{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:IHGOQYY52RVR7YENXYAYU7SD6V","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":"491543ad854185a2748d3c49a6229148a19e07163385d40e3a6fe35258e9f5f9","cross_cats_sorted":["cs.DM","math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2012-01-20T18:12:44Z","title_canon_sha256":"a264b88aa98e49444a822405a25aba2cb165e3118beed36a64037c45236baea3"},"schema_version":"1.0","source":{"id":"1201.4357","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1201.4357","created_at":"2026-05-18T03:51:25Z"},{"alias_kind":"arxiv_version","alias_value":"1201.4357v2","created_at":"2026-05-18T03:51:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1201.4357","created_at":"2026-05-18T03:51:25Z"},{"alias_kind":"pith_short_12","alias_value":"IHGOQYY52RVR","created_at":"2026-05-18T12:27:09Z"},{"alias_kind":"pith_short_16","alias_value":"IHGOQYY52RVR7YEN","created_at":"2026-05-18T12:27:09Z"},{"alias_kind":"pith_short_8","alias_value":"IHGOQYY5","created_at":"2026-05-18T12:27:09Z"}],"graph_snapshots":[{"event_id":"sha256:8e2b1517aba90a9a90aad08368e0a68b36ff64622daf4a97c9c2335414850a69","target":"graph","created_at":"2026-05-18T03:51:25Z","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":"The Riemann-Roch theorem on a graph G is related to Alexander duality in combinatorial commutive algebra. We study the lattice ideal given by chip firing on G and the initial ideal whose standard monomials are the G-parking functions. When G is a saturated graph, these ideals are generic and the Scarf complex is a minimal free resolution. Otherwise, syzygies are obtained by degeneration. We also develop a self-contained Riemann-Roch theory for artinian monomial ideals.","authors_text":"Bernd Sturmfels, Madhusudan Manjunath","cross_cats":["cs.DM","math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2012-01-20T18:12:44Z","title":"Monomials, Binomials, and Riemann-Roch"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.4357","kind":"arxiv","version":2},"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":"sha256:6385bf3e0afe5408a2649b3fc062364df8d4ba4fb3f7b92fed5ae4c82c9d7fb5","target":"record","created_at":"2026-05-18T03:51:25Z","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":"491543ad854185a2748d3c49a6229148a19e07163385d40e3a6fe35258e9f5f9","cross_cats_sorted":["cs.DM","math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2012-01-20T18:12:44Z","title_canon_sha256":"a264b88aa98e49444a822405a25aba2cb165e3118beed36a64037c45236baea3"},"schema_version":"1.0","source":{"id":"1201.4357","kind":"arxiv","version":2}},"canonical_sha256":"41cce8631dd46b1fe08dbe018a7e43f575d4240e3852a03ffb16d87f113f27f2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"41cce8631dd46b1fe08dbe018a7e43f575d4240e3852a03ffb16d87f113f27f2","first_computed_at":"2026-05-18T03:51:25.454700Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:51:25.454700Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"AabZ/B8PtaeoLxWxjtK12abdHEbtQuSHIbEisFDfiNs9LOlffedQIcwc6//0flwksoUpliLl3jP4m2GALd0ABA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:51:25.455197Z","signed_message":"canonical_sha256_bytes"},"source_id":"1201.4357","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6385bf3e0afe5408a2649b3fc062364df8d4ba4fb3f7b92fed5ae4c82c9d7fb5","sha256:8e2b1517aba90a9a90aad08368e0a68b36ff64622daf4a97c9c2335414850a69"],"state_sha256":"5b3842abdfad9d288805a9b3b14c1f76af307b76673b437f116b0eadd9b9ccd7"}