{"paper":{"title":"Factorization patterns on nonlinear families of univariate polynomials over a finite field","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Guillermo Matera, Mariana P\\'erez, Melina Privitelli","submitted_at":"2018-07-20T23:09:18Z","abstract_excerpt":"We estimate the number $|\\mathcal{A}_{\\boldsymbol\\lambda}|$ of elements on a nonlinear family $\\mathcal{A}$ of monic polynomials of $\\mathbb{F}_q[T]$ of degree $r$ having factorization pattern $\\boldsymbol\\lambda:=1^{\\lambda_1}2^{\\lambda_2}\\cdots r^{\\lambda_r}$. We show that $|\\mathcal{A}_{\\boldsymbol\\lambda}|= \\mathcal{T}(\\boldsymbol\\lambda)\\,q^{r-m}+\\mathcal{O}(q^{r-m-{1}/{2}})$, where $\\mathcal{T}(\\boldsymbol\\lambda)$ is the proportion of elements of the symmetric group of $r$ elements with cycle pattern $\\boldsymbol\\lambda$ and $m$ is the codimension of $\\mathcal{A}$. We provide explicit u"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.08052","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"}