{"paper":{"title":"On approximate Gauss-Lucas theorems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Trevor Richards","submitted_at":"2017-06-16T19:28:34Z","abstract_excerpt":"The Gauss--Lucas theorem states that any convex set $K\\subset\\mathbb{C}$ which contains all $n$ zeros of a degree $n$ polynomial $p\\in\\mathbb{C}[z]$ must also contain all $n-1$ critical points of $p$. In this paper we explore the following question: for which choices of positive integers $n$ and $k$, and positive real number $\\epsilon$, will it follow that for every degree $n$ polynomial $p$ with at least $k$ zeros lying in $K$, $p$ will have at least $k-1$ critical points lying in the $\\epsilon$-neighborhood of $K$. We supply an inequality relating $n$, $k$, and $\\epsilon$ which, when satisfi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.05410","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":""},"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"}