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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 $Q=\\sum_{i=1}^q l_i^{d-d_i}m_i$"},"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":"1305.1219","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2013-05-06T15:21:34Z","cross_cats_sorted":[],"title_canon_sha256":"7223768ad72e777d47a12deb552a446a16000b4d55b36b34fce68e330fbbd93d","abstract_canon_sha256":"f1e115fd0f9f8b4ca29c5b6aaf4a0c7f175f0ee5f5ad85c853164ac4849490f2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:26:23.779609Z","signature_b64":"EabfrP1TV+59HeNJvT6e7P2iaK/vyTVC+jINVaZh+NUeFYY2Xg9rMNRIEu7/rT7wtoWcrR+d5MuqNdT0FppDCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e19bff9a01f701bc55f74c2b0205df12e687e0f02e5a242f61ac597543c8cbb4","last_reissued_at":"2026-05-18T03:26:23.778869Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:26:23.778869Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1305.1219","created_at":"2026-05-18T03:26:23.779000+00:00"},{"alias_kind":"arxiv_version","alias_value":"1305.1219v1","created_at":"2026-05-18T03:26:23.779000+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1305.1219","created_at":"2026-05-18T03:26:23.779000+00:00"},{"alias_kind":"pith_short_12","alias_value":"4GN77GQB64A3","created_at":"2026-05-18T12:27:34.582898+00:00"},{"alias_kind":"pith_short_16","alias_value":"4GN77GQB64A3YVPX","created_at":"2026-05-18T12:27:34.582898+00:00"},{"alias_kind":"pith_short_8","alias_value":"4GN77GQB","created_at":"2026-05-18T12:27:34.582898+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/4GN77GQB64A3YVPXJQVQEBO7CL","json":"https://pith.science/pith/4GN77GQB64A3YVPXJQVQEBO7CL.json","graph_json":"https://pith.science/api/pith-number/4GN77GQB64A3YVPXJQVQEBO7CL/graph.json","events_json":"https://pith.science/api/pith-number/4GN77GQB64A3YVPXJQVQEBO7CL/events.json","paper":"https://pith.science/paper/4GN77GQB"},"agent_actions":{"view_html":"https://pith.science/pith/4GN77GQB64A3YVPXJQVQEBO7CL","download_json":"https://pith.science/pith/4GN77GQB64A3YVPXJQVQEBO7CL.json","view_paper":"https://pith.science/paper/4GN77GQB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1305.1219&json=true","fetch_graph":"https://pith.science/api/pith-number/4GN77GQB64A3YVPXJQVQEBO7CL/graph.json","fetch_events":"https://pith.science/api/pith-number/4GN77GQB64A3YVPXJQVQEBO7CL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/4GN77GQB64A3YVPXJQVQEBO7CL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/4GN77GQB64A3YVPXJQVQEBO7CL/action/storage_attestation","attest_author":"https://pith.science/pith/4GN77GQB64A3YVPXJQVQEBO7CL/action/author_attestation","sign_citation":"https://pith.science/pith/4GN77GQB64A3YVPXJQVQEBO7CL/action/citation_signature","submit_replication":"https://pith.science/pith/4GN77GQB64A3YVPXJQVQEBO7CL/action/replication_record"}},"created_at":"2026-05-18T03:26:23.779000+00:00","updated_at":"2026-05-18T03:26:23.779000+00:00"}