{"paper":{"title":"Polynomials vanishing on grids: The Elekes-R\\'onyai problem revisited","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.CG","authors_text":"J\\'ozsef Solymosi, Micha Sharir, Orit E. Raz","submitted_at":"2014-01-29T07:09:37Z","abstract_excerpt":"In this paper we characterize real bivariate polynomials which have a small range over large Cartesian products. We show that for every constant-degree bivariate real polynomial $f$, either $|f(A,B)|=\\Omega(n^{4/3})$, for every pair of finite sets $A,B\\subset{\\mathbb R}$, with $|A|=|B|=n$ (where the constant of proportionality depends on ${\\rm deg} f$), or else $f$ must be of one of the special forms $f(u,v)=h(\\varphi(u)+\\psi(v))$, or $f(u,v)=h(\\varphi(u)\\cdot\\psi(v))$, for some univariate polynomials $\\varphi,\\psi,h$ over ${\\mathbb R}$. This significantly improves a result of Elekes and R\\'on"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.7419","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":""},"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"}