{"paper":{"title":"The Distance to a Squarefree Polynomial Over $\\mathbb{F}_2[x]$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Michael Filaseta, Richard A. Moy","submitted_at":"2019-06-19T04:05:16Z","abstract_excerpt":"In this paper, we examine how far a polynomial in $\\mathbb{F}_2[x]$ can be from a squarefree polynomial. For any $\\epsilon>0$, we prove that for any polynomial $f(x)\\in\\mathbb{F}_2[x]$ with degree $n$, there exists a squarefree polynomial $g(x)\\in\\mathbb{F}_2[x]$ such that $\\mathrm{deg} (g) \\le n$ and $L_{2}(f-g)<(\\ln n)^{2\\ln(2)+\\epsilon}$ (where $L_{2}$ is a norm to be defined). As a consequence, the analogous result holds for polynomials $f(x)$ and $g(x)$ in $\\mathbb{Z}[x]$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1906.07904","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"}