Conservation Laws for the Density of Roots of Polynomials under Differentiation

Let $p_n(x)$ be a polynomial of degree $n$ having $n$ distinct, real roots distributed according to a nice probability distribution $u(0,x)dx$ on $\mathbb{R}$. One natural problem is to understand the density $u(t,x)$ of the roots of the $(t\cdot n)-$th derivative of $p_n$ where $0<t<1$ as $n \rightarrow \infty$. We derive an $\mathit{\text{infinite}}$ number of conversation laws for the evolution of $u(t,x)$. The first three are \begin{align*} \int_{$\mathbb{R}$}{ u(t,x) ~ dx} = 1-t, \qquad \qquad \int_{$\mathbb{R}$}{ u(t,x) x ~ dx} = \left(1-t\right)\int_{$\mathbb{R}$}{ u(0,x) x~ dx}, \qquad \int_{$\mathbb{R}$} \int_{$\mathbb{R}$} u(t,x) (x-y)^2 u(t,y) ~ dx dy = (1-t)^3 \int_{$\mathbb{R}$} \int_{$\mathbb{R}$} u(0,x) (x-y)^2 u(0,y) ~ dx dy. \end{align*} The author suggested that $u(t,x)$ might evolve according to a nonlocal evolution equation involving the Hilbert transform; this has been verified for two special closed form solutions – these conservation laws thus point to interesting identities for the Hilbert transform. We discuss many open problems.