{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:PJRAK3JWE7SD747R3GXFHSVMPX","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":"11f2b9e73eefa7c369aa477a1de25161e7b24bf15e13f3ccf49c04f6914aa4ac","cross_cats_sorted":["math.AG","math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2018-05-28T13:57:57Z","title_canon_sha256":"a89fc76cb6892ee2e23acfd4a74a76eed317930593694d6919860b2a8cd431c4"},"schema_version":"1.0","source":{"id":"1805.10923","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1805.10923","created_at":"2026-05-18T00:14:48Z"},{"alias_kind":"arxiv_version","alias_value":"1805.10923v1","created_at":"2026-05-18T00:14:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1805.10923","created_at":"2026-05-18T00:14:48Z"},{"alias_kind":"pith_short_12","alias_value":"PJRAK3JWE7SD","created_at":"2026-05-18T12:32:43Z"},{"alias_kind":"pith_short_16","alias_value":"PJRAK3JWE7SD747R","created_at":"2026-05-18T12:32:43Z"},{"alias_kind":"pith_short_8","alias_value":"PJRAK3JW","created_at":"2026-05-18T12:32:43Z"}],"graph_snapshots":[{"event_id":"sha256:1b4c75b2500354f8ce61cd9059e99ab3cf54f94311bd8f4a55c3dac4c916f3a5","target":"graph","created_at":"2026-05-18T00:14:48Z","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":"We study the Castelnuovo-Mumford regularity of the vanishing ideal over a bipartite graph endowed with a decomposition of its edge set. We prove that, under certain conditions, the regularity of the vanishing ideal over a bipartite graph obtained from a graph by attaching a path of length $\\ell$ increases by $\\lfloor \\frac{\\ell}{2}\\rfloor (q-2)$, where $q$ is the order of the field of coefficients. We use this result to show that the regularity of the vanishing ideal over a bipartite graph, $G$, endowed with a weak nested ear decomposition is equal to $$\\textstyle \\frac{|V_G|+ \\epsilon -3}{2}(","authors_text":"Jorge Neves","cross_cats":["math.AG","math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2018-05-28T13:57:57Z","title":"Regularity of the vanishing ideal over a bipartite nested ear decomposition"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.10923","kind":"arxiv","version":1},"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":"sha256:841f940b9ad013082c75eff195f2d07c19af4f71dedf65f6025ba5ccb48b0c97","target":"record","created_at":"2026-05-18T00:14:48Z","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":"11f2b9e73eefa7c369aa477a1de25161e7b24bf15e13f3ccf49c04f6914aa4ac","cross_cats_sorted":["math.AG","math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2018-05-28T13:57:57Z","title_canon_sha256":"a89fc76cb6892ee2e23acfd4a74a76eed317930593694d6919860b2a8cd431c4"},"schema_version":"1.0","source":{"id":"1805.10923","kind":"arxiv","version":1}},"canonical_sha256":"7a62056d3627e43ff3f1d9ae53caac7dd805c2370d22c0100bc1d8bfadc168ea","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7a62056d3627e43ff3f1d9ae53caac7dd805c2370d22c0100bc1d8bfadc168ea","first_computed_at":"2026-05-18T00:14:48.805561Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:14:48.805561Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"4hr2B3nEFQK/tpSy9H847WBug/ubQMZtgvYh5kNv7uDNbApNsNwuPemhmg3aiEZ30oGggV95hxkBIpsfrQLvAw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:14:48.806262Z","signed_message":"canonical_sha256_bytes"},"source_id":"1805.10923","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:841f940b9ad013082c75eff195f2d07c19af4f71dedf65f6025ba5ccb48b0c97","sha256:1b4c75b2500354f8ce61cd9059e99ab3cf54f94311bd8f4a55c3dac4c916f3a5"],"state_sha256":"7d9ef69118f95d3b8d9cd609ae70ebe4ad4ff20f730228230241a5be10b86916"}