{"paper":{"title":"New lower bounds for the rank of matrix multiplication","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG","math.RT"],"primary_cat":"cs.CC","authors_text":"J. M. Landsberg","submitted_at":"2012-06-07T15:36:36Z","abstract_excerpt":"The rank of the matrix multiplication operator for nxn matrices is one of the most studied quantities in algebraic complexity theory. I prove that the rank is at least n^2-o(n^2). More precisely, for any integer p\\leq n -1, the rank is at least (3- 1/(p+1))n^2-(1+2p\\binom{2p}{p-1})n. The previous lower bound, due to Blaser, was 5n^2/2-3n (the case p=1).\n  The new bounds improve Blaser's bound for all n>84. I also prove lower bounds for rectangular matrices significantly better than the the previous bound."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.1530","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"}