{"paper":{"title":"A constructive approach to Schaeffer's conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Oleg Szehr, Rachid Zarouf","submitted_at":"2017-05-30T15:41:39Z","abstract_excerpt":"J.J. Schaeffer proved that for $any$ induced matrix norm and $any$ invertible $T=T(n)$ the inequality \\[\\left|\\det T\\right|\\left\\Vert T^{-1}\\right\\Vert \\leq\\mathcal{S}\\left\\Vert T\\right\\Vert ^{n-1}\\] holds with $\\mathcal{S}=\\mathcal{S}(n)\\leq\\sqrt{en}$. He conjectured that the best $\\mathcal{S}$ was actually bounded. This was rebutted by Gluskin-Meyer-Pajor and subsequent contributions by J. Bourgain and H. Queffelec that successively improved lower estimates on $\\mathcal{S}$. These articles rely on a link to the theory of power sums of complex numbers. A probabilistic or number theoretic anal"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.10704","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1705.10704/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}