{"paper":{"title":"S-parts of values of univariate polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Maurizio Moreschi","submitted_at":"2019-07-18T18:40:52Z","abstract_excerpt":"Let $S=\\{p_1,\\dots,p_s\\}$ be a finite non-empty set of distinct prime numbers, let $f\\in \\mathbb{Z}[X]$ be a polynomial of degree $n\\ge 1$, and let $S'\\subseteq S$ be the subset of all $p\\in S$ such that $f$ has a root in $\\mathbb{Z}_p$. For any non-zero integer $y$, write $y=p_1^{k_1}\\dots p_s^{k_s}y_0$, where $k_1,\\dots,k_s$ are non-negative integers and $y_0$ is an integer coprime to $p_1,\\dots,p_s$. We define the $f$-normalized $S$-part of $y$ by $[y]_{f,S}:=p_1^{k_1 r_{p_1,S}(f)}\\dots p_s^{k_s r_{p_s,S}(f)}$, with $r_{p,S}(f)=1$ if $p\\in S\\setminus S'$ and $r_{p,S}(f)=R_{S'}(f)/R_{p}(f)$ "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.08239","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"}