{"paper":{"title":"Typical ranks of semi-tall real 3-tensors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC","math.AG"],"primary_cat":"math.RA","authors_text":"Mitsuhiro Miyazaki, Toshio Sakata, Toshio Sumi","submitted_at":"2017-05-08T08:09:37Z","abstract_excerpt":"Let $m$, $n$ and $p$ be integers with $3\\leq m\\leq n$ and $(m-1)(n-1)+1\\leq p\\leq (m-1)m$. We showed in previous papers that if $p\\geq (m-1)(n-1)+2$, then typical ranks of $p\\times n\\times m$-tensors over the real number field are $p$ and $p+1$ if and only if there exists a nonsingular bilinear map $\\mathbb{R}^m\\times \\mathbb{R}^n\\to\\mathbb{R}^{mn-p}$. We also showed that the \"if\" part also valid in the case where $p=(m-1)(n-1)+1$. In this paper, we consider the case where $p=(m-1)(n-1)+1$ and show that the typical ranks of $p\\times n\\times m$-tensors over the real number field are $p$ and $p+"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.02768","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"}