{"year":"2025","title":"Zeros of polynomial powers under the heat flow","status":"public","author":[{"full_name":"Höfert, Antonia","first_name":"Antonia","last_name":"Höfert"},{"full_name":"Jalowy, Jonas","orcid":"0000-0001-9624-2685","first_name":"Jonas","last_name":"Jalowy","id":"113768"},{"full_name":"Kabluchko, Zakhar","last_name":"Kabluchko","first_name":"Zakhar"}],"date_updated":"2025-12-22T08:36:46Z","language":[{"iso":"eng"}],"_id":"63393","user_id":"113768","publication":"arXiv:2512.17808","citation":{"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} }","ama":"Höfert A, Jalowy J, Kabluchko Z. Zeros of polynomial powers under the heat flow. <i>arXiv:251217808</i>. Published online 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).","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.","apa":"Höfert, A., Jalowy, J., &#38; Kabluchko, Z. (2025). Zeros of polynomial powers under the heat flow. In <i>arXiv:2512.17808</i>."},"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."}],"external_id":{"arxiv":["2512.17808"]},"date_created":"2025-12-22T08:36:24Z","type":"preprint","department":[{"_id":"94"}]}