Expanding polynomials and the Elekes-Szabó problem using proximity

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Elekes and Rónyai proved that if P(x,y) is a polynomial, then either P has a very special structure (for example P(x,y) = x+y or P(x,y) = xy), or else for every pair of sets A, B \subset R of size n, P(A x B) has cardinality substantially larger than n. This result was generalized by Elekes and Szabó, who proved that if Z \subset R^3 is an algebraic variety, then either Z has a very special structure (for example Z is a plane), or else for every triples of sets A, B, C \subset R of size n, Z \cap (A x B x C) has cardinality substantially smaller than n^2.

The Elekes-Rónyai and Elekes-Szabó theorems have applications to extremal questions in incidence geometry. There have been several generalizations of the above results, and several quantitative improvements that have strengthened the exponent in the Elekes-Szabó theorem from n^2 to n^{2-c} for various explicit values of c > 0 (and similarly for the Elekes-Rónyai theorem). I will discuss some ideas that give further quantitative improvements for this problem, leading to the exponent c = 2/7. This is joint work with Jozsef Solymosi.

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