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Hence $X:= \\ell_O(C)\\subset \\mathbb{P}^n$ is a linearly normal cuspidal curve with degree $n+1$. For any $P = \\ell_O(B)$, $B\\in \\mathbb{P}^{n+1}$, the $X$-rank $r_X(P)$ of $P$ is the minimal cardinality of a set $S\\subset X$ whose linear span contains $P$. Here we describe $r_X(P)$ in terms of the schemes computing the $C$-rank or the border $C$-rank of $B$."},"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":"1007.2822","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2010-07-16T17:59:50Z","cross_cats_sorted":[],"title_canon_sha256":"ff8550d8e850ab258d952d461f51ecff103ac79c23d9288fea51e7077151cd23","abstract_canon_sha256":"d46bd42d530a22a6f63b1a76412ceeaf09b0c72d65af4cebfc407f71fc068854"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:05:35.965459Z","signature_b64":"2M/eZkCTMkrM6r7nOMHhcrjfzacdYbIoINZ3A5fQEvrmXwljhqfvhJUudDuhkm8cfehFGGGeYDQ3Ya/pPnRdBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2f7b1552090102e1ae429d1022120d45e31c5843f3cdc58ec5a69d4c910f26b6","last_reissued_at":"2026-05-18T03:05:35.964789Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:05:35.964789Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Minimal decomposition of binary forms with respect to tangential projections","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Alessandra Bernardi, Edoardo Ballico","submitted_at":"2010-07-16T17:59:50Z","abstract_excerpt":"Let $C\\subset \\mathbb{P}^n$ be a rational normal curve and let $\\ell_O:\\mathbb{P}^{n+1}\\dashrightarrow \\mathbb{P}^n$ be any tangential projection form a point $O\\in T_AC$ where $A\\in C$. Hence $X:= \\ell_O(C)\\subset \\mathbb{P}^n$ is a linearly normal cuspidal curve with degree $n+1$. For any $P = \\ell_O(B)$, $B\\in \\mathbb{P}^{n+1}$, the $X$-rank $r_X(P)$ of $P$ is the minimal cardinality of a set $S\\subset X$ whose linear span contains $P$. Here we describe $r_X(P)$ in terms of the schemes computing the $C$-rank or the border $C$-rank of $B$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1007.2822","kind":"arxiv","version":2},"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":"1007.2822","created_at":"2026-05-18T03:05:35.964893+00:00"},{"alias_kind":"arxiv_version","alias_value":"1007.2822v2","created_at":"2026-05-18T03:05:35.964893+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1007.2822","created_at":"2026-05-18T03:05:35.964893+00:00"},{"alias_kind":"pith_short_12","alias_value":"F55RKUQJAEBO","created_at":"2026-05-18T12:26:06.534383+00:00"},{"alias_kind":"pith_short_16","alias_value":"F55RKUQJAEBODLSC","created_at":"2026-05-18T12:26:06.534383+00:00"},{"alias_kind":"pith_short_8","alias_value":"F55RKUQJ","created_at":"2026-05-18T12:26:06.534383+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/F55RKUQJAEBODLSCTUICEEQNIX","json":"https://pith.science/pith/F55RKUQJAEBODLSCTUICEEQNIX.json","graph_json":"https://pith.science/api/pith-number/F55RKUQJAEBODLSCTUICEEQNIX/graph.json","events_json":"https://pith.science/api/pith-number/F55RKUQJAEBODLSCTUICEEQNIX/events.json","paper":"https://pith.science/paper/F55RKUQJ"},"agent_actions":{"view_html":"https://pith.science/pith/F55RKUQJAEBODLSCTUICEEQNIX","download_json":"https://pith.science/pith/F55RKUQJAEBODLSCTUICEEQNIX.json","view_paper":"https://pith.science/paper/F55RKUQJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1007.2822&json=true","fetch_graph":"https://pith.science/api/pith-number/F55RKUQJAEBODLSCTUICEEQNIX/graph.json","fetch_events":"https://pith.science/api/pith-number/F55RKUQJAEBODLSCTUICEEQNIX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/F55RKUQJAEBODLSCTUICEEQNIX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/F55RKUQJAEBODLSCTUICEEQNIX/action/storage_attestation","attest_author":"https://pith.science/pith/F55RKUQJAEBODLSCTUICEEQNIX/action/author_attestation","sign_citation":"https://pith.science/pith/F55RKUQJAEBODLSCTUICEEQNIX/action/citation_signature","submit_replication":"https://pith.science/pith/F55RKUQJAEBODLSCTUICEEQNIX/action/replication_record"}},"created_at":"2026-05-18T03:05:35.964893+00:00","updated_at":"2026-05-18T03:05:35.964893+00:00"}