preprint Antonia Höfert author 115802 Jonas Jalowy author Zakhar Kabluchko author 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. 2025 eng 2512.17808 Höfert, A., Jalowy, J., &#38; Kabluchko, Z. (2025). Zeros of polynomial powers under the heat flow. In <i>arXiv:2512.17808</i>. Höfert A, Jalowy J, Kabluchko Z. Zeros of polynomial powers under the heat flow. <i>arXiv:251217808</i>. Published online 2025. Höfert, Antonia, et al. “Zeros of Polynomial Powers under the Heat Flow.” <i>ArXiv:2512.17808</i>, 2025. @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} } A. Höfert, J. Jalowy, Z. Kabluchko, ArXiv:2512.17808 (2025). Höfert, Antonia, Jonas Jalowy, and Zakhar Kabluchko. “Zeros of Polynomial Powers under the Heat Flow.” <i>ArXiv:2512.17808</i>, 2025. A. Höfert, J. Jalowy, and Z. Kabluchko, “Zeros of polynomial powers under the heat flow,” <i>arXiv:2512.17808</i>. 2025. 650702026-03-20T13:24:50Z2026-03-20T13:28:16Z