A Nonlocal Transport Equation Describing Roots of Polynomials Under Differentiation

Abstract

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.