{"paper":{"title":"Almost sure behavior of the zeros of iterated derivatives of random polynomials","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CA","math.CV"],"primary_cat":"math.PR","authors_text":"Marcus Michelen, Xuan-Truong Vu","submitted_at":"2023-07-13T14:58:53Z","abstract_excerpt":"Let $Z_1,\\, Z_2,\\dots$ be independent and identically distributed complex random variables with common distribution $\\mu$ and set $$ P_n(z) := (z - Z_1)\\cdots (z - Z_n)\\,. $$ Recently, Angst, Malicet and Poly proved that the critical points of $P_n$ converge in an almost-sure sense to the measure $\\mu$ as $n$ tends to infinity, thereby confirming a conjecture of Cheung-Ng-Yam and Kabluchko. In this short note, we prove for any fixed $k\\in \\mathbb{N}$, the empirical measure of zeros of the $k$th derivative of $P_n$ converges to $\\mu$ in the almost sure sense, as conjectured by Angst-Malicet-Pol"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2307.06788","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2307.06788/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}