Projective geometries arising from Elekes-Szabó problems

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 Erdős-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.

48 pages. To appear in ASENS. v2: Minor improvements in presentation. v3: Fixed some typos

References in corpus (3)

• Polynomials vanishing on Cartesian products: The Elekes-Szabó Theorem revisited
• A Note on Generic Types
• Model Theory of Compact Complex Manifolds with an Automorphism

Cited by in corpus (5)

• Improved Elekes-Szabó type estimates using proximity
• On the dimension of exceptional parameters for nonlinear projections, and the discretized Elekes-Rónyai theorem
• Incidence bounds in positive characteristic via valuations and distality
• Semialgebraic methods and generalized sum-product phenomena
• Dimension-expanding polynomials and the discretized Elekes-Rónyai theorem