{"paper":{"title":"On the typical rank of real polynomials (or symmetric tensors) with a fixed border rank","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Edoardo Ballico","submitted_at":"2013-07-09T15:21:14Z","abstract_excerpt":"Let $\\sigma_b(X_{m,d}(\\mathbb {C}))(\\mathbb {R})$, $b(m+1) < \\binom{m+d}{m}$, denote the set of all degree $d$ real homogeneous polynomials in $m+1$ variables (i.e. real symmetric tensors of format $(m+1)\\times ... \\times (m+1)$, $d$ times) which have border rank $b$ over $\\mathbb {C}$. It has a partition into manifolds of real dimension $\\le b(m+1)-1$ in which the real rank is constant. A typical rank of $\\sigma_b(X_{m,d}(\\mathbb {C}))(\\mathbb {R})$ is a rank associated to an open part of dimension $b(m+1)-1$. Here we classify all typical ranks when $b\\le 7$ and $d, m$ are not too small. For "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.2490","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"}