Model-theoretic Elekes-Szab\'o for stable and o-minimal hypergraphs

TL;DR

This paper generalizes the Elekes-Szabó theorem to higher arity relations in stable and o-minimal structures, providing bounds and characterizations for algebraic and definable hypergraphs with large intersections.

Contribution

It extends the Elekes-Szabó theorem to relations of any arity in stable and o-minimal structures, introducing new combinatorial and model-theoretic tools.

Findings

Explicit bounds on power saving exponents in non-group cases

Higher arity abelian group configuration theorem

Characterization of Latin hypercubes from abelian groups

Abstract

A theorem of Elekes and Szab\'{o} 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 non-group 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.