{"paper":{"title":"The distance to square-free polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Art\\=uras Dubickas, Min Sha","submitted_at":"2018-01-04T04:06:43Z","abstract_excerpt":"In this paper, we consider a variant of Tur\\'an's problem on the distance from an integer polynomial in $\\mathbb{Z}[x]$ to the nea\\-rest irreducible polynomial in $\\mathbb{Z}[x]$. We prove that for any polynomial $f \\in \\mathbb{Z}[x]$, there exist infinitely many square-free polynomials $g\\in \\mathbb{Z}[x]$ such that $L(f-g) \\le 2$, where $L(f-g)$ denotes the sum of the absolute values of the coefficients of $f-g$. On the other hand, we show that this inequality cannot be replaced by $L(f-g) \\le 1$. For this, for each integer $d \\geq 16$ we construct infinitely many polynomials $f \\in \\mathbb{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.01240","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"}