Model-theoretic Elekes-Szabó for stable and o-minimal hypergraphs

Abstract

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. ContentsDate: 2021-04-07 00:43:29Z. 1 2 ARTEM CHERNIKOV, YA'ACOV PETERZIL, AND SERGEI STARCHENKO 5.1. On the notion of p-dimension 42 5.2. Fiber-algebraic relations and p-irreducibility 45 5.3. On general position 46 5.4. Main theorem: the statement and some reductions 49 5.5. Proof of Theorem 5.31: the two cases 54 5.6. Case 1: dim p (Z) < s − 2. 56 5.7. Case 2: dim p (Z) = s − 2. 57 5.8. Proof of Theorem 5.24 for ternary Q 58 5.9. Discussion and some applications 61 6. Main theorem in the o-minimal case 64 6.1. Main theorem and some reductions 64 6.2. The proof of Theorem 6.8 66 6.3. Obtaining a nice Q-relation 70 6.4. Discussion and some applications 72 References 75