{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:XDFZFUJNUNQMIREY7L24WUIWMO","short_pith_number":"pith:XDFZFUJN","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"},"canonical_sha256":"b8cb92d12da360c44498faf5cb51166383d79bfbd45fcdb4df7b451b0c4c83ee","source":{"kind":"arxiv","id":"1611.07506","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1611.07506","created_at":"2026-05-18T00:02:18Z"},{"alias_kind":"arxiv_version","alias_value":"1611.07506v3","created_at":"2026-05-18T00:02:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.07506","created_at":"2026-05-18T00:02:18Z"},{"alias_kind":"pith_short_12","alias_value":"XDFZFUJNUNQM","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_16","alias_value":"XDFZFUJNUNQMIREY","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_8","alias_value":"XDFZFUJN","created_at":"2026-05-18T12:30:51Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:XDFZFUJNUNQMIREY7L24WUIWMO","target":"record","payload":{"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"},"canonical_sha256":"b8cb92d12da360c44498faf5cb51166383d79bfbd45fcdb4df7b451b0c4c83ee","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"},"source_kind":"arxiv","source_id":"1611.07506","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:02:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lduRhp49Xd8TwztP1VA6GWBlsaSh3qNsFV1u8clIg1G82sbUn0ICcGLdGmCvgFNPHje7OT2TSVILb7Vz7o8SAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T19:54:17.788063Z"},"content_sha256":"9f5f99852e78fceaeb95d50e5c1a23f21a2ad57d2c2d0128ce01a922c0f3cc4f","schema_version":"1.0","event_id":"sha256:9f5f99852e78fceaeb95d50e5c1a23f21a2ad57d2c2d0128ce01a922c0f3cc4f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:XDFZFUJNUNQMIREY7L24WUIWMO","target":"graph","payload":{"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:02:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qjiafC1loZtBLaIvRwOtIq497vE/psvPySyULsZgmqd8mjkWKgixSCjj86tx5kvYOjQBAr7UFInJyxPMmQSIBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T19:54:17.788850Z"},"content_sha256":"71d62bdf4161e8d3f4d1d7dd29a696ce840752433a0e471dd966ceee0265b309","schema_version":"1.0","event_id":"sha256:71d62bdf4161e8d3f4d1d7dd29a696ce840752433a0e471dd966ceee0265b309"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XDFZFUJNUNQMIREY7L24WUIWMO/bundle.json","state_url":"https://pith.science/pith/XDFZFUJNUNQMIREY7L24WUIWMO/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XDFZFUJNUNQMIREY7L24WUIWMO/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-09T19:54:17Z","links":{"resolver":"https://pith.science/pith/XDFZFUJNUNQMIREY7L24WUIWMO","bundle":"https://pith.science/pith/XDFZFUJNUNQMIREY7L24WUIWMO/bundle.json","state":"https://pith.science/pith/XDFZFUJNUNQMIREY7L24WUIWMO/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XDFZFUJNUNQMIREY7L24WUIWMO/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:XDFZFUJNUNQMIREY7L24WUIWMO","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"84d628c9a7082ce22ea90c4f2175671136c6fde5f313f532944e38abaa92ec63","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2016-11-22T20:43:12Z","title_canon_sha256":"f0cde7de6a1c32de52be52289c792fd2af2a6f27c19c2ebb98bf5a98613666a0"},"schema_version":"1.0","source":{"id":"1611.07506","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1611.07506","created_at":"2026-05-18T00:02:18Z"},{"alias_kind":"arxiv_version","alias_value":"1611.07506v3","created_at":"2026-05-18T00:02:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.07506","created_at":"2026-05-18T00:02:18Z"},{"alias_kind":"pith_short_12","alias_value":"XDFZFUJNUNQM","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_16","alias_value":"XDFZFUJNUNQMIREY","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_8","alias_value":"XDFZFUJN","created_at":"2026-05-18T12:30:51Z"}],"graph_snapshots":[{"event_id":"sha256:71d62bdf4161e8d3f4d1d7dd29a696ce840752433a0e471dd966ceee0265b309","target":"graph","created_at":"2026-05-18T00:02:18Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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.","authors_text":"Yairon Cid-Ruiz","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2016-11-22T20:43:12Z","title":"Bounding the degrees of a minimal $\\mu$-basis for a rational surface parametrization"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.07506","kind":"arxiv","version":3},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:9f5f99852e78fceaeb95d50e5c1a23f21a2ad57d2c2d0128ce01a922c0f3cc4f","target":"record","created_at":"2026-05-18T00:02:18Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"84d628c9a7082ce22ea90c4f2175671136c6fde5f313f532944e38abaa92ec63","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2016-11-22T20:43:12Z","title_canon_sha256":"f0cde7de6a1c32de52be52289c792fd2af2a6f27c19c2ebb98bf5a98613666a0"},"schema_version":"1.0","source":{"id":"1611.07506","kind":"arxiv","version":3}},"canonical_sha256":"b8cb92d12da360c44498faf5cb51166383d79bfbd45fcdb4df7b451b0c4c83ee","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b8cb92d12da360c44498faf5cb51166383d79bfbd45fcdb4df7b451b0c4c83ee","first_computed_at":"2026-05-18T00:02:18.580651Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:02:18.580651Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"66DlknfpS/nRKpM5HRmqgUEpEJF8ocFi/S7CW+1Tai3zY/lLT+uYWo/4qD6V9vVVg5JETGdE9GtPxOvfPNfeAA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:02:18.581017Z","signed_message":"canonical_sha256_bytes"},"source_id":"1611.07506","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9f5f99852e78fceaeb95d50e5c1a23f21a2ad57d2c2d0128ce01a922c0f3cc4f","sha256:71d62bdf4161e8d3f4d1d7dd29a696ce840752433a0e471dd966ceee0265b309"],"state_sha256":"8e2abaec193f8d249f17841d0a4d3582f9f6f415fdf482d3fadc1041b7721081"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hqNay6OyYDrq+hoKBsgMxSY+tQI97ZhV8VtxCYFoaQcgjZ8o2DIemfvJc29Q/0pO6uo9tLAZWCj+XdC5ATLACA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-09T19:54:17.793167Z","bundle_sha256":"7702391e5923bba7e0dc0c6a9f532ee0530116366b74df24eac816b000cb0292"}}