A nonlocal transport equation describing roots of polynomials under differentiation

Steinerberger, Stefan

doi:10.1090/proc/14699

A nonlocal transport equation describing roots of polynomials under differentiation
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Proc. Amer. Math. Soc. 147 (2019), 4733-4744 Request permission

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 \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 $H$ 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 \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.

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