  Let $p_n:\mathbb{C} \rightarrow \mathbb{C}$ be a random complex polynomial
whose roots are sampled i.i.d. from a radial distribution $u(r) r dr$ in the
complex plane. A natural question is how the distribution of roots evolves
under repeated (say $n/2-$times) differentiation of the polynomial. We
conjecture a mean-field expansion for the evolution of $\psi(s) = u(s) s$ $$
\frac{\partial \psi}{\partial t} = \frac{\partial}{\partial x} \left( \left(
\frac{1}{x} \int_{0}^{x} \psi(s) ds \right)^{-1} \psi(x) \right).$$ The
evolution of $\psi(s) \equiv 1$ corresponds to the evolution of random Taylor
polynomials $$ p_n(z) = \sum_{k=0}^{n}{ \gamma_k \frac{z^k}{k!}} \quad
\mbox{where} \quad \gamma_k \sim \mathcal{N}_{\mathbb{C}}(0,1).$$ We discuss
some numerical examples suggesting that this particular solution may be stable.
We prove that the solution is linearly stable. The linear stability analysis
reduces to the classical Hardy integral inequality. Many open problems are
discussed.
