{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:GH4D6FSGEWZZZHWCXFCBNJKL6X","short_pith_number":"pith:GH4D6FSG","canonical_record":{"source":{"id":"1310.7979","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2013-10-29T22:48:13Z","cross_cats_sorted":["math.AC"],"title_canon_sha256":"edf34879a386106deb38fec8df48e9d09fe8d832ffc98336982bd9b8ab75397b","abstract_canon_sha256":"380d02622ca86b65e9ee5737ad2e20ce1a1d49c4da34780140a359d5818cd510"},"schema_version":"1.0"},"canonical_sha256":"31f83f164625b39c9ec2b94416a54bf5d0cfcdce673fe0b00c5c36db55a6165b","source":{"kind":"arxiv","id":"1310.7979","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1310.7979","created_at":"2026-05-18T02:12:11Z"},{"alias_kind":"arxiv_version","alias_value":"1310.7979v2","created_at":"2026-05-18T02:12:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1310.7979","created_at":"2026-05-18T02:12:11Z"},{"alias_kind":"pith_short_12","alias_value":"GH4D6FSGEWZZ","created_at":"2026-05-18T12:27:45Z"},{"alias_kind":"pith_short_16","alias_value":"GH4D6FSGEWZZZHWC","created_at":"2026-05-18T12:27:45Z"},{"alias_kind":"pith_short_8","alias_value":"GH4D6FSG","created_at":"2026-05-18T12:27:45Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:GH4D6FSGEWZZZHWCXFCBNJKL6X","target":"record","payload":{"canonical_record":{"source":{"id":"1310.7979","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2013-10-29T22:48:13Z","cross_cats_sorted":["math.AC"],"title_canon_sha256":"edf34879a386106deb38fec8df48e9d09fe8d832ffc98336982bd9b8ab75397b","abstract_canon_sha256":"380d02622ca86b65e9ee5737ad2e20ce1a1d49c4da34780140a359d5818cd510"},"schema_version":"1.0"},"canonical_sha256":"31f83f164625b39c9ec2b94416a54bf5d0cfcdce673fe0b00c5c36db55a6165b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:12:11.675840Z","signature_b64":"GHBM1sF2IUyb9jR2sMgaN2ot6hUCx5M2N/g/Gp9s62qY4qZwRfTTXgNz5AmrshIqxXAOVvNZheuM0dVnGn7MCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"31f83f164625b39c9ec2b94416a54bf5d0cfcdce673fe0b00c5c36db55a6165b","last_reissued_at":"2026-05-18T02:12:11.674945Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:12:11.674945Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1310.7979","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":"2026-05-18T02:12:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Rc0BH2EJ9uYjq98xnZPnTkX4bvTn1HgyBS7/pKdya//TvEnw5P1bPEvwsfhdBEwhOHeKlmqgjb+Q4kZV9iWtBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T10:46:11.382570Z"},"content_sha256":"6dd1ae0f4bd25c1cc2cbbecc21d93ed6e50e09c370bfc70b987de08b840f316a","schema_version":"1.0","event_id":"sha256:6dd1ae0f4bd25c1cc2cbbecc21d93ed6e50e09c370bfc70b987de08b840f316a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:GH4D6FSGEWZZZHWCXFCBNJKL6X","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On mixed multiplicities of ideals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC"],"primary_cat":"math.AG","authors_text":"A. G. Khovanskii, Kiumars Kaveh","submitted_at":"2013-10-29T22:48:13Z","abstract_excerpt":"Let R be the local ring of a point on a variety X over an algebraically closed field k. We make a connection between the notion of mixed (Samuel) multiplicity of m-primary ideals in R and intersection theory of subspaces of rational functions on X which deals with the number of solutions of systems of equations. From this we readily deduce several properties of mixed multiplicities. In particular, we prove a (reverse) Alexandrov-Fenchel inequality for mixed multiplicities due to Teissier and Rees-Sharp. As an application in convex geometry we obtain a proof of a (reverse) Alexandrov-Fenchel in"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.7979","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":""},"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":"2026-05-18T02:12:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"um6GcQdOQCYelx9hH29Zo0QMux0A66bQWqBJxEBvH5JNKMAiQYD1irPkunrj2Wrw+QjXSeqAFzX6LDrvr983CA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T10:46:11.383260Z"},"content_sha256":"d5d490aacc095d9762263511beed863637efa4133f842a6e15b139b0d79173cb","schema_version":"1.0","event_id":"sha256:d5d490aacc095d9762263511beed863637efa4133f842a6e15b139b0d79173cb"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GH4D6FSGEWZZZHWCXFCBNJKL6X/bundle.json","state_url":"https://pith.science/pith/GH4D6FSGEWZZZHWCXFCBNJKL6X/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GH4D6FSGEWZZZHWCXFCBNJKL6X/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":"2026-06-10T10:46:11Z","links":{"resolver":"https://pith.science/pith/GH4D6FSGEWZZZHWCXFCBNJKL6X","bundle":"https://pith.science/pith/GH4D6FSGEWZZZHWCXFCBNJKL6X/bundle.json","state":"https://pith.science/pith/GH4D6FSGEWZZZHWCXFCBNJKL6X/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GH4D6FSGEWZZZHWCXFCBNJKL6X/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:GH4D6FSGEWZZZHWCXFCBNJKL6X","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":"380d02622ca86b65e9ee5737ad2e20ce1a1d49c4da34780140a359d5818cd510","cross_cats_sorted":["math.AC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2013-10-29T22:48:13Z","title_canon_sha256":"edf34879a386106deb38fec8df48e9d09fe8d832ffc98336982bd9b8ab75397b"},"schema_version":"1.0","source":{"id":"1310.7979","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1310.7979","created_at":"2026-05-18T02:12:11Z"},{"alias_kind":"arxiv_version","alias_value":"1310.7979v2","created_at":"2026-05-18T02:12:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1310.7979","created_at":"2026-05-18T02:12:11Z"},{"alias_kind":"pith_short_12","alias_value":"GH4D6FSGEWZZ","created_at":"2026-05-18T12:27:45Z"},{"alias_kind":"pith_short_16","alias_value":"GH4D6FSGEWZZZHWC","created_at":"2026-05-18T12:27:45Z"},{"alias_kind":"pith_short_8","alias_value":"GH4D6FSG","created_at":"2026-05-18T12:27:45Z"}],"graph_snapshots":[{"event_id":"sha256:d5d490aacc095d9762263511beed863637efa4133f842a6e15b139b0d79173cb","target":"graph","created_at":"2026-05-18T02:12:11Z","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 R be the local ring of a point on a variety X over an algebraically closed field k. We make a connection between the notion of mixed (Samuel) multiplicity of m-primary ideals in R and intersection theory of subspaces of rational functions on X which deals with the number of solutions of systems of equations. From this we readily deduce several properties of mixed multiplicities. In particular, we prove a (reverse) Alexandrov-Fenchel inequality for mixed multiplicities due to Teissier and Rees-Sharp. As an application in convex geometry we obtain a proof of a (reverse) Alexandrov-Fenchel in","authors_text":"A. G. Khovanskii, Kiumars Kaveh","cross_cats":["math.AC"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2013-10-29T22:48:13Z","title":"On mixed multiplicities of ideals"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.7979","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":"sha256:6dd1ae0f4bd25c1cc2cbbecc21d93ed6e50e09c370bfc70b987de08b840f316a","target":"record","created_at":"2026-05-18T02:12:11Z","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":"380d02622ca86b65e9ee5737ad2e20ce1a1d49c4da34780140a359d5818cd510","cross_cats_sorted":["math.AC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2013-10-29T22:48:13Z","title_canon_sha256":"edf34879a386106deb38fec8df48e9d09fe8d832ffc98336982bd9b8ab75397b"},"schema_version":"1.0","source":{"id":"1310.7979","kind":"arxiv","version":2}},"canonical_sha256":"31f83f164625b39c9ec2b94416a54bf5d0cfcdce673fe0b00c5c36db55a6165b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"31f83f164625b39c9ec2b94416a54bf5d0cfcdce673fe0b00c5c36db55a6165b","first_computed_at":"2026-05-18T02:12:11.674945Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:12:11.674945Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"GHBM1sF2IUyb9jR2sMgaN2ot6hUCx5M2N/g/Gp9s62qY4qZwRfTTXgNz5AmrshIqxXAOVvNZheuM0dVnGn7MCg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:12:11.675840Z","signed_message":"canonical_sha256_bytes"},"source_id":"1310.7979","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6dd1ae0f4bd25c1cc2cbbecc21d93ed6e50e09c370bfc70b987de08b840f316a","sha256:d5d490aacc095d9762263511beed863637efa4133f842a6e15b139b0d79173cb"],"state_sha256":"f23e7fd4a43d5767021d6b855e1e6d5610ebe5e739f3fb1fa432621747ed66ae"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nUrlqySFnP9wrGkCM2imkp6b5XcMcND/ittGJJamwljJuF9soThrW1dOVtU0/jgIZj8iKzRdsF5tO+vB0pRDAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T10:46:11.388697Z","bundle_sha256":"87f2fc286cd346af23ab10a59613b08ab30a0dee0091a2ee2a408540d1331752"}}