Projective geometries arising from Elekes-Szab\'o problems

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.