{"paper":{"title":"The Common Limit of the Linear Statistics of Zeros of Random Polynomials and Their Derivatives","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Chih-Chung Chang, I-Shing Hu","submitted_at":"2017-01-14T16:44:02Z","abstract_excerpt":"Let $ p_n(x) $ be a random polynomial of degree $n$ and $\\{Z^{(n)}_j\\}_{j=1}^n$ and $\\{X^{n, k}_j\\}_{j=1}^{n-k}, k<n$, be the zeros of $p_n$ and $p_n^{(k)}$, the $k$th derivative of $p_n$, respectively. We show that if the linear statistics $\\frac{1}{a_n} \\left[ f\\left( \\frac {Z^{(n)}_1}{b_n} \\right)+ \\cdots + f \\left(\\frac {Z^{(n)}_n}{b_n} \\right) \\right]$ associated with $\\{Z^{(n)}_j\\}$ has a limit as $n\\to\\infty$ at some mode of convergence, the linear statistics associated with $\\{X^{n, k}_j\\}$ converges to the same limit at the same mode. Similar statement also holds for the centered line"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.03946","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"}