{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:P6JT7T3ZWXYN5HZU2UAIJMUZYQ","short_pith_number":"pith:P6JT7T3Z","schema_version":"1.0","canonical_sha256":"7f933fcf79b5f0de9f34d50084b299c418dccf2c862dd688deeb0354a2b2a38e","source":{"kind":"arxiv","id":"1210.2694","version":1},"attestation_state":"computed","paper":{"title":"From Spline Approximation to Roth's Equation and Schur Functors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC"],"primary_cat":"math.NA","authors_text":"Jan Minac, Stefan O. Tohaneanu","submitted_at":"2012-10-09T19:07:37Z","abstract_excerpt":"Alfeld and Schumaker provide a formula for the dimension of the space of piecewise polynomial functions, called splines, of degree $d$ and smoothness $r$ on a generic triangulation of a planar simplicial complex $\\Delta$, for $d \\geq 3r+1$. Schenck and Stiller conjectured that this formula actually holds for all $d \\geq 2r+1$. Up to this moment there was not known a single example where one could show that the bound $d\\geq 2r +1$ is sharp. However, in 2005, a possible such example was constructed to show that this bound is the best possible (i.e., the Alfeld-Schumaker formula does not hold if "},"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":"1210.2694","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2012-10-09T19:07:37Z","cross_cats_sorted":["math.AC"],"title_canon_sha256":"624a572194b3eb787039ceac71e3a2ab9508a09b9d82644dc48ce1fcd11747ab","abstract_canon_sha256":"752198738dec3795bcc2f4f3b9b7553055cad75d7f7a8dc1296ab759807fc972"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:43:40.908009Z","signature_b64":"3FFGoYXcc8LQUcaboYRzHsb7QtEKbq2U/mCga6YCU+mBDJNr/g870f89lHMiYQNfvMxTEQC0gU/eYzQB9EokDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7f933fcf79b5f0de9f34d50084b299c418dccf2c862dd688deeb0354a2b2a38e","last_reissued_at":"2026-05-18T03:43:40.907510Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:43:40.907510Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"From Spline Approximation to Roth's Equation and Schur Functors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC"],"primary_cat":"math.NA","authors_text":"Jan Minac, Stefan O. Tohaneanu","submitted_at":"2012-10-09T19:07:37Z","abstract_excerpt":"Alfeld and Schumaker provide a formula for the dimension of the space of piecewise polynomial functions, called splines, of degree $d$ and smoothness $r$ on a generic triangulation of a planar simplicial complex $\\Delta$, for $d \\geq 3r+1$. Schenck and Stiller conjectured that this formula actually holds for all $d \\geq 2r+1$. Up to this moment there was not known a single example where one could show that the bound $d\\geq 2r +1$ is sharp. However, in 2005, a possible such example was constructed to show that this bound is the best possible (i.e., the Alfeld-Schumaker formula does not hold if "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.2694","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":"1210.2694","created_at":"2026-05-18T03:43:40.907580+00:00"},{"alias_kind":"arxiv_version","alias_value":"1210.2694v1","created_at":"2026-05-18T03:43:40.907580+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.2694","created_at":"2026-05-18T03:43:40.907580+00:00"},{"alias_kind":"pith_short_12","alias_value":"P6JT7T3ZWXYN","created_at":"2026-05-18T12:27:18.751474+00:00"},{"alias_kind":"pith_short_16","alias_value":"P6JT7T3ZWXYN5HZU","created_at":"2026-05-18T12:27:18.751474+00:00"},{"alias_kind":"pith_short_8","alias_value":"P6JT7T3Z","created_at":"2026-05-18T12:27:18.751474+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/P6JT7T3ZWXYN5HZU2UAIJMUZYQ","json":"https://pith.science/pith/P6JT7T3ZWXYN5HZU2UAIJMUZYQ.json","graph_json":"https://pith.science/api/pith-number/P6JT7T3ZWXYN5HZU2UAIJMUZYQ/graph.json","events_json":"https://pith.science/api/pith-number/P6JT7T3ZWXYN5HZU2UAIJMUZYQ/events.json","paper":"https://pith.science/paper/P6JT7T3Z"},"agent_actions":{"view_html":"https://pith.science/pith/P6JT7T3ZWXYN5HZU2UAIJMUZYQ","download_json":"https://pith.science/pith/P6JT7T3ZWXYN5HZU2UAIJMUZYQ.json","view_paper":"https://pith.science/paper/P6JT7T3Z","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1210.2694&json=true","fetch_graph":"https://pith.science/api/pith-number/P6JT7T3ZWXYN5HZU2UAIJMUZYQ/graph.json","fetch_events":"https://pith.science/api/pith-number/P6JT7T3ZWXYN5HZU2UAIJMUZYQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/P6JT7T3ZWXYN5HZU2UAIJMUZYQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/P6JT7T3ZWXYN5HZU2UAIJMUZYQ/action/storage_attestation","attest_author":"https://pith.science/pith/P6JT7T3ZWXYN5HZU2UAIJMUZYQ/action/author_attestation","sign_citation":"https://pith.science/pith/P6JT7T3ZWXYN5HZU2UAIJMUZYQ/action/citation_signature","submit_replication":"https://pith.science/pith/P6JT7T3ZWXYN5HZU2UAIJMUZYQ/action/replication_record"}},"created_at":"2026-05-18T03:43:40.907580+00:00","updated_at":"2026-05-18T03:43:40.907580+00:00"}