{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:7G5ZVPBGBNF44W7JW6KWIFHPNI","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":"34c6d5deccb947444466d3ee055846a0401f85d2314ac96e2a144ce49f1c4249","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2018-09-20T04:01:06Z","title_canon_sha256":"76563ab40802ff65f860bbfe5862b639d4c3200d61d937dc4833e780dbb54514"},"schema_version":"1.0","source":{"id":"1809.07464","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1809.07464","created_at":"2026-05-18T00:05:16Z"},{"alias_kind":"arxiv_version","alias_value":"1809.07464v1","created_at":"2026-05-18T00:05:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1809.07464","created_at":"2026-05-18T00:05:16Z"},{"alias_kind":"pith_short_12","alias_value":"7G5ZVPBGBNF4","created_at":"2026-05-18T12:32:11Z"},{"alias_kind":"pith_short_16","alias_value":"7G5ZVPBGBNF44W7J","created_at":"2026-05-18T12:32:11Z"},{"alias_kind":"pith_short_8","alias_value":"7G5ZVPBG","created_at":"2026-05-18T12:32:11Z"}],"graph_snapshots":[{"event_id":"sha256:15334cd66d3954508236985f8993ab138084280e5ff25b5fe3c46f87cf2ab0e9","target":"graph","created_at":"2026-05-18T00:05:16Z","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":"This is an exposition of some new results on associated primes and the depth of different kinds of powers of monomial ideals in order to show a deep connection between commutative algebra and some objects in combinatorics such as simplicial complexes, integral points in polytopes and graphs.","authors_text":"Le Tuan Hoa","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2018-09-20T04:01:06Z","title":"Powers of Monomial Ideals and Combinatorics"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.07464","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:e9f68f8cbf58add760254c3ee50d771b1122eea02519946496deac906d9db744","target":"record","created_at":"2026-05-18T00:05:16Z","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":"34c6d5deccb947444466d3ee055846a0401f85d2314ac96e2a144ce49f1c4249","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2018-09-20T04:01:06Z","title_canon_sha256":"76563ab40802ff65f860bbfe5862b639d4c3200d61d937dc4833e780dbb54514"},"schema_version":"1.0","source":{"id":"1809.07464","kind":"arxiv","version":1}},"canonical_sha256":"f9bb9abc260b4bce5be9b7956414ef6a1c73d5023c4c1cb1e243b21c853404f4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f9bb9abc260b4bce5be9b7956414ef6a1c73d5023c4c1cb1e243b21c853404f4","first_computed_at":"2026-05-18T00:05:16.244089Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:05:16.244089Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"j1WZjzxk4XY8T9cFC07JezV0HDg+HHsQ6d9m9HfF1jS/OVBIoqJ1abXqIZ7ntE3m5EMV/aqDohKtNXmQfN6xDw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:05:16.244705Z","signed_message":"canonical_sha256_bytes"},"source_id":"1809.07464","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e9f68f8cbf58add760254c3ee50d771b1122eea02519946496deac906d9db744","sha256:15334cd66d3954508236985f8993ab138084280e5ff25b5fe3c46f87cf2ab0e9"],"state_sha256":"a0b2f78c59ad22ee35ffdacc175fb49d54f2f5aa9dd959c2691d2cf0b6121d8d"}