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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 $\\mathbb {P}^n$."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1903.10188","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2019-03-25T09:01:53Z","cross_cats_sorted":[],"title_canon_sha256":"c8bb025c8c521647db4c3b274fbd7e5f44c6b3f3d889410875bc9c8a0916a4c0","abstract_canon_sha256":"0c0438d6634734c1e2fa79bd4a6b2da83c7747248091f425ea4aa71679f77981"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:50:30.413672Z","signature_b64":"x+dXSm+5c/gPnXmm7ZSLgSS/fPJAMqjTTIzVgLA185jKhQQVo2nMjMnRS9MW2l3cD6cgXgZ+LQlS6AarV/KpBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5753066cd5e2a4cf26caa7bb175cc41116d3f1a8da3f6ab3cf33ededce01bf29","last_reissued_at":"2026-05-17T23:50:30.413064Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:50:30.413064Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $q$. 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