  Let $p_n$ be a polynomial of degree $n$ having all its roots on the real line
distributed according to a smooth function $u(0,x)$. One could wonder how the
distribution of roots behaves under iterated differentation of the function,
i.e. how the density of roots of $p_n^{(k)}$ evolves. We derive a nonlinear
transport equation with nonlocal flux $$ u_t + \frac{1}{\pi}\left( \arctan{
\left( \frac{Hu}{ u}\right)} \right)_x = 0,$$ where $H$ is the Hilbert
transform. This equation has three very different compactly supported
solutions: (1) the arcsine distribution $u(t,x) = (1-x^2)^{-1/2}
\chi_{(-1,1)}$, (2) the family of semicircle distributions $$ u(t,x) =
\frac{2}{\pi} \sqrt{(T-t) - x^2}$$ and (3) a family of solutions contained in
the Marchenko-Pastur law.
