  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 \textit{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.
