{"paper":{"title":"Distances between zeroes and critical points for random polynomials with i.i.d. zeroes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Hauke Seidel, Zakhar Kabluchko","submitted_at":"2018-07-05T18:34:09Z","abstract_excerpt":"Consider a random polynomial $Q_n$ of degree $n+1$ whose zeroes are i.i.d. random variables $\\xi_0,\\xi_1,\\ldots,\\xi_n$ in the complex plane. We study the pairing between the zeroes of $Q_n$ and its critical points, i.e. the zeroes of its derivative $Q_n'$. In the asymptotic regime when $n\\to\\infty$, with high probability there is a critical point of $Q_n$ which is very close to $\\xi_0$. We localize the position of this critical point by proving that the difference between $\\xi_0$ and the critical point has approximately complex Gaussian distribution with mean $1/(nf(\\xi_0))$ and variance of or"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.02140","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"}