{"paper":{"title":"On the minimum trace norm of (0,1)-matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Natalia Agudelo, Vladimir Nikiforov","submitted_at":"2017-03-02T17:21:56Z","abstract_excerpt":"The trace norm of a matrix is the sum of its singular values. This paper presents results on the minimum trace norm $\\psi_{n}\\left( m\\right) $ of $\\left( 0,1\\right) $-matrices of size $n\\times n$ with exactly $m$ ones. It is shown that:\n  (1) if $n\\geq2$ and $n<m\\leq2n,$ then $\\psi_{n}\\left( m\\right) \\leq \\sqrt{m+\\sqrt{2\\left( m-1\\right) }}$ , with equality if and only if $m$ is a prime;\n  (2) if $n\\geq4$ and $2n<m\\leq3n,$ then $\\psi_{n}\\left( m\\right) \\leq \\sqrt{m+2\\sqrt{2\\left\\lfloor m/3\\right\\rfloor }}$ , with equality if and only if $m$ is a prime or a double of a prime;\n  (3) if $3n<m\\leq"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.00859","kind":"arxiv","version":3},"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"}