{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2023:D2ITNJEMVP6XZB3B5HWO7R7RCZ","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":"b360080680c547fe87c13f974ff51736330380d7998d93b34d1eea406e8511a7","cross_cats_sorted":["math.GR"],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.NT","submitted_at":"2023-11-17T13:47:37Z","title_canon_sha256":"5e1991ef5108afe2e9909f99ba7d70c427f3039a071e772f0e92f4f3e1991f0a"},"schema_version":"1.0","source":{"id":"2311.10527","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2311.10527","created_at":"2026-07-05T11:55:59Z"},{"alias_kind":"arxiv_version","alias_value":"2311.10527v2","created_at":"2026-07-05T11:55:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2311.10527","created_at":"2026-07-05T11:55:59Z"},{"alias_kind":"pith_short_12","alias_value":"D2ITNJEMVP6X","created_at":"2026-07-05T11:55:59Z"},{"alias_kind":"pith_short_16","alias_value":"D2ITNJEMVP6XZB3B","created_at":"2026-07-05T11:55:59Z"},{"alias_kind":"pith_short_8","alias_value":"D2ITNJEM","created_at":"2026-07-05T11:55:59Z"}],"graph_snapshots":[{"event_id":"sha256:02f49dfebe252c0bb0dbac65d9a3ebaa29660092972afdd4c44fdd5f98a6549a","target":"graph","created_at":"2026-07-05T11:55:59Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2311.10527/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Continuing our work on group-theoretic generalizations of the prime Ax-Katz Theorem, we give a lower bound on the $p$-adic divisibility of the cardinality of the set of simultaneous zeros $Z(f_1,f_2,\\ldots,f_r)$ of $r$ maps $f_j:A\\rightarrow B_j$ between arbitrary finite commutative groups $A$ and $B_j$ in terms of the invariant factors of $A, B_1,B_2,\\dotsc,B_r$ and the \\emph{functional degrees} of the maps $f_1,f_2,\\dotsc,f_r$.","authors_text":"Pete L. Clark, Uwe Schauz","cross_cats":["math.GR"],"headline":"","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.NT","submitted_at":"2023-11-17T13:47:37Z","title":"Functional degrees and arithmetic applications III: Beyond Prime Exponent"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2311.10527","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:c923b69ffb1fd41bcb2485c58ed791f361b39c8590e002cb84383635c5953bef","target":"record","created_at":"2026-07-05T11:55:59Z","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":"b360080680c547fe87c13f974ff51736330380d7998d93b34d1eea406e8511a7","cross_cats_sorted":["math.GR"],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.NT","submitted_at":"2023-11-17T13:47:37Z","title_canon_sha256":"5e1991ef5108afe2e9909f99ba7d70c427f3039a071e772f0e92f4f3e1991f0a"},"schema_version":"1.0","source":{"id":"2311.10527","kind":"arxiv","version":2}},"canonical_sha256":"1e9136a48cabfd7c8761e9ecefc7f1167484a875d166eef23eaeb82f673c36e2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1e9136a48cabfd7c8761e9ecefc7f1167484a875d166eef23eaeb82f673c36e2","first_computed_at":"2026-07-05T11:55:59.287841Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:55:59.287841Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"fsE7HWbaEgGS4DWDWfnB35xaSxlXeCx4H3VMfgyJyzkXifdUYCX3K0aSSUSwo64bqro3qg0lHAEn+lUtWQ77Bw==","signature_status":"signed_v1","signed_at":"2026-07-05T11:55:59.288475Z","signed_message":"canonical_sha256_bytes"},"source_id":"2311.10527","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c923b69ffb1fd41bcb2485c58ed791f361b39c8590e002cb84383635c5953bef","sha256:02f49dfebe252c0bb0dbac65d9a3ebaa29660092972afdd4c44fdd5f98a6549a"],"state_sha256":"3b7227bb2328f46da1c4b4262f12dd29e8044da101a79f1eefbcaf209e4cb3c9"}