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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"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"},"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"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2211.02908","created_at":"2026-07-05T08:56:01.645700+00:00"},{"alias_kind":"arxiv_version","alias_value":"2211.02908v2","created_at":"2026-07-05T08:56:01.645700+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2211.02908","created_at":"2026-07-05T08:56:01.645700+00:00"},{"alias_kind":"pith_short_12","alias_value":"CCKREJV4R7EK","created_at":"2026-07-05T08:56:01.645700+00:00"},{"alias_kind":"pith_short_16","alias_value":"CCKREJV4R7EKD3NF","created_at":"2026-07-05T08:56:01.645700+00:00"},{"alias_kind":"pith_short_8","alias_value":"CCKREJV4","created_at":"2026-07-05T08:56:01.645700+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/CCKREJV4R7EKD3NF3HQOMIQTJ2","json":"https://pith.science/pith/CCKREJV4R7EKD3NF3HQOMIQTJ2.json","graph_json":"https://pith.science/api/pith-number/CCKREJV4R7EKD3NF3HQOMIQTJ2/graph.json","events_json":"https://pith.science/api/pith-number/CCKREJV4R7EKD3NF3HQOMIQTJ2/events.json","paper":"https://pith.science/paper/CCKREJV4"},"agent_actions":{"view_html":"https://pith.science/pith/CCKREJV4R7EKD3NF3HQOMIQTJ2","download_json":"https://pith.science/pith/CCKREJV4R7EKD3NF3HQOMIQTJ2.json","view_paper":"https://pith.science/paper/CCKREJV4","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2211.02908&json=true","fetch_graph":"https://pith.science/api/pith-number/CCKREJV4R7EKD3NF3HQOMIQTJ2/graph.json","fetch_events":"https://pith.science/api/pith-number/CCKREJV4R7EKD3NF3HQOMIQTJ2/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CCKREJV4R7EKD3NF3HQOMIQTJ2/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CCKREJV4R7EKD3NF3HQOMIQTJ2/action/storage_attestation","attest_author":"https://pith.science/pith/CCKREJV4R7EKD3NF3HQOMIQTJ2/action/author_attestation","sign_citation":"https://pith.science/pith/CCKREJV4R7EKD3NF3HQOMIQTJ2/action/citation_signature","submit_replication":"https://pith.science/pith/CCKREJV4R7EKD3NF3HQOMIQTJ2/action/replication_record"}},"created_at":"2026-07-05T08:56:01.645700+00:00","updated_at":"2026-07-05T08:56:01.645700+00:00"}