{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:XDFZFUJNUNQMIREY7L24WUIWMO","short_pith_number":"pith:XDFZFUJN","schema_version":"1.0","canonical_sha256":"b8cb92d12da360c44498faf5cb51166383d79bfbd45fcdb4df7b451b0c4c83ee","source":{"kind":"arxiv","id":"1611.07506","version":3},"attestation_state":"computed","paper":{"title":"Bounding the degrees of a minimal $\\mu$-basis for a rational surface parametrization","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Yairon Cid-Ruiz","submitted_at":"2016-11-22T20:43:12Z","abstract_excerpt":"In this paper, we study how the degrees of the elements in a minimal $\\mu$-basis of a parametrized surface behave. For an arbitrary rational surface parametrization $P(s,t)=(a_1(s,t),a_2(s,t),a_3(s,t),a_4(s,t)) \\in \\mathbb{F}[s,t]^4$ over an infinite field $\\mathbb{F}$, we show the existence of a $\\mu$-basis with polynomials bounded in degree by $O(d^{33})$, where $d=\\max(\\text{deg}(a_1),\\text{deg}(a_2), \\text{deg}(a_3), \\text{deg}(a_4))$. Under additional assumptions we can obtain tighter bounds."},"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":"1611.07506","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2016-11-22T20:43:12Z","cross_cats_sorted":[],"title_canon_sha256":"f0cde7de6a1c32de52be52289c792fd2af2a6f27c19c2ebb98bf5a98613666a0","abstract_canon_sha256":"84d628c9a7082ce22ea90c4f2175671136c6fde5f313f532944e38abaa92ec63"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:02:18.581017Z","signature_b64":"66DlknfpS/nRKpM5HRmqgUEpEJF8ocFi/S7CW+1Tai3zY/lLT+uYWo/4qD6V9vVVg5JETGdE9GtPxOvfPNfeAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b8cb92d12da360c44498faf5cb51166383d79bfbd45fcdb4df7b451b0c4c83ee","last_reissued_at":"2026-05-18T00:02:18.580651Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:02:18.580651Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Bounding the degrees of a minimal $\\mu$-basis for a rational surface parametrization","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Yairon Cid-Ruiz","submitted_at":"2016-11-22T20:43:12Z","abstract_excerpt":"In this paper, we study how the degrees of the elements in a minimal $\\mu$-basis of a parametrized surface behave. For an arbitrary rational surface parametrization $P(s,t)=(a_1(s,t),a_2(s,t),a_3(s,t),a_4(s,t)) \\in \\mathbb{F}[s,t]^4$ over an infinite field $\\mathbb{F}$, we show the existence of a $\\mu$-basis with polynomials bounded in degree by $O(d^{33})$, where $d=\\max(\\text{deg}(a_1),\\text{deg}(a_2), \\text{deg}(a_3), \\text{deg}(a_4))$. Under additional assumptions we can obtain tighter bounds."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.07506","kind":"arxiv","version":3},"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":"1611.07506","created_at":"2026-05-18T00:02:18.580706+00:00"},{"alias_kind":"arxiv_version","alias_value":"1611.07506v3","created_at":"2026-05-18T00:02:18.580706+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.07506","created_at":"2026-05-18T00:02:18.580706+00:00"},{"alias_kind":"pith_short_12","alias_value":"XDFZFUJNUNQM","created_at":"2026-05-18T12:30:51.357362+00:00"},{"alias_kind":"pith_short_16","alias_value":"XDFZFUJNUNQMIREY","created_at":"2026-05-18T12:30:51.357362+00:00"},{"alias_kind":"pith_short_8","alias_value":"XDFZFUJN","created_at":"2026-05-18T12:30:51.357362+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/XDFZFUJNUNQMIREY7L24WUIWMO","json":"https://pith.science/pith/XDFZFUJNUNQMIREY7L24WUIWMO.json","graph_json":"https://pith.science/api/pith-number/XDFZFUJNUNQMIREY7L24WUIWMO/graph.json","events_json":"https://pith.science/api/pith-number/XDFZFUJNUNQMIREY7L24WUIWMO/events.json","paper":"https://pith.science/paper/XDFZFUJN"},"agent_actions":{"view_html":"https://pith.science/pith/XDFZFUJNUNQMIREY7L24WUIWMO","download_json":"https://pith.science/pith/XDFZFUJNUNQMIREY7L24WUIWMO.json","view_paper":"https://pith.science/paper/XDFZFUJN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1611.07506&json=true","fetch_graph":"https://pith.science/api/pith-number/XDFZFUJNUNQMIREY7L24WUIWMO/graph.json","fetch_events":"https://pith.science/api/pith-number/XDFZFUJNUNQMIREY7L24WUIWMO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XDFZFUJNUNQMIREY7L24WUIWMO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XDFZFUJNUNQMIREY7L24WUIWMO/action/storage_attestation","attest_author":"https://pith.science/pith/XDFZFUJNUNQMIREY7L24WUIWMO/action/author_attestation","sign_citation":"https://pith.science/pith/XDFZFUJNUNQMIREY7L24WUIWMO/action/citation_signature","submit_replication":"https://pith.science/pith/XDFZFUJNUNQMIREY7L24WUIWMO/action/replication_record"}},"created_at":"2026-05-18T00:02:18.580706+00:00","updated_at":"2026-05-18T00:02:18.580706+00:00"}