Model-theoretic Elekes-Szab\'o for stable and o-minimal hypergraphs
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 ); and 2) -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.
Cite
@article{arxiv.2104.02235,
title = {Model-theoretic Elekes-Szab\'o for stable and o-minimal hypergraphs},
author = {Artem Chernikov and Ya'acov Peterzil and Sergei Starchenko},
journal= {arXiv preprint arXiv:2104.02235},
year = {2023}
}
Comments
78 pages; many minor corrections and clarifications throughout the article following referees' suggestions; the main theorem is now presented with additional uniformity in families; accepted to the Duke Mathematical Journal