{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:2CODX7PQY45D5YUDWROEG2XFAU","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":"4d79cb239499475e5cbd360b92157129803e2bd27d489bace84f17bdc606a629","cross_cats_sorted":["math.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2011-07-27T17:10:49Z","title_canon_sha256":"2f12f8cb80a83c702a985a647a08b133a6331a691cca9fd500ff700d92deadcd"},"schema_version":"1.0","source":{"id":"1107.5542","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1107.5542","created_at":"2026-05-18T04:16:42Z"},{"alias_kind":"arxiv_version","alias_value":"1107.5542v1","created_at":"2026-05-18T04:16:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1107.5542","created_at":"2026-05-18T04:16:42Z"},{"alias_kind":"pith_short_12","alias_value":"2CODX7PQY45D","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_16","alias_value":"2CODX7PQY45D5YUD","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_8","alias_value":"2CODX7PQ","created_at":"2026-05-18T12:26:18Z"}],"graph_snapshots":[{"event_id":"sha256:ec11cb5138a1848c0afb5c627f737711787c2e6d8bde1100c51244507cd29d6a","target":"graph","created_at":"2026-05-18T04:16:42Z","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":"To construct mirror symmetric Landau-Ginzburg models, P.Berglund, T.H\\\"ubsch and M.Henningson considered a pair $(f,G)$ consisting of an invertible polynomial $f$ and an abelian group $G$ of its symmetries together with a dual pair $(\\widetilde{f}, \\widetilde{G})$. Here we study the reduced orbifold Euler characteristics of the Milnor fibres of $f$ and $\\widetilde f$ with the actions of the groups $G$ and $\\widetilde G$ respectively and show that they coincide up to a sign.","authors_text":"Sabir M. Gusein-Zade, Wolfgang Ebeling","cross_cats":["math.GT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2011-07-27T17:10:49Z","title":"Orbifold Euler characteristics for dual invertible polynomials"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.5542","kind":"arxiv","version":1},"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:fa6cc6d255c38afc09007eb6026a51ec87aaf2f988ef573601a8ba07da89faa8","target":"record","created_at":"2026-05-18T04:16:42Z","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":"4d79cb239499475e5cbd360b92157129803e2bd27d489bace84f17bdc606a629","cross_cats_sorted":["math.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2011-07-27T17:10:49Z","title_canon_sha256":"2f12f8cb80a83c702a985a647a08b133a6331a691cca9fd500ff700d92deadcd"},"schema_version":"1.0","source":{"id":"1107.5542","kind":"arxiv","version":1}},"canonical_sha256":"d09c3bfdf0c73a3ee283b45c436ae505182c2476fd75fd8a1e903066301340d3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d09c3bfdf0c73a3ee283b45c436ae505182c2476fd75fd8a1e903066301340d3","first_computed_at":"2026-05-18T04:16:42.911539Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:16:42.911539Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"wBIUR3oWvc37dyfJG3U1fGUZe0KNu6BsV4nepkPadbwuq85eKlWv3fuFgpS+Gz9RQZ4o/pBirQaJcq68/fUvCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:16:42.912207Z","signed_message":"canonical_sha256_bytes"},"source_id":"1107.5542","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fa6cc6d255c38afc09007eb6026a51ec87aaf2f988ef573601a8ba07da89faa8","sha256:ec11cb5138a1848c0afb5c627f737711787c2e6d8bde1100c51244507cd29d6a"],"state_sha256":"246f56d76b60146a39a95ffd81991318b4018afbed6fa20fb543c726382b8890"}