A Survey of Elekes-Rónyai-Type Problems

Abstract

We give an overview of recent progress around a problem introduced by Elekes and Rónyai. The prototype problem is to show that a polynomial f ∈ R[x, y] has a large image on a Cartesian product A×B ⊂ R 2 , unless f has a group-related special form. We discuss this problem and a number of variants and generalizations. This includes the Elekes-Szabó problem, which generalizes the Elekes-Rónyai 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.

Convexity, Squeezing, and the Elekes-Szabó Theorem

“…The bound (3) is also tight, up to constant and logarithmic factors, as is illustrated by the case when A = [n] for some positive integer n and f (x) = x 2 . Moreover, a recent paper of Bradshaw [1] removed the logarithmic factors from (3).…”

Convexity, Squeezing, and the Elekes-Szabó Theorem

“…The development of the Elekes-Szabó Theorem has largely been motivated by applications to problems in discrete geometry. See the survey of de Zeeuw [3] for more background on this problem and its applications. A recent paper of Roche-Newton [16] applied the Elekes-Szabó Theorem in order to prove new results in sum-product theory.…”

“…The bound (3) is also tight, up to constant and logarithmic factors, as is illustrated by the case when A = [n] for some positive integer n and f (x) = x 2 . Moreover, a recent paper of Bradshaw [1] removed the logarithmic factors from (3).…”

Sums, products and dilates on sparse graphs

“…The assumption that λ = 1, −1 means that the expression is square brackets is non-zero. Therefore, (14) equals zero only when Y 2 = λX 2 , so it does not vanish on any nontrivial open set. This gives the required contradiction, which completes the proof of the claim and also the theorem.…”