{N}$ associated to the root set of the $\lfloor t N\rfloor$-th derivative of $P_N$. A question posed by O'Rourke and Steinerberger [21], reformulated as a conjecture by Hoskins and Kabluchko [10], and recently reaffirmed by Campbell, O'Rourke and Renfrew [5], states that under suitable conditions of regularity on $\mu_0$, for an i.i.d. sample, $\mu_t^{N}$ converges to a rotationally invariant probability measure $\mu_t$ when $N$ tends to infinity, and that $(1-t)\mu_t$ has a radial density $x \mapsto \psi(x,t)$ satisfying the following partial differential equation: \begin{equation} \label{PDErotational} \frac{ \partial \psi(x,t) }{\partial t} = \frac{ \partial}{\partial x} \left( \frac{ \psi(x,t) }{ \frac{1}{x} \int_0^x \psi(y,t) dy } \right). \end{equation} In [10], this equation is reformulated as an equation on the distribution function $\Psi_t$ of the radial part of $(1-t) \mu_t$: \begin{equation} \label{equationPsixtabstract} \frac{\partial \Psi_t (x)}{\partial t} = x \frac{\frac{\partial \Psi_t (x)}{\partial x} } {\Psi_t(x)} - 1. \end{equation} Restricting our study to a specific family of $N$-samplings, we are able to prove a variant of the conjecture above. We also emphasize the important differences between the two-dimensional setting and the one-dimensional setting, illustrated in our Theorem 2.1.