MODEL-THEORETIC ELEKES–SZABÓ FOR STABLE AND O-MINIMAL HYPERGRAPHS

A theorem of Elekes and Szabó recognizes algebraic groups among certain complex algebraic varieties with maximal size intersections with finite grids. We establish a generalization to relations of any arity and dimension definable in: (1) stable structures with distal expansions (includes algebraically and differentially closed fields of characteristic 0) and (2) o-minimal expansions of groups. Our methods provide explicit bounds on the power-saving exponent in the nongroup case. Ingredients of the proof include a higher arity generalization of the abelian group configuration theorem in stable structures (along with a purely combinatorial variant characterizing Latin hypercubes that arise from abelian groups) and Zarankiewicz-style bounds for hypergraphs definable in distal structures.