A Nonlocal Transport Equation Modeling Complex Roots of Polynomials under Differentiation

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.