{"paper":{"title":"Tensor Rank, Invariants, Inequalities, and Applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR","math.ST","q-bio.PE","stat.TH"],"primary_cat":"math.AG","authors_text":"Elizabeth S. Allman, Jeremy G. Sumner, John A. Rhodes, Peter D. Jarvis","submitted_at":"2012-11-14T23:41:01Z","abstract_excerpt":"Though algebraic geometry over $\\mathbb C$ is often used to describe the closure of the tensors of a given size and complex rank, this variety includes tensors of both smaller and larger rank. Here we focus on the $n\\times n\\times n$ tensors of rank $n$ over $\\mathbb C$, which has as a dense subset the orbit of a single tensor under a natural group action. We construct polynomial invariants under this group action whose non-vanishing distinguishes this orbit from points only in its closure. Together with an explicit subset of the defining polynomials of the variety, this gives a semialgebraic "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.3461","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"}