Title: Around the Elekes-Szabo's Theorem
Abstract: The Elekes-Szabo's Theorem roughly says the following: Let R be an algebraic ternary relation in W_1*W_2*W_3 defined in a field K of characteristic 0, such that any two coordinate is interalgebraic with the third one, for example the collear relation for three points in a curve. Suppose there are arbitrarily large finite subsets X_i of W_i with each X_i of size n and has bounded intersection with any subvariety of W_i, such that the intersection of R with X_1*X_2*X_3 has size approximately n^2, then R must be essentially the graph of addition of some commutative algebraic group G. With Martin Bays and Jan Dobrowolski, we are trying to remove the assumption of having bounded intersection with any subvarieties. This assumption is called in general position. Along the way, we develop a group action version of the Elekes-Szabo's Theorem, which seems necessary if one wants to remove the general position assumption. In this talk, I will give an overview of the Elekes-Szabo's Theorem and the model theoretic approach towards it.