{"paper":{"title":"Sweeping words and the length of a generic vector subspace of M_n(F)","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Igor Klep, \\v{S}pela \\v{S}penko","submitted_at":"2015-06-10T20:03:20Z","abstract_excerpt":"The main result of this short note is a generic version of Paz's conjecture on the lengths of generating sets in matrix algebras. Consider a generic g-tuple A=(A_1,..., A_g) of nxn matrices over a field. We show that whenever $g^{2d}\\geq n^2$, the set of all words of degree 2d in A spans the full nxn matrix algebra. Our proofs use generic matrices, are combinatorial and depend on the construction of a special kind of directed multigraphs with few edge-disjoint walks."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.03460","kind":"arxiv","version":1},"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"}