Zeros of polynomial powers under the heat flow Höfert, Antonia Jalowy, Jonas Kabluchko, Zakhar 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 info:eu-repo/semantics/preprint doc-type:preprint text http://purl.org/coar/resource_type/c_816b https://ris.uni-paderborn.de/record/65070 Höfert A, Jalowy J, Kabluchko Z. Zeros of polynomial powers under the heat flow. <i>arXiv:251217808</i>. Published online 2025. eng info:eu-repo/semantics/altIdentifier/arxiv/2512.17808 info:eu-repo/semantics/closedAccess