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A Survey of Elekes-Rónyai-Type Problems

  • Frank de ZeeuwEmail author
Chapter
Part of the Bolyai Society Mathematical Studies book series (BSMS, volume 27)

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 \(f\in \mathbb R[x,y]\) 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 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.

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Copyright information

© János Bolyai Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsÉcole Polytechnique Fédérale de LausanneLausanneSwitzerland

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