{"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. This bound improves the previous one $(3-\\frac{1}{p+1})n^2-\\big(1+2p\\binom{2p}{p}\\big)n$ due to Landsberg. 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 $n\\geq 24$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.6320","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"}