A survey of Elekes-R\'onyai-type problems

TL;DR

This survey reviews recent advances in Elekes-Rónyai-type problems, exploring the structure of polynomials with large images on Cartesian products and their implications in combinatorics, algebra, and geometry.

Contribution

It summarizes recent progress, variants, and generalizations of Elekes-Rónyai problems, highlighting open questions and applications in combinatorial geometry and additive combinatorics.

Findings

Progress in bounding polynomial images on Cartesian products

Connections between algebraic structure and combinatorial properties

Open problems in algebraic and geometric intersections

Abstract

We give an overview of recent progress around a problem introduced by Elekes and R\'onyai. The prototype problem is to show that a polynomial $f\in \mathbb{R}[x,y]$ has a large image on a Cartesian product $A\times B\subset \mathbb{R}^2$, unless $f$ has a group-related special form. We discuss a number of variants and generalizations. This includes the Elekes-Szab\'o problem, which generalizes the Elekes-R\'onyai problem to a question about an upper bound on the intersection of an algebraic surface with a Cartesian product, and curve variants, where we ask the same questions for Cartesian products of finite subsets of algebraic curves. These problems lie at the crossroads of combinatorics, algebra, and geometry: They ask combinatorial questions about algebraic objects, whose answers turn out to have applications to geometric questions involving basic objects like distances, lines, and circles, as well as to sum-product-type questions from additive combinatorics. As part of a recent surge of algebraic techniques in combinatorial geometry, a number of quantitative and qualitative steps have been made within this framework. Nevertheless, many tantalizing open questions remain.