Submission date: 9 June 2018

Abstract:

We generalise the Elekes-Szabó 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 Erdös-Szemerédi sum-product phenomenon to elliptic curves. Our approach is based on Hrushovski's framework of pseudo-finite dimensions and the abelian group configuration theorem.

Mathematics Subject Classification:

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Full text arXiv 1806.03422: pdf, ps.