{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:5ASN42RV5SHNSYM2JQILARBUJ2","short_pith_number":"pith:5ASN42RV","canonical_record":{"source":{"id":"1408.2136","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-08-09T17:04:17Z","cross_cats_sorted":["math.AC"],"title_canon_sha256":"e592d606ea7f25af144ae01b1a1d6845e86573aeed08c2a77d675feb5be5ef6a","abstract_canon_sha256":"30d6cf3edbab1193e0d0925ea5f4e0abcb3c89991653d13b766b514ae2c149ca"},"schema_version":"1.0"},"canonical_sha256":"e824de6a35ec8ed9619a4c10b044344eb62da7044de9c4ab3564791c48a67f64","source":{"kind":"arxiv","id":"1408.2136","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1408.2136","created_at":"2026-05-18T01:04:41Z"},{"alias_kind":"arxiv_version","alias_value":"1408.2136v2","created_at":"2026-05-18T01:04:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1408.2136","created_at":"2026-05-18T01:04:41Z"},{"alias_kind":"pith_short_12","alias_value":"5ASN42RV5SHN","created_at":"2026-05-18T12:28:14Z"},{"alias_kind":"pith_short_16","alias_value":"5ASN42RV5SHNSYM2","created_at":"2026-05-18T12:28:14Z"},{"alias_kind":"pith_short_8","alias_value":"5ASN42RV","created_at":"2026-05-18T12:28:14Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:5ASN42RV5SHNSYM2JQILARBUJ2","target":"record","payload":{"canonical_record":{"source":{"id":"1408.2136","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-08-09T17:04:17Z","cross_cats_sorted":["math.AC"],"title_canon_sha256":"e592d606ea7f25af144ae01b1a1d6845e86573aeed08c2a77d675feb5be5ef6a","abstract_canon_sha256":"30d6cf3edbab1193e0d0925ea5f4e0abcb3c89991653d13b766b514ae2c149ca"},"schema_version":"1.0"},"canonical_sha256":"e824de6a35ec8ed9619a4c10b044344eb62da7044de9c4ab3564791c48a67f64","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:04:41.833835Z","signature_b64":"+1YJ7ICnKBMbZS8FkwLlai0Z6SZcmSyKIbjK/NTS4raEWvzns0mqUouFgvd63cT9KHSqzR6MBd9/CTzLLs2HCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e824de6a35ec8ed9619a4c10b044344eb62da7044de9c4ab3564791c48a67f64","last_reissued_at":"2026-05-18T01:04:41.833377Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:04:41.833377Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1408.2136","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":"2026-05-18T01:04:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dED7fIKGsm3LNMwoHUngfYlXdPLtT9cUJLZz5xXoztTTK4VDfaW/XytE6HW1S2ApZ0mXr9aYt8mgexmjKpNMBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T17:17:47.057408Z"},"content_sha256":"6afea4b6457c2359abe7cc626282dd97d1c8d056ed736278414528c6c8448509","schema_version":"1.0","event_id":"sha256:6afea4b6457c2359abe7cc626282dd97d1c8d056ed736278414528c6c8448509"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:5ASN42RV5SHNSYM2JQILARBUJ2","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Determinants of incidence and Hessian matrices arising from the vector space lattice","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC"],"primary_cat":"math.CO","authors_text":"Alexandra Seceleanu, Junzo Watanabe, Saeed Nasseh","submitted_at":"2014-08-09T17:04:17Z","abstract_excerpt":"Let $\\mathcal{V}=\\bigsqcup_{i=0}^n\\mathcal{V}_i$ be the lattice of subspaces of the $n$-dimensional vector space over the finite field $\\mathbb{F}_q$ and let $\\mathcal{A}$ be the graded Gorenstein algebra defined over $\\mathbb{Q}$ which has $\\mathcal{V}$ as a $\\mathbb{Q}$ basis. Let $F$ be the Macaulay dual generator for $\\mathcal{A}$. We compute explicitly the Hessian determinant $|\\frac{\\partial ^2F}{\\partial X_i \\partial X_j}|$ evaluated at the point $X_1 = X_2 = \\cdots = X_N=1$ and relate it to the determinant of the incidence matrix between $\\mathcal{V}_1$ and $\\mathcal{V}_{n-1}$. Our exp"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.2136","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":"2026-05-18T01:04:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"U+1wakriPcloEnxGK3P0EUWHIRxlwpoOsN8lEmMj0rL38m/55neR2PGTId6UOpPWT88zPUBx1iiq6k06OSrnCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T17:17:47.058169Z"},"content_sha256":"562029aa87382b9aa5e8f397a7d43d260b4876430240d807b999001641a18174","schema_version":"1.0","event_id":"sha256:562029aa87382b9aa5e8f397a7d43d260b4876430240d807b999001641a18174"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/5ASN42RV5SHNSYM2JQILARBUJ2/bundle.json","state_url":"https://pith.science/pith/5ASN42RV5SHNSYM2JQILARBUJ2/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/5ASN42RV5SHNSYM2JQILARBUJ2/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":"2026-05-31T17:17:47Z","links":{"resolver":"https://pith.science/pith/5ASN42RV5SHNSYM2JQILARBUJ2","bundle":"https://pith.science/pith/5ASN42RV5SHNSYM2JQILARBUJ2/bundle.json","state":"https://pith.science/pith/5ASN42RV5SHNSYM2JQILARBUJ2/state.json","well_known_bundle":"https://pith.science/.well-known/pith/5ASN42RV5SHNSYM2JQILARBUJ2/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:5ASN42RV5SHNSYM2JQILARBUJ2","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":"30d6cf3edbab1193e0d0925ea5f4e0abcb3c89991653d13b766b514ae2c149ca","cross_cats_sorted":["math.AC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-08-09T17:04:17Z","title_canon_sha256":"e592d606ea7f25af144ae01b1a1d6845e86573aeed08c2a77d675feb5be5ef6a"},"schema_version":"1.0","source":{"id":"1408.2136","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1408.2136","created_at":"2026-05-18T01:04:41Z"},{"alias_kind":"arxiv_version","alias_value":"1408.2136v2","created_at":"2026-05-18T01:04:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1408.2136","created_at":"2026-05-18T01:04:41Z"},{"alias_kind":"pith_short_12","alias_value":"5ASN42RV5SHN","created_at":"2026-05-18T12:28:14Z"},{"alias_kind":"pith_short_16","alias_value":"5ASN42RV5SHNSYM2","created_at":"2026-05-18T12:28:14Z"},{"alias_kind":"pith_short_8","alias_value":"5ASN42RV","created_at":"2026-05-18T12:28:14Z"}],"graph_snapshots":[{"event_id":"sha256:562029aa87382b9aa5e8f397a7d43d260b4876430240d807b999001641a18174","target":"graph","created_at":"2026-05-18T01:04:41Z","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 $\\mathcal{V}=\\bigsqcup_{i=0}^n\\mathcal{V}_i$ be the lattice of subspaces of the $n$-dimensional vector space over the finite field $\\mathbb{F}_q$ and let $\\mathcal{A}$ be the graded Gorenstein algebra defined over $\\mathbb{Q}$ which has $\\mathcal{V}$ as a $\\mathbb{Q}$ basis. Let $F$ be the Macaulay dual generator for $\\mathcal{A}$. We compute explicitly the Hessian determinant $|\\frac{\\partial ^2F}{\\partial X_i \\partial X_j}|$ evaluated at the point $X_1 = X_2 = \\cdots = X_N=1$ and relate it to the determinant of the incidence matrix between $\\mathcal{V}_1$ and $\\mathcal{V}_{n-1}$. Our exp","authors_text":"Alexandra Seceleanu, Junzo Watanabe, Saeed Nasseh","cross_cats":["math.AC"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-08-09T17:04:17Z","title":"Determinants of incidence and Hessian matrices arising from the vector space lattice"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.2136","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":"sha256:6afea4b6457c2359abe7cc626282dd97d1c8d056ed736278414528c6c8448509","target":"record","created_at":"2026-05-18T01:04:41Z","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":"30d6cf3edbab1193e0d0925ea5f4e0abcb3c89991653d13b766b514ae2c149ca","cross_cats_sorted":["math.AC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-08-09T17:04:17Z","title_canon_sha256":"e592d606ea7f25af144ae01b1a1d6845e86573aeed08c2a77d675feb5be5ef6a"},"schema_version":"1.0","source":{"id":"1408.2136","kind":"arxiv","version":2}},"canonical_sha256":"e824de6a35ec8ed9619a4c10b044344eb62da7044de9c4ab3564791c48a67f64","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e824de6a35ec8ed9619a4c10b044344eb62da7044de9c4ab3564791c48a67f64","first_computed_at":"2026-05-18T01:04:41.833377Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:04:41.833377Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"+1YJ7ICnKBMbZS8FkwLlai0Z6SZcmSyKIbjK/NTS4raEWvzns0mqUouFgvd63cT9KHSqzR6MBd9/CTzLLs2HCg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:04:41.833835Z","signed_message":"canonical_sha256_bytes"},"source_id":"1408.2136","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6afea4b6457c2359abe7cc626282dd97d1c8d056ed736278414528c6c8448509","sha256:562029aa87382b9aa5e8f397a7d43d260b4876430240d807b999001641a18174"],"state_sha256":"8c1dcf996614c74a3ad6b3d5c18c5bf8aa47e402f186c2aafdfad5691eb69363"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XPn5/HO4nBkyr9YtFz9+IITrhPGGFOrQC89xJHdkq2QkllwsbqCxUHO12D2b5qsEq+eZUZg6JXkYhqvqE8iCDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T17:17:47.062453Z","bundle_sha256":"ce0067d51e90a7a8efb8fe88bffa8858616aa1d34b3e30567f5c5ddd63b6bd65"}}