{"paper":{"title":"Reconstruction of a homogeneous polynomial from its additive decompositions when identifiability fails","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Edoardo Ballico","submitted_at":"2019-03-25T09:01:53Z","abstract_excerpt":"Let $X\\subset \\mathbb {P}^r$ be an integral and non-degenerate variety. For any $q\\in \\mathbb {P}^r$ let $r_X(q)$ be its $X$-rank and $\\mathcal {S} (X,q)$ the set of all finite subsets of $X$ such that $|S|=r_X(q)$ and $q\\in \\langle S\\rangle$, where $\\langle \\ \\ \\rangle$ denotes the linear span. We consider the case $|\\mathcal {S} (X,q)|>1$ (i.e. when $q$ is not $X$-identifiable) and study the set $W(X)_q:= \\cap _{S\\in\\mathcal {S}}\\langle S\\rangle$, which we call the non-uniqueness set of $q$. We study the case $\\dim X=1$ and the case $X$ a Veronese embedding of $\\mathbb {P}^n$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1903.10188","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"}