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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."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1311.6189","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2013-11-25T00:48:19Z","cross_cats_sorted":[],"title_canon_sha256":"66f46a4efd47ad5f9e816c9a41c5f0dc8bb5dbf9904b7d353784abf72e9b17fc","abstract_canon_sha256":"eabaab91747b7d1faa8236c69677bc1ba99087cad5bd1a89866fc737b11a8d6e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:21:23.703090Z","signature_b64":"9tvL17MqefJiZmQxE+L+8M9eA0ELJlHpFQ/aATP/zkZ+dEYbq4w7+FxzN7QLSsnX4sEZ5uPg7O0l2+pOLMSUDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3f67ba4a09c80ba8773c92ee23c657ab9be293a950a7765f38ec935fc340c811","last_reissued_at":"2026-05-18T01:21:23.702447Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:21:23.702447Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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