{"paper":{"title":"Growth Rates of Algebras, II: Wiegold Dichotomy","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Agnes Szendrei, Emil W. Kiss, Keith A. Kearnes","submitted_at":"2013-11-25T00:48:19Z","abstract_excerpt":"We investigate the function $d_\\mathbf{A}(n)$, which gives the size of a least size generating set for $\\mathbf{A}^n$, in the case where $\\mathbf{A}$ has a cube term. We show that if $\\mathbf{A}$ has a $k$-cube term and $\\mathbf{A}^k$ is finitely generated, then $d_\\mathbf{A}(n) \\in O(\\log(n))$ if $\\mathbf{A}$ is perfect and $d_\\mathbf{A}(n) \\in O(n)$ if $\\mathbf{A}$ is imperfect. When $\\mathbf{A}$ is finite, then one may replace \"Big Oh\" with \"Big Theta\" in these estimates."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.6189","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"}