Let
p
n
p_n
be a polynomial of degree
n
n
having all its roots on the real line distributed according to a smooth function
u
(
0
,
x
)
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
)
p_n^{(k)}
evolves. We derive a nonlinear transport equation with nonlocal flux
u
t
+
1
π
(
arctan
(
H
u
u
)
)
x
=
0
on
supp
{
u
>
0
}
,
\begin{equation*} u_t + \frac {1}{\pi }\left ( \arctan { \left ( \frac {Hu}{ u}\right )} \right )_x = 0 \qquad \text {on} ~\operatorname {supp} \left \{u>0\right \}, \end{equation*}
where
H
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
χ
(
−
1
,
1
)
u(t,x) = (1-x^2)^{-1/2} \chi _{(-1,1)}
, (2) the family of semicircle distributions
u
(
t
,
x
)
=
2
π
(
T
−
t
)
−
x
2
,
\begin{equation*} u(t,x) = \frac {2}{\pi } \sqrt {(T-t) - x^2}, \end{equation*}
and (3) a family of solutions contained in the Marchenko–Pastur law.