{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2022:CCKREJV4R7EKD3NF3HQOMIQTJ2","short_pith_number":"pith:CCKREJV4","canonical_record":{"source":{"id":"2211.02908","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2022-11-05T13:46:54Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"a1b2be9034cb8bb44a54aab955ef31cd56c61daa64b40e482f0714194e38bfb7","abstract_canon_sha256":"b6e7f35983e5e801db7489f5d9e746ec9079fe35584951dc27c5e480f79f58c2"},"schema_version":"1.0"},"canonical_sha256":"10951226bc8fc8a1eda5d9e0e622134e923879bd6d491a7148930e992641ab90","source":{"kind":"arxiv","id":"2211.02908","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2211.02908","created_at":"2026-07-05T08:56:01Z"},{"alias_kind":"arxiv_version","alias_value":"2211.02908v2","created_at":"2026-07-05T08:56:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2211.02908","created_at":"2026-07-05T08:56:01Z"},{"alias_kind":"pith_short_12","alias_value":"CCKREJV4R7EK","created_at":"2026-07-05T08:56:01Z"},{"alias_kind":"pith_short_16","alias_value":"CCKREJV4R7EKD3NF","created_at":"2026-07-05T08:56:01Z"},{"alias_kind":"pith_short_8","alias_value":"CCKREJV4","created_at":"2026-07-05T08:56:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2022:CCKREJV4R7EKD3NF3HQOMIQTJ2","target":"record","payload":{"canonical_record":{"source":{"id":"2211.02908","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2022-11-05T13:46:54Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"a1b2be9034cb8bb44a54aab955ef31cd56c61daa64b40e482f0714194e38bfb7","abstract_canon_sha256":"b6e7f35983e5e801db7489f5d9e746ec9079fe35584951dc27c5e480f79f58c2"},"schema_version":"1.0"},"canonical_sha256":"10951226bc8fc8a1eda5d9e0e622134e923879bd6d491a7148930e992641ab90","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:56:01.646059Z","signature_b64":"J+4FMV1yIiQfEH4U+5yxCT0oLXLdX7lA6SbRTToOafLQo7vEo4fJlD2SPsfzLlwf8I5czPCNtAde4ZRyyDyKBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"10951226bc8fc8a1eda5d9e0e622134e923879bd6d491a7148930e992641ab90","last_reissued_at":"2026-07-05T08:56:01.645632Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:56:01.645632Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2211.02908","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":"2026-07-05T08:56:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nT+K0mNDQkrdKjH2d7jnbWLGAbNXprXUO+MRCe39LY44PIJbMkEqMUXLh540O4b/SeGX7cOVoeW5htaBdk73Aw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-26T22:34:16.613435Z"},"content_sha256":"2a5bf25c377cf8d5ad1d835849d684d412900d92e88eaa026ff44ae77160c241","schema_version":"1.0","event_id":"sha256:2a5bf25c377cf8d5ad1d835849d684d412900d92e88eaa026ff44ae77160c241"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2022:CCKREJV4R7EKD3NF3HQOMIQTJ2","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Paucity phenomena for polynomial products","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.NT","authors_text":"Max Wenqiang Xu, Victor Y. Wang","submitted_at":"2022-11-05T13:46:54Z","abstract_excerpt":"Let $P(x)\\in \\mathbb{Z}[x]$ be a polynomial with at least two distinct complex roots. We prove that the number of solutions $(x_1, \\dots, x_k, y_1, \\dots, y_k)\\in [N]^{2k}$ to the equation \\[\n  \\prod_{1\\le i \\le k} P(x_i) = \\prod_{1\\le j \\le k} P(y_j)\\neq 0 \\] (for any $k\\ge 1$) is asymptotically $k!N^{k}$ as $N\\to +\\infty$. This solves a question first proposed and studied by Najnudel. The result can also be interpreted as saying that all even moments of random partial sums $\\frac{1}{\\sqrt{N}}\\sum_{n\\le N}f(P(n))$ match standard complex Gaussian moments as $N\\to +\\infty$, where $f$ is the Ste"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2211.02908","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2211.02908/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T08:56:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nE4Tzzgef4wLXcoto39zAD/YzsIy046ZB258zObIPt+dgf91KjDZNtJWOt4K1v7Z91FdYFma8uhZKtAJFBd9DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-26T22:34:16.613832Z"},"content_sha256":"605488dcc3e4c27c41ca205838e5734c5c314b93362812454fa4b454daa05550","schema_version":"1.0","event_id":"sha256:605488dcc3e4c27c41ca205838e5734c5c314b93362812454fa4b454daa05550"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CCKREJV4R7EKD3NF3HQOMIQTJ2/bundle.json","state_url":"https://pith.science/pith/CCKREJV4R7EKD3NF3HQOMIQTJ2/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CCKREJV4R7EKD3NF3HQOMIQTJ2/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":"2026-07-26T22:34:16Z","links":{"resolver":"https://pith.science/pith/CCKREJV4R7EKD3NF3HQOMIQTJ2","bundle":"https://pith.science/pith/CCKREJV4R7EKD3NF3HQOMIQTJ2/bundle.json","state":"https://pith.science/pith/CCKREJV4R7EKD3NF3HQOMIQTJ2/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CCKREJV4R7EKD3NF3HQOMIQTJ2/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2022:CCKREJV4R7EKD3NF3HQOMIQTJ2","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":"b6e7f35983e5e801db7489f5d9e746ec9079fe35584951dc27c5e480f79f58c2","cross_cats_sorted":["math.PR"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2022-11-05T13:46:54Z","title_canon_sha256":"a1b2be9034cb8bb44a54aab955ef31cd56c61daa64b40e482f0714194e38bfb7"},"schema_version":"1.0","source":{"id":"2211.02908","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2211.02908","created_at":"2026-07-05T08:56:01Z"},{"alias_kind":"arxiv_version","alias_value":"2211.02908v2","created_at":"2026-07-05T08:56:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2211.02908","created_at":"2026-07-05T08:56:01Z"},{"alias_kind":"pith_short_12","alias_value":"CCKREJV4R7EK","created_at":"2026-07-05T08:56:01Z"},{"alias_kind":"pith_short_16","alias_value":"CCKREJV4R7EKD3NF","created_at":"2026-07-05T08:56:01Z"},{"alias_kind":"pith_short_8","alias_value":"CCKREJV4","created_at":"2026-07-05T08:56:01Z"}],"graph_snapshots":[{"event_id":"sha256:605488dcc3e4c27c41ca205838e5734c5c314b93362812454fa4b454daa05550","target":"graph","created_at":"2026-07-05T08:56:01Z","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/2211.02908/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $P(x)\\in \\mathbb{Z}[x]$ be a polynomial with at least two distinct complex roots. We prove that the number of solutions $(x_1, \\dots, x_k, y_1, \\dots, y_k)\\in [N]^{2k}$ to the equation \\[\n  \\prod_{1\\le i \\le k} P(x_i) = \\prod_{1\\le j \\le k} P(y_j)\\neq 0 \\] (for any $k\\ge 1$) is asymptotically $k!N^{k}$ as $N\\to +\\infty$. This solves a question first proposed and studied by Najnudel. The result can also be interpreted as saying that all even moments of random partial sums $\\frac{1}{\\sqrt{N}}\\sum_{n\\le N}f(P(n))$ match standard complex Gaussian moments as $N\\to +\\infty$, where $f$ is the Ste","authors_text":"Max Wenqiang Xu, Victor Y. Wang","cross_cats":["math.PR"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2022-11-05T13:46:54Z","title":"Paucity phenomena for polynomial products"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2211.02908","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":"sha256:2a5bf25c377cf8d5ad1d835849d684d412900d92e88eaa026ff44ae77160c241","target":"record","created_at":"2026-07-05T08:56:01Z","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":"b6e7f35983e5e801db7489f5d9e746ec9079fe35584951dc27c5e480f79f58c2","cross_cats_sorted":["math.PR"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2022-11-05T13:46:54Z","title_canon_sha256":"a1b2be9034cb8bb44a54aab955ef31cd56c61daa64b40e482f0714194e38bfb7"},"schema_version":"1.0","source":{"id":"2211.02908","kind":"arxiv","version":2}},"canonical_sha256":"10951226bc8fc8a1eda5d9e0e622134e923879bd6d491a7148930e992641ab90","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"10951226bc8fc8a1eda5d9e0e622134e923879bd6d491a7148930e992641ab90","first_computed_at":"2026-07-05T08:56:01.645632Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T08:56:01.645632Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"J+4FMV1yIiQfEH4U+5yxCT0oLXLdX7lA6SbRTToOafLQo7vEo4fJlD2SPsfzLlwf8I5czPCNtAde4ZRyyDyKBQ==","signature_status":"signed_v1","signed_at":"2026-07-05T08:56:01.646059Z","signed_message":"canonical_sha256_bytes"},"source_id":"2211.02908","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2a5bf25c377cf8d5ad1d835849d684d412900d92e88eaa026ff44ae77160c241","sha256:605488dcc3e4c27c41ca205838e5734c5c314b93362812454fa4b454daa05550"],"state_sha256":"c6b865da8cdde2fbcf83bcb8730aff6c4dcb70689e3ef281c570e67629fb9530"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8bhCE21f1YQdYzRg7POVXdpT75aQnMlgGCKmHgZy9bhdwSsED+6HV/bE+LzJZVOjL4L+Q1Uh/M4zY9l/GE0jAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-26T22:34:16.616718Z","bundle_sha256":"4f09cd1d09c1b1a0393f4af2dcb647661ae016ddeea9c4d5d636d559b719b955"}}