{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:EJUPYNTEI3RKSHBGLXQD2MSGX6","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":"6080d6d89001f3e89d11201d6c72fcc8cc4709ee512f5dba29bfa1dddde67272","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2013-03-10T09:59:02Z","title_canon_sha256":"b8a31523bd6af11d7261752f9969e03b2f78afe83f6f9cfab3d94713c65c4159"},"schema_version":"1.0","source":{"id":"1303.2303","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1303.2303","created_at":"2026-05-18T00:52:29Z"},{"alias_kind":"arxiv_version","alias_value":"1303.2303v4","created_at":"2026-05-18T00:52:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1303.2303","created_at":"2026-05-18T00:52:29Z"},{"alias_kind":"pith_short_12","alias_value":"EJUPYNTEI3RK","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_16","alias_value":"EJUPYNTEI3RKSHBG","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_8","alias_value":"EJUPYNTE","created_at":"2026-05-18T12:27:43Z"}],"graph_snapshots":[{"event_id":"sha256:4c32120e7899249657dbae907f0490005b9001014a7c84275f485c892bc95201","target":"graph","created_at":"2026-05-18T00:52:29Z","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":"Let $L\\subset \\mathbb{Z}^n$ be a lattice and $I_L=\\langle x^{\\bf u}-x^{\\bf v}:\\ {\\bf u}-{\\bf v}\\in L\\rangle$ be the corresponding lattice ideal in $\\Bbbk[x_1,\\ldots, x_n]$, where $\\Bbbk$ is a field. In this paper we describe minimal binomial generating sets of $I_L$ and their invariants. We use as a main tool a graph construction on equivalence classes of fibers of $I_L$. As one application of the theory developed we characterize binomial complete intersection lattice ideals, a longstanding open problem in the case of non-positive lattices.","authors_text":"Apostolos Thoma, Hara Charalambous, Marius Vladoiu","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2013-03-10T09:59:02Z","title":"Minimal Generating Sets of Lattice Ideals"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1303.2303","kind":"arxiv","version":4},"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":"sha256:06576424c1cad6c194b86c1484ea7d2fae3cad9ea246f3a717c2db4dfb1a41aa","target":"record","created_at":"2026-05-18T00:52:29Z","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":"6080d6d89001f3e89d11201d6c72fcc8cc4709ee512f5dba29bfa1dddde67272","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2013-03-10T09:59:02Z","title_canon_sha256":"b8a31523bd6af11d7261752f9969e03b2f78afe83f6f9cfab3d94713c65c4159"},"schema_version":"1.0","source":{"id":"1303.2303","kind":"arxiv","version":4}},"canonical_sha256":"2268fc366446e2a91c265de03d3246bfbf22bef14f4c42c6d5656080d6a44613","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2268fc366446e2a91c265de03d3246bfbf22bef14f4c42c6d5656080d6a44613","first_computed_at":"2026-05-18T00:52:29.522450Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:52:29.522450Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Sxe3wSK5lrsFsUDAbTodzPuUjOs9fkq4Z8iRJrNEu1MDmwF3WXyzpzgeOjXtI7yH6y8lQFSSsrUbh9bYxyzQCw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:52:29.522919Z","signed_message":"canonical_sha256_bytes"},"source_id":"1303.2303","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:06576424c1cad6c194b86c1484ea7d2fae3cad9ea246f3a717c2db4dfb1a41aa","sha256:4c32120e7899249657dbae907f0490005b9001014a7c84275f485c892bc95201"],"state_sha256":"1537a09d12eadb46adf13258c103690d7245bc5fd2dac5e8fd38a118093b363e"}