An asymmetric version of Elekes-Szab\'o 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 $|F(A)| = O(|A|^{1+\eta})$ 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 $\eta>0$. Our conclusions generalise the Elekes-R\'onyai and Elekes-Szab\'o 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-R\'onyai and Elekes-Szab\'o, with explicit bounds on exponents. Our methods originate in model theory.