Projective geometries arising from Elekes-Szab\'o problems

TL;DR

This paper generalizes the Elekes-Szabó theorem to higher dimensions and arities, characterizes varieties without power saving, and extends sum-product phenomena to elliptic curves using model-theoretic methods.

Contribution

It introduces a broad generalization of the Elekes-Szabó theorem and extends sum-product results to elliptic curves, employing Hrushovski's pseudo-finite dimensions.

Findings

Characterization of complex algebraic varieties without power saving.

Extension of sum-product phenomena to elliptic curves.

Application of model-theoretic frameworks to algebraic geometry.

Abstract

We generalise the Elekes-Szab\'o theorem to arbitrary arity and dimension and characterise the complex algebraic varieties without power saving. The characterisation involves certain algebraic subgroups of commutative algebraic groups endowed with an extra structure arising from a skew field of endomorphisms. We also extend the Erd\H{o}s-Szemer\'edi sum-product phenomenon to elliptic curves. Our approach is based on Hrushovski's framework of pseudo-finite dimensions and the abelian group configuration theorem.