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Preprint Number 1961

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1961. Artem Chernikov, Ya'acov Peterzil and Sergei Starchenko
Model-theoretic Elekes-Szabó for stable and o-minimal hypergraphs

Submission date: 6 April 2021


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 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.

Mathematics Subject Classification: 03C45, 52C10

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Full text arXiv 2104.02235: pdf, ps.

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