{"paper":{"title":"Typical ranks for 3-tensors, nonsingular bilinear maps and determinantal ideals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC"],"primary_cat":"math.RA","authors_text":"Mitsuhiro Miyazaki, Toshio Sakata, Toshio Sumi","submitted_at":"2015-12-28T16:44:31Z","abstract_excerpt":"Let $m,n\\geq 3$, $(m-1)(n-1)+2\\leq p\\leq mn$, and $u=mn-p$. The set $\\mathbb{R}^{u\\times n\\times m}$ of all real tensors with size $u\\times n\\times m$ is one to one corresponding to the set of bilinear maps $\\mathbb{R}^m\\times \\mathbb{R}^n\\to \\mathbb{R}^u$. We show that $\\mathbb{R}^{m\\times n\\times p}$ has plural typical ranks $p$ and $p+1$ if and only if there exists a nonsingular bilinear map $\\mathbb{R}^m\\times\\mathbb{R}^n\\to\\mathbb{R}^{u}$. We show that there is a dense open subset $\\mathscr{O}$ of $\\mathbb{R}^{u\\times n\\times m}$ such that for any $Y\\in\\mathscr{O}$, the ideal of maximal m"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.08452","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"}