An asymmetric version of Elekes-Szabó via group actions

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 $\mathrm{\mid }F(A)\mathrm{\mid }=O(\mathrm{\mid }A{\mathrm{\mid }}^{1+\eta })$ in their actions on finite sets $A\subseteq \mathbb{C}$ with $\mathrm{\mid }F\mathrm{\mid }\gg \mathrm{\mid }A{\mathrm{\mid }}^{\text{eps}}\gg 1$, for a fixed $\text{eps}>0$ and arbitrarily small $\eta >0$. 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 $\mathrm{\mid }F\mathrm{\mid }=\mathrm{\mid }A\mathrm{\mid }$. 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.