Erdos and Szemeredi observed the following sum-productphenomenon: there is some c>0 such that for any finite set A of reals, max{|A+A|, |A*A|} > |A|^{1+c}. Later, Elekes and Ronyai generalized this by showing that for any polynomial f(x,y) we must have |f(A*A)|>|A|^{1+c}, unless f is either additive or multiplicative (i.e. of the form g(h(x) + i(y)) or g(h(x) *i(y)) for some univariate polynomials g,h,i respectively). A remarkable theorem of Elekes and Szabo provides a conceptual generalization, showing that for any polynomial F(x,y,z) such that its set of solutions has dimension 2, if F has a maximal possible number of zeroes n^2 on finite n-by-n-by-n grids, then F is in a finite-to-finite correspondence with the graph of multiplication of an algebraic group (in the special case above, either the additive or the multiplicative group of the field).

We present a generalization of this result to hypergraphs of any arity definable in a large class of stable structures, including differentially closed fields, as well as a version for o-minimal structures. We will discuss in detail some of the main ingredients of the proof, including local distality and its connections to Szemeredi-Trotter type incidence bounds, as well as a local version of the abelian group configuration theorem.

Joint work with Kobi Peterzil and Sergei Starchenko.