Polynomials vanishing on grids: The Elekes-Rónyai problem revisited

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)| = Ω(n^4/3), for every pair of finite sets A,B ⊂ ℝ, with |A| = |B| = n (where the constant of proportionality depends on deg f), or else f must be of one of the special forms f(u,v)=h(ϕ(u)+ ψ(v)), or f(u,v)=h(ϕ(u)·ψ (v)), for some univariate polynomials ϕ,ψ,h over ℝ. This significantly improves a result of Elekes and Rónyai (2000). Our results are cast in a more general form, in which we give an upper bound for the number of zeros of z = f(x,y) on a triple Cartesian product A× B×C, when the sizes |A|, |B|, |C| need not be the same; the upper bound is O(n^11/6) when |A| = |B| = |C| = n, where the constant of proportionality depends on degf, unless f has one of the aforementioned special forms. This result provides a unified tool for improving bounds in various Erdős-type problems in geometry and additive combinatorics. Several applications of our results to problems of these kinds are presented. For example, we show that the number of distinct distances between n points lying on a constant-degree algebraic curve that has a polynomial parameterization, and that does not contain a line, in any dimension, is Ω(n^4/3), extending the result of Pach and de Zeeuw (2014) and improving the bound of Charalambides (2014), for the special case where the curve under consideration has a polynomial parameterization. We also derive improved lower bounds for several variants of the sum-product problem in additive combinatorics.