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A nonlocal transport equation describing roots of polynomials under differentiation


Author: Stefan Steinerberger
Journal: Proc. Amer. Math. Soc. 147 (2019), 4733-4744
MSC (2010): Primary 35Q70, 44A15; Secondary 26C10, 31A99, 37F10
DOI: https://doi.org/10.1090/proc/14699
Published electronically: July 30, 2019
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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

$\displaystyle u_t + \frac {1}{\pi }\left ( \arctan { \left ( \frac {Hu}{ u}\right )} \right )_x = 0$$\displaystyle \qquad \text {on} ~\operatorname {supp} \left \{u>0\right \},$    

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

$\displaystyle u(t,x) = \frac {2}{\pi } \sqrt {(T-t) - x^2},$    

and (3) a family of solutions contained in the Marchenko-Pastur law.

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Additional Information

Stefan Steinerberger
Affiliation: Department of Mathematics, Yale University, New Haven, Connecticut 06511
Email: stefan.steinerberger@yale.edu

DOI: https://doi.org/10.1090/proc/14699
Keywords: Roots, polynomials, arcsine distribution, semicircle law, Marchenko--Pastur law.
Received by editor(s): December 15, 2018
Published electronically: July 30, 2019
Additional Notes: The author was partially supported by the NSF (DMS-1763179) and by the Alfred P. Sloan Foundation.
Communicated by: Mourad Ismail
Article copyright: © Copyright 2019 American Mathematical Society