---
_id: '59209'
abstract:
- lang: eng
  text: "We start with a random polynomial $P^{N}$ of degree $N$ with independent\r\ncoefficients
    and consider a new polynomial $P_{t}^{N}$ obtained by repeated\r\napplications
    of a fraction differential operator of the form $z^{a}%\r\n(d/dz)^{b},$ where
    $a$ and $b$ are real numbers. When $b>0,$ we compute the\r\nlimiting root distribution
    $\\mu_{t}$ of $P_{t}^{N}$ as $N\\rightarrow\\infty.$ We\r\nshow that $\\mu_{t}$
    is the push-forward of the limiting root distribution of\r\n$P^{N}$ under a transport
    map $T_{t}$. The map $T_{t}$ is defined by flowing\r\nalong the characteristic
    curves of the PDE satisfied by the log potential of\r\n$\\mu_{t}.$ In the special
    case of repeated differentiation, our results may be\r\ninterpreted as saying
    that the roots evolve radially \\textit{with constant\r\nspeed} until they hit
    the origin, at which point, they cease to exist. For\r\ngeneral $a$ and $b,$ the
    transport map $T_{t}$ has a free probability\r\ninterpretation as multiplication
    of an $R$-diagonal operator by an $R$-diagonal\r\n\"transport operator.\" As an
    application, we obtain a push-forward\r\ncharacterization of the free self-convolution
    semigroup $\\oplus$ of radial\r\nmeasures on $\\mathbb{C}$.\r\n  We also consider
    the case $b<0,$ which includes the case of repeated\r\nintegration. More complicated
    behavior of the roots can occur in this case."
author:
- first_name: Brian C.
  full_name: Hall, Brian C.
  last_name: Hall
- first_name: Ching-Wei
  full_name: Ho, Ching-Wei
  last_name: Ho
- first_name: Jonas
  full_name: Jalowy, Jonas
  id: '113768'
  last_name: Jalowy
  orcid: 0000-0001-9624-2685
- first_name: Zakhar
  full_name: Kabluchko, Zakhar
  last_name: Kabluchko
citation:
  ama: Hall BC, Ho C-W, Jalowy J, Kabluchko Z. Roots of polynomials under repeated
    differentiation and repeated  applications of fractional differential operators.
    <i>arXiv:231214883</i>. Published online 2023.
  apa: Hall, B. C., Ho, C.-W., Jalowy, J., &#38; Kabluchko, Z. (2023). Roots of polynomials
    under repeated differentiation and repeated  applications of fractional differential
    operators. In <i>arXiv:2312.14883</i>.
  bibtex: '@article{Hall_Ho_Jalowy_Kabluchko_2023, title={Roots of polynomials under
    repeated differentiation and repeated  applications of fractional differential
    operators}, journal={arXiv:2312.14883}, author={Hall, Brian C. and Ho, Ching-Wei
    and Jalowy, Jonas and Kabluchko, Zakhar}, year={2023} }'
  chicago: Hall, Brian C., Ching-Wei Ho, Jonas Jalowy, and Zakhar Kabluchko. “Roots
    of Polynomials under Repeated Differentiation and Repeated  Applications of Fractional
    Differential Operators.” <i>ArXiv:2312.14883</i>, 2023.
  ieee: B. C. Hall, C.-W. Ho, J. Jalowy, and Z. Kabluchko, “Roots of polynomials under
    repeated differentiation and repeated  applications of fractional differential
    operators,” <i>arXiv:2312.14883</i>. 2023.
  mla: Hall, Brian C., et al. “Roots of Polynomials under Repeated Differentiation
    and Repeated  Applications of Fractional Differential Operators.” <i>ArXiv:2312.14883</i>,
    2023.
  short: B.C. Hall, C.-W. Ho, J. Jalowy, Z. Kabluchko, ArXiv:2312.14883 (2023).
date_created: 2025-03-31T07:15:40Z
date_updated: 2025-04-23T14:38:56Z
external_id:
  arxiv:
  - '2312.14883'
language:
- iso: eng
publication: arXiv:2312.14883
status: public
title: Roots of polynomials under repeated differentiation and repeated  applications
  of fractional differential operators
type: preprint
user_id: '113768'
year: '2023'
...