Elekes-R\'{o}nyai Theorem revisited

TL;DR

This paper improves the Elekes-Rónyai Theorem by establishing a lower bound on the size of the image set of a rational function over finite real subsets, except for specific algebraic forms, and extends the result to complex functions.

Contribution

It refines the Elekes-Rónyai Theorem with a stronger bound and generalizes it to rational functions over complex numbers, broadening its applicability.

Findings

Established a lower bound of (|A_1|^{4/3}) for the image size of rational functions.

Identified specific algebraic forms where the bound does not hold.

Extended the theorem to functions over complex variables.

Abstract

In this paper it is proven that for any $f\in\mathbb{R}(x_1,x_2)$ and $A_1,A_2$ nonempty finite subsets of $\mathbb{R}$ such that $|A_1|=|A_2|$ and $f$ is defined in $A_1\times A_2$, we have that \begin{equation*} |f(A_1,A_2)|=\Omega\left(|A_1|^{\frac{4}{3}}\right) \end{equation*} unless there are $g,l_1,l_2\in\mathbb{R}(x)$ such that $f(x_1,x_2)=g(l_1(x_1)+l_2(x_2)), f(x_1,x_2)=g(l_1(x_1)\cdot l_2(x_2))$ or $f(x_1,x_2)=g\left(\frac{l_1(x_1)+l_2(x_2)}{1-l_1(x_1)\cdot l_2(x_2)}\right)$. This result improves Elekes-R\'{o}nyai Theorem and it generalizes a result of Raz-Sharir-Solymosi proven for $f\in\mathbb{R}[x_1,x_2]$. Furthermore, an analogous result is proven for $f\in\mathbb{C}(x_1,x_2)$ and $A_1,A_2$ subsets of $\mathbb{C}$.