{"paper":{"title":"Unique decomposition for a polynomial of low rank","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"A. Bernardi, E. Ballico","submitted_at":"2013-05-06T15:21:34Z","abstract_excerpt":"Let $F$ be a homogeneous polynomial of degree $d$ in $m+1$ variables defined over an algebraically closed field of characteristic 0 and suppose that $F$ belongs to the $s$-th secant variety of the $d$-uple Veronese embedding of $\\mathbb{P}^m$ into $ \\PP {{m+d\\choose d}-1}$ but that its minimal decomposition as a sum of $d$-th powers of linear forms requires more than $s$ addenda. We show that if $s\\leq d$ then $F$ can be uniquely written as $F=M_1^d+\\cdots + M_t^d+Q$, where $M_1, \\ldots, M_t$ are linear forms with $t\\leq (d-1)/2$, and $Q$ a binary form such that $Q=\\sum_{i=1}^q l_i^{d-d_i}m_i$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.1219","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"}