---
_id: '65070'
abstract:
- lang: eng
  text: 'We study the evolution of zeros of high polynomial powers under the heat
    flow. For any fixed polynomial $P(z)$, we prove that the empirical zero distribution
    of its heat-evolved $n$-th power converges to a distribution on the complex plane
    as $n$ tends to infinity. We describe this limit distribution $μ_t$ as a function
    of the time parameter $t$ of the heat evolution: For small time, zeros start to
    spread out in approximately semicircular distributions, then intricate curves
    start to form and merge, until for large time, the zero distribution approaches
    a widespread semicircle law through the initial center of mass. The Stieltjes
    transform of the limit distribution $μ_t$ satisfies a self-consistent equation
    and a Burgers'' equation. The present paper deals with general complex-rooted
    polynomials for which, in contrast to the real-rooted case, no free-probabilistic
    representation for $μ_t$ is available.'
author:
- first_name: Antonia
  full_name: Höfert, Antonia
  id: '115802'
  last_name: Höfert
- first_name: Jonas
  full_name: Jalowy, Jonas
  last_name: Jalowy
- first_name: Zakhar
  full_name: Kabluchko, Zakhar
  last_name: Kabluchko
citation:
  ama: Höfert A, Jalowy J, Kabluchko Z. Zeros of polynomial powers under the heat
    flow. <i>arXiv:251217808</i>. Published online 2025.
  apa: Höfert, A., Jalowy, J., &#38; Kabluchko, Z. (2025). Zeros of polynomial powers
    under the heat flow. In <i>arXiv:2512.17808</i>.
  bibtex: '@article{Höfert_Jalowy_Kabluchko_2025, title={Zeros of polynomial powers
    under the heat flow}, journal={arXiv:2512.17808}, author={Höfert, Antonia and
    Jalowy, Jonas and Kabluchko, Zakhar}, year={2025} }'
  chicago: Höfert, Antonia, Jonas Jalowy, and Zakhar Kabluchko. “Zeros of Polynomial
    Powers under the Heat Flow.” <i>ArXiv:2512.17808</i>, 2025.
  ieee: A. Höfert, J. Jalowy, and Z. Kabluchko, “Zeros of polynomial powers under
    the heat flow,” <i>arXiv:2512.17808</i>. 2025.
  mla: Höfert, Antonia, et al. “Zeros of Polynomial Powers under the Heat Flow.” <i>ArXiv:2512.17808</i>,
    2025.
  short: A. Höfert, J. Jalowy, Z. Kabluchko, ArXiv:2512.17808 (2025).
date_created: 2026-03-20T13:24:50Z
date_updated: 2026-03-20T13:28:16Z
external_id:
  arxiv:
  - '2512.17808'
language:
- iso: eng
publication: arXiv:2512.17808
status: public
title: Zeros of polynomial powers under the heat flow
type: preprint
user_id: '115802'
year: '2025'
...