{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:NNAWLJCW4CEHVUCIOI57UBU2ZR","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":"09f5b211c128aa0a114c2346447e04fe50ae9770049106d58d173eb07be3a220","cross_cats_sorted":["math.AC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2010-07-28T08:15:08Z","title_canon_sha256":"441e647fa04136265ce62c012dbe29be62c3cbb517d608fc24d3b953ca672aff"},"schema_version":"1.0","source":{"id":"1007.4891","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1007.4891","created_at":"2026-05-18T03:43:53Z"},{"alias_kind":"arxiv_version","alias_value":"1007.4891v2","created_at":"2026-05-18T03:43:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1007.4891","created_at":"2026-05-18T03:43:53Z"},{"alias_kind":"pith_short_12","alias_value":"NNAWLJCW4CEH","created_at":"2026-05-18T12:26:10Z"},{"alias_kind":"pith_short_16","alias_value":"NNAWLJCW4CEHVUCI","created_at":"2026-05-18T12:26:10Z"},{"alias_kind":"pith_short_8","alias_value":"NNAWLJCW","created_at":"2026-05-18T12:26:10Z"}],"graph_snapshots":[{"event_id":"sha256:7498be327ed45b64d1fb1588c25a751647266f18a3244c91f882f1d405c2210a","target":"graph","created_at":"2026-05-18T03:43:53Z","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":"A result by Macaulay states that an Artinian graded Gorenstein ring R of socle dimension one and socle degree b can be realized as the apolar ring of a homogeneous polynomial f of degree b. If R is the Jacobian ring of a smooth hypersurface g=0, then b is just equal to the degree of the Hessian polynomial of g. In this paper we investigate the relationship between f and the Hessian polynomial of g.","authors_text":"Elisa Postinghel, Lorenzo Di Biagio","cross_cats":["math.AC"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2010-07-28T08:15:08Z","title":"Apolarity, Hessian and Macaulay polynomials"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1007.4891","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":"sha256:68cacdadd2b1dda8378b47d0c06ea18febe1f1c7571442201ab0105895b7ff29","target":"record","created_at":"2026-05-18T03:43:53Z","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":"09f5b211c128aa0a114c2346447e04fe50ae9770049106d58d173eb07be3a220","cross_cats_sorted":["math.AC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2010-07-28T08:15:08Z","title_canon_sha256":"441e647fa04136265ce62c012dbe29be62c3cbb517d608fc24d3b953ca672aff"},"schema_version":"1.0","source":{"id":"1007.4891","kind":"arxiv","version":2}},"canonical_sha256":"6b4165a456e0887ad048723bfa069acc768053d6fe393c7b5662203ae80f9d1b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6b4165a456e0887ad048723bfa069acc768053d6fe393c7b5662203ae80f9d1b","first_computed_at":"2026-05-18T03:43:53.068964Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:43:53.068964Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"g7T79kTYsa/XxREAJsxH1obIu6IMDFXkKwIowGQO5VyXEK7GuZNko4sIUOfk8CKtrcT29IcnSNq33kx8ynwjAg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:43:53.069728Z","signed_message":"canonical_sha256_bytes"},"source_id":"1007.4891","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:68cacdadd2b1dda8378b47d0c06ea18febe1f1c7571442201ab0105895b7ff29","sha256:7498be327ed45b64d1fb1588c25a751647266f18a3244c91f882f1d405c2210a"],"state_sha256":"cb596878ee8638000986f6e259577ab1cbb9f8485952dd31eb3e7ea372ca2fbd"}