{"paper":{"title":"The distribution of factorization patterns on linear families of polynomials over a finite field","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Eda Cesaratto, Guillermo Matera, Mariana P\\'erez","submitted_at":"2014-08-29T13:27:54Z","abstract_excerpt":"We obtain estimates on the number $|\\mathcal{A}_{\\boldsymbol{\\lambda}}|$ of elements on a linear family $\\mathcal{A}$ of monic polynomials of $\\mathbb{F}_q[T]$ of degree $n$ having factorization pattern $\\boldsymbol{\\lambda}:=1^{\\lambda_1}2^{\\lambda_2}\\cdots n^{\\lambda_n}$. We show that $|\\mathcal{A}_{\\boldsymbol{\\lambda}}|= \\mathcal{T}(\\boldsymbol{\\lambda})\\,q^{n-m}+\\mathcal{O}(q^{n-m-{1}/{2}})$, where $\\mathcal{T}(\\boldsymbol{\\lambda})$ is the proportion of elements of the symmetric group of $n$ elements with cycle pattern $\\boldsymbol{\\lambda}$ and $m$ is the codimension of $\\mathcal{A}$. F"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.7014","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"}