Projective geometries arising from Elekes-Szab\'o problems
Combinatorics
2022-09-13 v3 Logic
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.
Cite
@article{arxiv.1806.03422,
title = {Projective geometries arising from Elekes-Szab\'o problems},
author = {Martin Bays and Emmanuel Breuillard},
journal= {arXiv preprint arXiv:1806.03422},
year = {2022}
}
Comments
48 pages. To appear in ASENS. v2: Minor improvements in presentation. v3: Fixed some typos