  This article is devoted to the study of a nonlinear and nonlocal parabolic
equation introduced by Stefan Steinerberger to study the roots of polynomials
under differentiation; it also appeared in a work by Dimitri Shlyakhtenko and
Terence Tao on free convolution. Rafael Granero-Belinch\'on obtained a global
well-posedness result for positive initial data small enough in a Wiener space,
and recently Alexander Kiselev and Changhui Tan proved a global well-posedness
result for any positive initial data in the Sobolev space $H^s(\mathbb{S})$
with $s>3/2$. In this paper, we consider the Cauchy problem in the critical
space $H^{1/2}(\mathbb{S})$. Two interesting new features, at this level of
regularity, are that the equation can be written in the form $$ \partial_t
u+V\partial_x u+\gamma \Lambda u=0, $$ where $\gamma$ is non-negative but not
bounded from below and $V/\sqrt{\gamma}$ is not bounded. Therefore, the
equation is only weakly parabolic. We prove that nevertheless the Cauchy
problem is well posed locally in time and that the solutions are smooth for
positive times. Combining this with the results of Kiselev and Tan, this gives
a global well-posedness result for any positive initial data in
$H^{1/2}(\mathbb{S})$. Our proof relies on sharp commutator estimates and
introduces a strategy to prove a local well-posedness result in a situation
where the lifespan depends on the profile of the initial data and not only on
its norm.
