Appears in collection : Model Theory and Combinatorics

Given a system of polynomial equations in m complex variables with solution set V of dimension d, if we take finite subsets Xi of C each of size N, then the number of solutions to the system whose i-th co-ordinate is in Xiis easily seen to be bounded as O(Nd). We ask: for what V is the exponent d in this bound optimal? Hrushovski developed a formalism in which such questions become amenable to the tools of model theory, and in particular observed that incidence bounds of Szemer´edi–Trotter type imply modularity of associated geometries, allowing application of the group configuration theorem. Exploiting this, we answer the question above, and treat a higher dimensional version. This generalises results of Elekes– Szab´o on the case (m = 3,d = 2). Part of a joint project with Emmanuel Breuillard.