An asymmetric version of Elekes-Szab\'o via group actions

Abstract

We prove a generalisation of the Elekes-Szab\'o Theorem on the existence of large intersections of a complex variety $V\subseteq W_1\times W_2\times W_3$ with finite grids $A\times B\times C$. We arrive at the same conclusion that $V$ is essentially addition in a commutative algebraic group, but we weaken the assumptions. Firstly, we allow arbitrarily unbalanced cardinalities $|A|^\epsilon= O(|B|)$. Secondly, we remove the assumption that $V$ projects dominantly onto $W_1\times W_3$. Thirdly, we weaken the general position assumption on $B$. As a corollary, we generalise the Elekes-R\'onyai Theorem to an arbitrarily unbalanced version, and more generally to the case of finite families $F$ of univariate polynomials exhibiting non-expansion $|\bigcup_{f\in F}f(A)|= O(|A|^{1+\eta})$ of their action on a finite set $A\subseteq \mathbb{C}$.

Locations

• arXiv (Cornell University) - View - PDF