{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2020:KR4X4MTBCN4IQQFI5QSYBMYV6L","short_pith_number":"pith:KR4X4MTB","canonical_record":{"source":{"id":"2006.01920","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2020-06-02T20:02:18Z","cross_cats_sorted":[],"title_canon_sha256":"3f38bedbc1fd3430566da900458fdd6b3f88e35150d50f02fd276b70058c7c5f","abstract_canon_sha256":"107514ce4c95d68abd9759bb5864c9a66ffec05ac11473e24bacdefabd520d40"},"schema_version":"1.0"},"canonical_sha256":"54797e326113788840a8ec2580b315f2f0fd524d2d26a04674945eac055b1666","source":{"kind":"arxiv","id":"2006.01920","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2006.01920","created_at":"2026-07-05T05:48:29Z"},{"alias_kind":"arxiv_version","alias_value":"2006.01920v2","created_at":"2026-07-05T05:48:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2006.01920","created_at":"2026-07-05T05:48:29Z"},{"alias_kind":"pith_short_12","alias_value":"KR4X4MTBCN4I","created_at":"2026-07-05T05:48:29Z"},{"alias_kind":"pith_short_16","alias_value":"KR4X4MTBCN4IQQFI","created_at":"2026-07-05T05:48:29Z"},{"alias_kind":"pith_short_8","alias_value":"KR4X4MTB","created_at":"2026-07-05T05:48:29Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2020:KR4X4MTBCN4IQQFI5QSYBMYV6L","target":"record","payload":{"canonical_record":{"source":{"id":"2006.01920","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2020-06-02T20:02:18Z","cross_cats_sorted":[],"title_canon_sha256":"3f38bedbc1fd3430566da900458fdd6b3f88e35150d50f02fd276b70058c7c5f","abstract_canon_sha256":"107514ce4c95d68abd9759bb5864c9a66ffec05ac11473e24bacdefabd520d40"},"schema_version":"1.0"},"canonical_sha256":"54797e326113788840a8ec2580b315f2f0fd524d2d26a04674945eac055b1666","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T05:48:29.731475Z","signature_b64":"40rSOaFwuUyibGf1NH+NrXTA+LeOjP3j/qtZDtlKMnH2bSmFeGbqkzjaj6o5aMw5oQR1XoCYFAdlZUHErB/FCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"54797e326113788840a8ec2580b315f2f0fd524d2d26a04674945eac055b1666","last_reissued_at":"2026-07-05T05:48:29.731044Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T05:48:29.731044Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2006.01920","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T05:48:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9BZOu/ORSwAe/+eTCV3JcfG1baZxbVP3X0LlqFVUENFUvGTfuLnClqY5qpkIM0YMHq5JVfxXeujepnRm7DqvAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-14T01:29:21.668008Z"},"content_sha256":"f5747ce51bc835bd5696dae71d10ce5e1ea36ed9a1d27191af01574a553b21ed","schema_version":"1.0","event_id":"sha256:f5747ce51bc835bd5696dae71d10ce5e1ea36ed9a1d27191af01574a553b21ed"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2020:KR4X4MTBCN4IQQFI5QSYBMYV6L","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Multivariate volume, Ehrhart, and $h^*$-polynomials of polytropes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Leon Zhang, Marie-Charlotte Brandenburg, Sophia Elia","submitted_at":"2020-06-02T20:02:18Z","abstract_excerpt":"The univariate Ehrhart and $h^*$-polynomials of lattice polytopes have been widely studied. We describe methods from toric geometry for computing multivariate versions of volume, Ehrhart and $h^*$-polynomials of lattice polytropes, which are both tropically and classically convex. These algorithms are applied to all polytropes of dimensions 2,3 and 4, yielding a large class of integer polynomials. We give a complete combinatorial description of the coefficients of volume polynomials of 3-dimensional polytropes in terms of regular central subdivisions of the fundamental polytope. Finally, we pr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2006.01920","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2006.01920/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T05:48:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fITs34rS3Yuv5QTfQvgBnkkhIVfmbbLRBpby0Lm2NFl2fTezEblPXWFasYJ2Eou7fzLgzOy4fKkCWTVvmfE9Dw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-14T01:29:21.668387Z"},"content_sha256":"ead5970c7d6c39dff3455df3fe880fb7aeb13669d3bc6c850c7ce74ecdb78bdc","schema_version":"1.0","event_id":"sha256:ead5970c7d6c39dff3455df3fe880fb7aeb13669d3bc6c850c7ce74ecdb78bdc"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/KR4X4MTBCN4IQQFI5QSYBMYV6L/bundle.json","state_url":"https://pith.science/pith/KR4X4MTBCN4IQQFI5QSYBMYV6L/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/KR4X4MTBCN4IQQFI5QSYBMYV6L/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":"2026-08-14T01:29:21Z","links":{"resolver":"https://pith.science/pith/KR4X4MTBCN4IQQFI5QSYBMYV6L","bundle":"https://pith.science/pith/KR4X4MTBCN4IQQFI5QSYBMYV6L/bundle.json","state":"https://pith.science/pith/KR4X4MTBCN4IQQFI5QSYBMYV6L/state.json","well_known_bundle":"https://pith.science/.well-known/pith/KR4X4MTBCN4IQQFI5QSYBMYV6L/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2020:KR4X4MTBCN4IQQFI5QSYBMYV6L","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":"107514ce4c95d68abd9759bb5864c9a66ffec05ac11473e24bacdefabd520d40","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2020-06-02T20:02:18Z","title_canon_sha256":"3f38bedbc1fd3430566da900458fdd6b3f88e35150d50f02fd276b70058c7c5f"},"schema_version":"1.0","source":{"id":"2006.01920","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2006.01920","created_at":"2026-07-05T05:48:29Z"},{"alias_kind":"arxiv_version","alias_value":"2006.01920v2","created_at":"2026-07-05T05:48:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2006.01920","created_at":"2026-07-05T05:48:29Z"},{"alias_kind":"pith_short_12","alias_value":"KR4X4MTBCN4I","created_at":"2026-07-05T05:48:29Z"},{"alias_kind":"pith_short_16","alias_value":"KR4X4MTBCN4IQQFI","created_at":"2026-07-05T05:48:29Z"},{"alias_kind":"pith_short_8","alias_value":"KR4X4MTB","created_at":"2026-07-05T05:48:29Z"}],"graph_snapshots":[{"event_id":"sha256:ead5970c7d6c39dff3455df3fe880fb7aeb13669d3bc6c850c7ce74ecdb78bdc","target":"graph","created_at":"2026-07-05T05:48:29Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2006.01920/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The univariate Ehrhart and $h^*$-polynomials of lattice polytopes have been widely studied. We describe methods from toric geometry for computing multivariate versions of volume, Ehrhart and $h^*$-polynomials of lattice polytropes, which are both tropically and classically convex. These algorithms are applied to all polytropes of dimensions 2,3 and 4, yielding a large class of integer polynomials. We give a complete combinatorial description of the coefficients of volume polynomials of 3-dimensional polytropes in terms of regular central subdivisions of the fundamental polytope. Finally, we pr","authors_text":"Leon Zhang, Marie-Charlotte Brandenburg, Sophia Elia","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2020-06-02T20:02:18Z","title":"Multivariate volume, Ehrhart, and $h^*$-polynomials of polytropes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2006.01920","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:f5747ce51bc835bd5696dae71d10ce5e1ea36ed9a1d27191af01574a553b21ed","target":"record","created_at":"2026-07-05T05:48:29Z","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":"107514ce4c95d68abd9759bb5864c9a66ffec05ac11473e24bacdefabd520d40","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2020-06-02T20:02:18Z","title_canon_sha256":"3f38bedbc1fd3430566da900458fdd6b3f88e35150d50f02fd276b70058c7c5f"},"schema_version":"1.0","source":{"id":"2006.01920","kind":"arxiv","version":2}},"canonical_sha256":"54797e326113788840a8ec2580b315f2f0fd524d2d26a04674945eac055b1666","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"54797e326113788840a8ec2580b315f2f0fd524d2d26a04674945eac055b1666","first_computed_at":"2026-07-05T05:48:29.731044Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T05:48:29.731044Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"40rSOaFwuUyibGf1NH+NrXTA+LeOjP3j/qtZDtlKMnH2bSmFeGbqkzjaj6o5aMw5oQR1XoCYFAdlZUHErB/FCA==","signature_status":"signed_v1","signed_at":"2026-07-05T05:48:29.731475Z","signed_message":"canonical_sha256_bytes"},"source_id":"2006.01920","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f5747ce51bc835bd5696dae71d10ce5e1ea36ed9a1d27191af01574a553b21ed","sha256:ead5970c7d6c39dff3455df3fe880fb7aeb13669d3bc6c850c7ce74ecdb78bdc"],"state_sha256":"710a7ffcb1f2d9f8dfbf733cf6c6dbb7bb393daecdc169bdb3cae2087921fe6d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VYFTdKTbs7K9Ggk/p4fOixUKf3gqrzoLWSkbXQfdiH70iGNB/7D8/URMnDNi9aYy3JerfY8y4DQPHNX7YsBnBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-14T01:29:21.671147Z","bundle_sha256":"3afc3a981a8b6ee79124a659482eaffbdda7c80374b071aa4af0d14072101df6"}}