{"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"}