{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2008:SGPT27D5SPMMCBY4YSVB4PTCN2","short_pith_number":"pith:SGPT27D5","canonical_record":{"source":{"id":"0809.2083","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2008-09-11T19:00:12Z","cross_cats_sorted":["cs.CC","cs.SC"],"title_canon_sha256":"e443037aeebf57c875e1a6c704ff1b391d9b30e1f6b725139b4dc57b4cfe7376","abstract_canon_sha256":"62d6ec9b89602b729898751aa8bc20fcaac99be8c019f9b604bc429a846d536c"},"schema_version":"1.0"},"canonical_sha256":"919f3d7c7d93d8c1071cc4aa1e3e626ea77e44fd71be4bae59e15c88ffa1eebb","source":{"kind":"arxiv","id":"0809.2083","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0809.2083","created_at":"2026-05-18T03:19:57Z"},{"alias_kind":"arxiv_version","alias_value":"0809.2083v3","created_at":"2026-05-18T03:19:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0809.2083","created_at":"2026-05-18T03:19:57Z"},{"alias_kind":"pith_short_12","alias_value":"SGPT27D5SPMM","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"SGPT27D5SPMMCBY4","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"SGPT27D5","created_at":"2026-05-18T12:25:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2008:SGPT27D5SPMMCBY4YSVB4PTCN2","target":"record","payload":{"canonical_record":{"source":{"id":"0809.2083","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2008-09-11T19:00:12Z","cross_cats_sorted":["cs.CC","cs.SC"],"title_canon_sha256":"e443037aeebf57c875e1a6c704ff1b391d9b30e1f6b725139b4dc57b4cfe7376","abstract_canon_sha256":"62d6ec9b89602b729898751aa8bc20fcaac99be8c019f9b604bc429a846d536c"},"schema_version":"1.0"},"canonical_sha256":"919f3d7c7d93d8c1071cc4aa1e3e626ea77e44fd71be4bae59e15c88ffa1eebb","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:19:57.213241Z","signature_b64":"4SPhiff8xEMYsduk6r2hWzk4GegI0Kt7r0zrGSKyEck+mTh87U49KHzALxtcyLW2x6EfVteTQcj3j5T/HKPdAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"919f3d7c7d93d8c1071cc4aa1e3e626ea77e44fd71be4bae59e15c88ffa1eebb","last_reissued_at":"2026-05-18T03:19:57.212519Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:19:57.212519Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0809.2083","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:19:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"43VFQR2DpUy+JpSTE+lagFq7O761t3+2LP35OxDao/8BUgOSiOvV6pre24U/P2thne3mgQbMCQ0Xaph7lX90BQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T14:03:17.062690Z"},"content_sha256":"fce744a6df4a4c90828ff256b2b38c79f1af06eeea993960e6b2037ff88661d5","schema_version":"1.0","event_id":"sha256:fce744a6df4a4c90828ff256b2b38c79f1af06eeea993960e6b2037ff88661d5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2008:SGPT27D5SPMMCBY4YSVB4PTCN2","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"How to Integrate a Polynomial over a Simplex","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC","cs.SC"],"primary_cat":"math.MG","authors_text":"Jesus De Loera, Matthias K\\\"oppe, Mich\\`ele Vergne (CMLS-EcolePolytechnique), Nicole Berline (CMLS-EcolePolytechnique), Velleda Baldoni","submitted_at":"2008-09-11T19:00:12Z","abstract_excerpt":"This paper settles the computational complexity of the problem of integrating a polynomial function f over a rational simplex. We prove that the problem is NP-hard for arbitrary polynomials via a generalization of a theorem of Motzkin and Straus. On the other hand, if the polynomial depends only on a fixed number of variables, while its degree and the dimension of the simplex are allowed to vary, we prove that integration can be done in polynomial time. As a consequence, for polynomials of fixed total degree, there is a polynomial time algorithm as well. We conclude the article with extensions"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0809.2083","kind":"arxiv","version":3},"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":"2026-05-18T03:19:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"C/mKFUh0vNYbS/AuPpvd1X9in3e/s6g8yhCZFg7pGq65k1/rPr3uaMGKJzNbNRvEKW411GPVMhNrZum4zpmcBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T14:03:17.063038Z"},"content_sha256":"d47b13bbad77f06ec637f45c1671896d646497b0ff9c1c25e3d4a79ebcec5401","schema_version":"1.0","event_id":"sha256:d47b13bbad77f06ec637f45c1671896d646497b0ff9c1c25e3d4a79ebcec5401"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/SGPT27D5SPMMCBY4YSVB4PTCN2/bundle.json","state_url":"https://pith.science/pith/SGPT27D5SPMMCBY4YSVB4PTCN2/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/SGPT27D5SPMMCBY4YSVB4PTCN2/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":"2026-06-05T14:03:17Z","links":{"resolver":"https://pith.science/pith/SGPT27D5SPMMCBY4YSVB4PTCN2","bundle":"https://pith.science/pith/SGPT27D5SPMMCBY4YSVB4PTCN2/bundle.json","state":"https://pith.science/pith/SGPT27D5SPMMCBY4YSVB4PTCN2/state.json","well_known_bundle":"https://pith.science/.well-known/pith/SGPT27D5SPMMCBY4YSVB4PTCN2/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2008:SGPT27D5SPMMCBY4YSVB4PTCN2","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":"62d6ec9b89602b729898751aa8bc20fcaac99be8c019f9b604bc429a846d536c","cross_cats_sorted":["cs.CC","cs.SC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2008-09-11T19:00:12Z","title_canon_sha256":"e443037aeebf57c875e1a6c704ff1b391d9b30e1f6b725139b4dc57b4cfe7376"},"schema_version":"1.0","source":{"id":"0809.2083","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0809.2083","created_at":"2026-05-18T03:19:57Z"},{"alias_kind":"arxiv_version","alias_value":"0809.2083v3","created_at":"2026-05-18T03:19:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0809.2083","created_at":"2026-05-18T03:19:57Z"},{"alias_kind":"pith_short_12","alias_value":"SGPT27D5SPMM","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"SGPT27D5SPMMCBY4","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"SGPT27D5","created_at":"2026-05-18T12:25:58Z"}],"graph_snapshots":[{"event_id":"sha256:d47b13bbad77f06ec637f45c1671896d646497b0ff9c1c25e3d4a79ebcec5401","target":"graph","created_at":"2026-05-18T03:19:57Z","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 paper settles the computational complexity of the problem of integrating a polynomial function f over a rational simplex. We prove that the problem is NP-hard for arbitrary polynomials via a generalization of a theorem of Motzkin and Straus. On the other hand, if the polynomial depends only on a fixed number of variables, while its degree and the dimension of the simplex are allowed to vary, we prove that integration can be done in polynomial time. As a consequence, for polynomials of fixed total degree, there is a polynomial time algorithm as well. We conclude the article with extensions","authors_text":"Jesus De Loera, Matthias K\\\"oppe, Mich\\`ele Vergne (CMLS-EcolePolytechnique), Nicole Berline (CMLS-EcolePolytechnique), Velleda Baldoni","cross_cats":["cs.CC","cs.SC"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2008-09-11T19:00:12Z","title":"How to Integrate a Polynomial over a Simplex"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0809.2083","kind":"arxiv","version":3},"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:fce744a6df4a4c90828ff256b2b38c79f1af06eeea993960e6b2037ff88661d5","target":"record","created_at":"2026-05-18T03:19:57Z","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":"62d6ec9b89602b729898751aa8bc20fcaac99be8c019f9b604bc429a846d536c","cross_cats_sorted":["cs.CC","cs.SC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2008-09-11T19:00:12Z","title_canon_sha256":"e443037aeebf57c875e1a6c704ff1b391d9b30e1f6b725139b4dc57b4cfe7376"},"schema_version":"1.0","source":{"id":"0809.2083","kind":"arxiv","version":3}},"canonical_sha256":"919f3d7c7d93d8c1071cc4aa1e3e626ea77e44fd71be4bae59e15c88ffa1eebb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"919f3d7c7d93d8c1071cc4aa1e3e626ea77e44fd71be4bae59e15c88ffa1eebb","first_computed_at":"2026-05-18T03:19:57.212519Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:19:57.212519Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"4SPhiff8xEMYsduk6r2hWzk4GegI0Kt7r0zrGSKyEck+mTh87U49KHzALxtcyLW2x6EfVteTQcj3j5T/HKPdAg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:19:57.213241Z","signed_message":"canonical_sha256_bytes"},"source_id":"0809.2083","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fce744a6df4a4c90828ff256b2b38c79f1af06eeea993960e6b2037ff88661d5","sha256:d47b13bbad77f06ec637f45c1671896d646497b0ff9c1c25e3d4a79ebcec5401"],"state_sha256":"1ffd8126700550d6dc790f735c3c22b6182b0e7cb88fe9c081b85eb6b07f20af"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Z6zrEb3bJHS1LVdUpPa2ZuSMsSI6+KHvKgP2Gsveu8tNXblV3uWU5F+JW9m1boYumym7pFa/rKGpQ+JSATL0Ag==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T14:03:17.065197Z","bundle_sha256":"70123639eae0e93a7533d5addf4433b3b02ccb0983ab3426e862286eaa6507f9"}}