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Furthermore our bound improves the classic bound $\\frac{5}{2}n^2-3n$, due to Bl\\\"aser, for every $n\\geq 132$. Finally, for $p = 2$, with a sligtly different strategy we menage to obtain the lower bound $\\frac{8}{3}n^2-7n$ which improves Bl\\\"aser's bound for any $n\\geq 24$."},"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":"1211.6320","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2012-11-27T14:58:37Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"a2e103aec773613ba95bf96a94d46fd51c259a326cbb25e8bd4d77dcd6a40363","abstract_canon_sha256":"d9f8db452ff0a4ac3d9d0387c38c80926dcb25276b932fd663c62c87162f3094"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:07:52.745998Z","signature_b64":"ZJY5CcpO+OvrWcAWvWLLD2a8QXYZsZ3HBxN0Uy4+eoPB33zhW8xY4pXgp+K+TjdEL+/R0qw4m88ZZhEpkIXRDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"13d4888686c64c07449fe5cf1d33a135fa4923c6ae6a4ccaf9c41572791b383d","last_reissued_at":"2026-05-18T03:07:52.745240Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:07:52.745240Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the rank of $n\\times n$ matrix multiplication","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"cs.CC","authors_text":"Alex Massarenti, Emanuele Raviolo","submitted_at":"2012-11-27T14:58:37Z","abstract_excerpt":"For every $p\\leq n$ positive integer we obtain the lower bound $(3-\\frac{1}{p+1})n^2-\\big(2\\binom{2p}{p+1}-\\binom{2p-2}{p-1}+2\\big)n$ for the rank of the $n\\times n$ matrix multiplication. 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