{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:QXIOEC6HUWPFPXLMNXDFLNGQZ2","short_pith_number":"pith:QXIOEC6H","canonical_record":{"source":{"id":"0912.1502","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2009-12-08T14:19:36Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"ef3ebff065d04f4fca0f0781abbd3ca616d03936178924f3a02e9626aac7b48e","abstract_canon_sha256":"e8d91d2e5097c3f1325d598dbd54661494aaa81332f80c404865fb44c46e8e31"},"schema_version":"1.0"},"canonical_sha256":"85d0e20bc7a59e57dd6c6dc655b4d0ce9b3eaeae506176b78555b755e0acfc25","source":{"kind":"arxiv","id":"0912.1502","version":5},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0912.1502","created_at":"2026-05-18T01:01:29Z"},{"alias_kind":"arxiv_version","alias_value":"0912.1502v5","created_at":"2026-05-18T01:01:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0912.1502","created_at":"2026-05-18T01:01:29Z"},{"alias_kind":"pith_short_12","alias_value":"QXIOEC6HUWPF","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_16","alias_value":"QXIOEC6HUWPFPXLM","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_8","alias_value":"QXIOEC6H","created_at":"2026-05-18T12:26:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:QXIOEC6HUWPFPXLMNXDFLNGQZ2","target":"record","payload":{"canonical_record":{"source":{"id":"0912.1502","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2009-12-08T14:19:36Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"ef3ebff065d04f4fca0f0781abbd3ca616d03936178924f3a02e9626aac7b48e","abstract_canon_sha256":"e8d91d2e5097c3f1325d598dbd54661494aaa81332f80c404865fb44c46e8e31"},"schema_version":"1.0"},"canonical_sha256":"85d0e20bc7a59e57dd6c6dc655b4d0ce9b3eaeae506176b78555b755e0acfc25","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:01:29.781456Z","signature_b64":"lJUS4RoYC8poA4ML7/lbdYR458/zgO2v4jdCdrmbeGy7s/dtjRYxZKembn3yszFRo+8oEE/d1SQs/fOEOVtlBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"85d0e20bc7a59e57dd6c6dc655b4d0ce9b3eaeae506176b78555b755e0acfc25","last_reissued_at":"2026-05-18T01:01:29.781011Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:01:29.781011Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0912.1502","source_version":5,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:01:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5JvQvrt2QI3RWddZPnc6FnmRxcX9dcsdzGfYVPdhZ32HGzZQQOW2eYBD3Ppp+uyCURVnOm6VRlO3MBSZ+EG0BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-24T00:48:11.161603Z"},"content_sha256":"cefa8c0c62767d1599caaab549e78372e0c7f2bc3676fc73ba44996ddf47fe7e","schema_version":"1.0","event_id":"sha256:cefa8c0c62767d1599caaab549e78372e0c7f2bc3676fc73ba44996ddf47fe7e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:QXIOEC6HUWPFPXLMNXDFLNGQZ2","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Border bases and order ideals: a polyhedral characterization","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.AC","authors_text":"G\\'abor Braun, Sebastian Pokutta","submitted_at":"2009-12-08T14:19:36Z","abstract_excerpt":"Border bases arise as a canonical generalization of Gr\\\"obner bases. We provide a polyhedral characterization of all order ideals (and hence border bases) that are supported by a zero-dimensional ideal: order ideals that support a border basis correspond one-to-one to integral points of the order ideal polytope. In particular, we establish a crucial connection between the ideal and its combinatorial structure. Based on this characterization we adapt the classical border basis algorithm to allow for computing border bases for arbitrary order ideals, which are independent of term orderings. We a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0912.1502","kind":"arxiv","version":5},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:01:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CqITBvPDOj2mIY4bA3U1MKoog6r89neR9y18dhsM4zJNshiS/7nzzYa2bsJWBX/VM9e8/YWqSWd59kvgShBrBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-24T00:48:11.162267Z"},"content_sha256":"ee0dc4c0ed2c44a5c3b359c04fa6406394145af8c992deaaac5d13015ae469e6","schema_version":"1.0","event_id":"sha256:ee0dc4c0ed2c44a5c3b359c04fa6406394145af8c992deaaac5d13015ae469e6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QXIOEC6HUWPFPXLMNXDFLNGQZ2/bundle.json","state_url":"https://pith.science/pith/QXIOEC6HUWPFPXLMNXDFLNGQZ2/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QXIOEC6HUWPFPXLMNXDFLNGQZ2/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-24T00:48:11Z","links":{"resolver":"https://pith.science/pith/QXIOEC6HUWPFPXLMNXDFLNGQZ2","bundle":"https://pith.science/pith/QXIOEC6HUWPFPXLMNXDFLNGQZ2/bundle.json","state":"https://pith.science/pith/QXIOEC6HUWPFPXLMNXDFLNGQZ2/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QXIOEC6HUWPFPXLMNXDFLNGQZ2/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:QXIOEC6HUWPFPXLMNXDFLNGQZ2","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":"e8d91d2e5097c3f1325d598dbd54661494aaa81332f80c404865fb44c46e8e31","cross_cats_sorted":["math.AG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2009-12-08T14:19:36Z","title_canon_sha256":"ef3ebff065d04f4fca0f0781abbd3ca616d03936178924f3a02e9626aac7b48e"},"schema_version":"1.0","source":{"id":"0912.1502","kind":"arxiv","version":5}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0912.1502","created_at":"2026-05-18T01:01:29Z"},{"alias_kind":"arxiv_version","alias_value":"0912.1502v5","created_at":"2026-05-18T01:01:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0912.1502","created_at":"2026-05-18T01:01:29Z"},{"alias_kind":"pith_short_12","alias_value":"QXIOEC6HUWPF","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_16","alias_value":"QXIOEC6HUWPFPXLM","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_8","alias_value":"QXIOEC6H","created_at":"2026-05-18T12:26:01Z"}],"graph_snapshots":[{"event_id":"sha256:ee0dc4c0ed2c44a5c3b359c04fa6406394145af8c992deaaac5d13015ae469e6","target":"graph","created_at":"2026-05-18T01:01:29Z","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":"Border bases arise as a canonical generalization of Gr\\\"obner bases. We provide a polyhedral characterization of all order ideals (and hence border bases) that are supported by a zero-dimensional ideal: order ideals that support a border basis correspond one-to-one to integral points of the order ideal polytope. In particular, we establish a crucial connection between the ideal and its combinatorial structure. Based on this characterization we adapt the classical border basis algorithm to allow for computing border bases for arbitrary order ideals, which are independent of term orderings. We a","authors_text":"G\\'abor Braun, Sebastian Pokutta","cross_cats":["math.AG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2009-12-08T14:19:36Z","title":"Border bases and order ideals: a polyhedral characterization"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0912.1502","kind":"arxiv","version":5},"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:cefa8c0c62767d1599caaab549e78372e0c7f2bc3676fc73ba44996ddf47fe7e","target":"record","created_at":"2026-05-18T01:01:29Z","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":"e8d91d2e5097c3f1325d598dbd54661494aaa81332f80c404865fb44c46e8e31","cross_cats_sorted":["math.AG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2009-12-08T14:19:36Z","title_canon_sha256":"ef3ebff065d04f4fca0f0781abbd3ca616d03936178924f3a02e9626aac7b48e"},"schema_version":"1.0","source":{"id":"0912.1502","kind":"arxiv","version":5}},"canonical_sha256":"85d0e20bc7a59e57dd6c6dc655b4d0ce9b3eaeae506176b78555b755e0acfc25","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"85d0e20bc7a59e57dd6c6dc655b4d0ce9b3eaeae506176b78555b755e0acfc25","first_computed_at":"2026-05-18T01:01:29.781011Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:01:29.781011Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"lJUS4RoYC8poA4ML7/lbdYR458/zgO2v4jdCdrmbeGy7s/dtjRYxZKembn3yszFRo+8oEE/d1SQs/fOEOVtlBA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:01:29.781456Z","signed_message":"canonical_sha256_bytes"},"source_id":"0912.1502","source_kind":"arxiv","source_version":5}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:cefa8c0c62767d1599caaab549e78372e0c7f2bc3676fc73ba44996ddf47fe7e","sha256:ee0dc4c0ed2c44a5c3b359c04fa6406394145af8c992deaaac5d13015ae469e6"],"state_sha256":"12fa1d7bb9d70a3edab97276fa4f814a0b37c3908c80ebcce3781ab0c49de600"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"K3uRHv9HtmSBKq4WmF/f1ikEzVUBGAOVH5cak7ZHVhiDvGNw1oiZNfdpnyLb+Zm1KLKelIM9jv0nc4mRhUx/AA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-24T00:48:11.165717Z","bundle_sha256":"f61b1955252900ff1531e41b7b2a06a7acca963b643592ef1fd7c7861c8bd4e5"}}