Tingxiang Zou (University of Bonn) – Around the Elekes-Szabó Theorem

The Elekes-Szabó Theorem roughly says the following: Let R be an algebraic ternary relation in W1*W2*W3 defined in a field K of characteristic 0, such that any two coordinate is interalgebraic with the third one, for example the collinear relation for three points in a curve.

Suppose there are arbitrarily large finite subsets Xi of Wi each of size n and has bounded intersection with any proper subvariety of Wi, such that the intersection of R with X1*X2*X3 has size approximately n^2, then R must be essentially the graph of addition of some commutative algebraic group G. In this talk, I will give an overview of several results (joint work with Martin Bays and Jan Dobrowolski) in the effort of removing the assumption of Xi having bounded intersection with proper subvarieties of Wi. This assumption is closely related to Wi being 1-dimensional. Our motivation is to find a genuine higher-dimensional generalisation of the Elekes-Szabó Theorem.