Elekes-R\'{o}nyai Theorem revisited

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 $\mathbb{C}$.