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A nonlocal transport equation modeling complex roots of polynomials under differentiation


Authors: Sean O’Rourke and Stefan Steinerberger
Journal: Proc. Amer. Math. Soc. 149 (2021), 1581-1592
MSC (2010): Primary 35Q70, 35Q82, 44A15, 82C70; Secondary 26C10, 31A99, 37F10
DOI: https://doi.org/10.1090/proc/15314
Published electronically: February 10, 2021
MathSciNet review: 4242313
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Abstract: Let $p_n:\mathbb {C} \rightarrow \mathbb {C}$ be a random complex polynomial whose roots are sampled i.i.d. from a radial distribution $2\pi r u(r) dr$ 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 $\psi (s) = u(s) s$: \begin{equation*} \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 ). \end{equation*} The evolution of $\psi (s) \equiv 1$ corresponds to the evolution of random Taylor polynomials \begin{equation*} p_n(z) = \sum _{k=0}^{n}{ \gamma _k \frac {z^k}{k!}} \quad \text {where} \quad \gamma _k \sim \mathcal {N}_{\mathbb {C}}(0,1). \end{equation*} 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.


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Additional Information

Sean O’Rourke
Affiliation: Department of Mathematics, University of Colorado, Campus Box 395, Boulder, Colorado 80309-0395
Email: sean.d.orourke@colorado.edu

Stefan Steinerberger
Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195
MR Author ID: 869041
ORCID: 0000-0002-7745-4217
Email: steinerb@uw.edu

Keywords: Roots, differentiation, transport equation, Hardy inequality
Received by editor(s): April 15, 2020
Received by editor(s) in revised form: August 7, 2020
Published electronically: February 10, 2021
Additional Notes: The first author was supported in part by NSF grants ECCS-1610003 and DMS-1810500. Part of the work was carried out while the first author was visiting Yale University, he is grateful for the hospitality.
The second author was partially supported by the NSF (DMS-1763179) and the Alfred P. Sloan Foundation.
Communicated by: Mourad Ismail
Article copyright: © Copyright 2021 American Mathematical Society