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Consider an algebraic variety $X$ in $\\mathbb{C}^4$ of degree $d$, such that the polynomials defining $X$ are not all of the form $F(x,y,s,t) = G(x,y)H(x,y,s,t) + K(s,t)L(x,y,s,t)$. Let $P$ and $Q$ be finite subsets of $\\mathbb{C}^2$ of size $n$. If $X$ has dimension one or two, then we prove $|X\\cap (P\\times Q)| = O_d(n)$, while if $X$ has dimension three, then $|X\\cap (P\\ti"},"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":"1507.08181","kind":"arxiv","version":5},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2015-07-29T15:19:16Z","cross_cats_sorted":[],"title_canon_sha256":"d06fed68157bfabffc6865aeda2ba94ce608e69b453f1b3f81b5da88e913800c","abstract_canon_sha256":"6b2aed67d05ec0c1f41c21448ef64d2f5e1f1119925331701d7e6b3b31beb4f7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:27:36.902917Z","signature_b64":"1D5/BX+1OqO3n8Y363AAU1SfduswML+Sp+quGyYvkSiOxUznuinx7+F4B3fodKn0Nk3mfwmSXkThHTUMG2ltDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a2677de61d5acfeec04ab4d004e00ab1d4651b7cb2c87ca361eadadffc4fd55d","last_reissued_at":"2026-05-18T00:27:36.901794Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:27:36.901794Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Schwartz-Zippel bounds for two-dimensional products","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Claudiu Valculescu, Frank de Zeeuw, Hossein Nassajian Mojarrad, Thang Pham","submitted_at":"2015-07-29T15:19:16Z","abstract_excerpt":"We prove bounds on intersections of algebraic varieties in $\\mathbb{C}^4$ with Cartesian products of finite sets from $\\mathbb{C}^2$, and we point out connections with several classic theorems from combinatorial geometry. Consider an algebraic variety $X$ in $\\mathbb{C}^4$ of degree $d$, such that the polynomials defining $X$ are not all of the form $F(x,y,s,t) = G(x,y)H(x,y,s,t) + K(s,t)L(x,y,s,t)$. Let $P$ and $Q$ be finite subsets of $\\mathbb{C}^2$ of size $n$. If $X$ has dimension one or two, then we prove $|X\\cap (P\\times Q)| = O_d(n)$, while if $X$ has dimension three, then $|X\\cap (P\\ti"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.08181","kind":"arxiv","version":5},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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":"1507.08181","created_at":"2026-05-18T00:27:36.901923+00:00"},{"alias_kind":"arxiv_version","alias_value":"1507.08181v5","created_at":"2026-05-18T00:27:36.901923+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.08181","created_at":"2026-05-18T00:27:36.901923+00:00"},{"alias_kind":"pith_short_12","alias_value":"UJTX3ZQ5LLH6","created_at":"2026-05-18T12:29:44.643036+00:00"},{"alias_kind":"pith_short_16","alias_value":"UJTX3ZQ5LLH65QCK","created_at":"2026-05-18T12:29:44.643036+00:00"},{"alias_kind":"pith_short_8","alias_value":"UJTX3ZQ5","created_at":"2026-05-18T12:29:44.643036+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/UJTX3ZQ5LLH65QCKWTIAJYAKWH","json":"https://pith.science/pith/UJTX3ZQ5LLH65QCKWTIAJYAKWH.json","graph_json":"https://pith.science/api/pith-number/UJTX3ZQ5LLH65QCKWTIAJYAKWH/graph.json","events_json":"https://pith.science/api/pith-number/UJTX3ZQ5LLH65QCKWTIAJYAKWH/events.json","paper":"https://pith.science/paper/UJTX3ZQ5"},"agent_actions":{"view_html":"https://pith.science/pith/UJTX3ZQ5LLH65QCKWTIAJYAKWH","download_json":"https://pith.science/pith/UJTX3ZQ5LLH65QCKWTIAJYAKWH.json","view_paper":"https://pith.science/paper/UJTX3ZQ5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1507.08181&json=true","fetch_graph":"https://pith.science/api/pith-number/UJTX3ZQ5LLH65QCKWTIAJYAKWH/graph.json","fetch_events":"https://pith.science/api/pith-number/UJTX3ZQ5LLH65QCKWTIAJYAKWH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/UJTX3ZQ5LLH65QCKWTIAJYAKWH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/UJTX3ZQ5LLH65QCKWTIAJYAKWH/action/storage_attestation","attest_author":"https://pith.science/pith/UJTX3ZQ5LLH65QCKWTIAJYAKWH/action/author_attestation","sign_citation":"https://pith.science/pith/UJTX3ZQ5LLH65QCKWTIAJYAKWH/action/citation_signature","submit_replication":"https://pith.science/pith/UJTX3ZQ5LLH65QCKWTIAJYAKWH/action/replication_record"}},"created_at":"2026-05-18T00:27:36.901923+00:00","updated_at":"2026-05-18T00:27:36.901923+00:00"}