{"paper":{"title":"Zeros of a growing number of derivatives of random polynomials with independent roots","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":"2022-12-22T17:01:56Z","abstract_excerpt":"Let $X_1,X_2,\\ldots$ be independent and identically distributed random variables in $\\mathbb{C}$ chosen from a probability measure $\\mu$ and define the random polynomial $$ P_n(z)=(z-X_1)\\ldots(z-X_n)\\,. $$ We show that for any sequence $k = k(n)$ satisfying $k \\leq \\log n / (5 \\log\\log n)$, the zeros of the $k$th derivative of $P_n$ are asymptotically distributed according to the same measure $\\mu$. This extends work of Kabluchko, which proved the $k = 1$ case, as well as Byun, Lee and Reddy who proved the fixed $k$ case."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2212.11867","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/2212.11867/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"}