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 $|F(A)| = O(|A|^{1+\eta})$ 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-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.
From: Martin Bays [view email]
[v1]
Mon, 26 Aug 2024 12:14:27 UTC (45 KB)
[v2]
Tue, 7 Jan 2025 21:39:23 UTC (57 KB)
[v3]
Sun, 2 Nov 2025 11:31:21 UTC (58 KB)
[v4]
Sun, 23 Aug 2026 11:01:10 UTC (69 KB)
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