An asymmetric version of Elekes-Szabó via group actions
Martin Bays, Tingxiang Zou
TL;DR
This paper advances Elekes–Szabó-type results by developing a model-theoretic, δ-dimension framework to address asymmetrical expansion via group actions on algebraic homogeneous spaces. It proves that large non-expansion forces the acting group to be commutative and the action to arise from a principal homogeneous space isomorphic to (G,G), enabling asymmetric Elekes–Szabó and Elekes–Rónyai statements with explicit exponent bounds. The approach combines Balog–Szemerédi–Gowers-type type arguments, group-configuration theorems, and Szemerédi–Trotter-style incidence bounds to recognise algebraic groups from combinatorial configurations and to derive finitary expansions for polynomial families. The results provide structural classifications for polynomial maps and correspondences, yielding both qualitative and quantitative expansion bounds with potential applications to incidence geometry and algebraic combinatorics.
Abstract
We consider when finite families $F \subseteq \mathbb{C}[t]$ of bounded degree polynomials, or more generally of bounded complexity finite-to-finite correspondences on $\mathbb{C}$, can exhibit non-expansion of the form $|F(A)| = O(|A|^{1+η})$ in their actions on finite sets $A \subseteq \mathbb{C}$ with $|F| \gg |A|^\eps \gg 1$, for a fixed $\eps>0$ and arbitrarily small $η>0$. Our conclusions generalise the Elekes-Rónyai and Elekes-Szabó theorems, which correspond to the case that $F$ is parametrised by a single complex variable and $|F|=|A|$. Our result also applies to families of correspondences between varieties of arbitrary dimension if we impose a general position assumption on $A$. In all cases, the conclusion is that a commutative algebraic group structure is responsible. As a special case, we obtain asymmetric versions of Elekes-Rónyai and Elekes-Szabó, with explicit bounds on exponents. Our methods originate in model theory.