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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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


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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Additional Information
  • Stefan Steinerberger
  • Affiliation: Department of Mathematics, Yale University, New Haven, Connecticut 06511
  • MR Author ID: 869041
  • ORCID: 0000-0002-7745-4217
  • Email:
  • 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
  • © Copyright 2019 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 147 (2019), 4733-4744
  • MSC (2010): Primary 35Q70, 44A15; Secondary 26C10, 31A99, 37F10
  • DOI:
  • MathSciNet review: 4011508