{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:HG3JSL4UORASZLB5FSZNGFPNR3","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":"046ddc640b2f676b69ca27f6a90ba492920dcf4bd992f0c219be07a97f083ebc","cross_cats_sorted":["math.AC","math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-12-22T03:53:21Z","title_canon_sha256":"efd5e3f96b23972abd4c4ead1738651601dcebcfbb1fc078de5581a06df1cfd5"},"schema_version":"1.0","source":{"id":"1312.6340","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1312.6340","created_at":"2026-05-18T02:47:24Z"},{"alias_kind":"arxiv_version","alias_value":"1312.6340v2","created_at":"2026-05-18T02:47:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1312.6340","created_at":"2026-05-18T02:47:24Z"},{"alias_kind":"pith_short_12","alias_value":"HG3JSL4UORAS","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_16","alias_value":"HG3JSL4UORASZLB5","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_8","alias_value":"HG3JSL4U","created_at":"2026-05-18T12:27:46Z"}],"graph_snapshots":[{"event_id":"sha256:9e0737d8edb316c15974431098faaae7cb49bdcb6f49a6f685b2953282ae3e1f","target":"graph","created_at":"2026-05-18T02:47:24Z","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 consider the normality or the integer decomposition property (IDP, for short) for Minkowski sums of integral convex polytopes. We discuss some properties on the toric rings associated with Minkowski sums of integral convex polytopes. We also study Minkowski sums of edge polytopes and give a sufficient condition for Minkowski sums of edge polytopes to have IDP.","authors_text":"Akihiro Higashitani","cross_cats":["math.AC","math.MG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-12-22T03:53:21Z","title":"Minkowski sum of polytopes and its normality"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.6340","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":"sha256:4adb17a12842c349d5abd38090d8ae4e91b4c5234e005fe8d3060b776144d1cd","target":"record","created_at":"2026-05-18T02:47:24Z","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":"046ddc640b2f676b69ca27f6a90ba492920dcf4bd992f0c219be07a97f083ebc","cross_cats_sorted":["math.AC","math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-12-22T03:53:21Z","title_canon_sha256":"efd5e3f96b23972abd4c4ead1738651601dcebcfbb1fc078de5581a06df1cfd5"},"schema_version":"1.0","source":{"id":"1312.6340","kind":"arxiv","version":2}},"canonical_sha256":"39b6992f9474412cac3d2cb2d315ed8ed4264e42a6b9d25c1674d41a4eccd9b9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"39b6992f9474412cac3d2cb2d315ed8ed4264e42a6b9d25c1674d41a4eccd9b9","first_computed_at":"2026-05-18T02:47:24.273040Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:47:24.273040Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"HxFgQNYb7fHcSFzkXCmfLktoNTiAqQ/aUvS2nbCAt3tOS0bHaGacoRKsyopjPAqNO8FTj4rDBW/d3wPwdBUsDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:47:24.273415Z","signed_message":"canonical_sha256_bytes"},"source_id":"1312.6340","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4adb17a12842c349d5abd38090d8ae4e91b4c5234e005fe8d3060b776144d1cd","sha256:9e0737d8edb316c15974431098faaae7cb49bdcb6f49a6f685b2953282ae3e1f"],"state_sha256":"c1b0f2d420e12dc1ba2e79737c163088472c3887836c3cd9fba5801bd8a98da3"}