{"paper":{"title":"Factorization Statistics of Restricted Polynomial Specializations over Large Finite Fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Alexei Entin","submitted_at":"2018-10-17T13:06:18Z","abstract_excerpt":"For a polynomial $F(t,A_1,\\ldots,A_n)\\in\\mathbf{F}_p[t,A_1,\\ldots,A_n]$ ($p$ being a prime number) we study the factorization statistics of its specializations $$F(t,a_1,\\ldots,a_n)\\in\\mathbf{F}_p[t]$$ with $(a_1,\\ldots,a_n)\\in S$, where $S\\subset\\mathbf{F}_p^n$ is a subset, in the limit $p\\to\\infty$ and $\\mathrm{deg} F$ fixed. We show that for a sufficiently large and regular subset $S\\subset\\mathbf{F}_p^n$, e.g. a product of $n$ intervals of length $H_1,\\ldots,H_n$ with $\\prod_{i=1}^nH_n>p^{n-1/2+\\epsilon}$, the factorization statistics is the same as for unrestricted specializations (i.e. $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.07512","kind":"arxiv","version":2},"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"}