Expanding polynomials: A generalization of the Elekes-R\'onyai theorem to $d$ variables

TL;DR

This paper generalizes the Elekes-Rónyai theorem to polynomials in $d \\ge 3$ variables, characterizing when such polynomials have large image sets or specific additive or multiplicative forms.

Contribution

It extends the Elekes-Rónyai theorem from 2 and 3 variables to any number of variables, providing a complete classification of polynomials with large image sets.

Findings

If $f$ depends non-trivially on each variable, then its image set size is at least proportional to $n^{3/2}$ for finite sets of size $n$.

Polynomials with smaller image sets must be of additive or multiplicative form involving univariate polynomials.

The result generalizes previous cases for $d=2$ and $d=3$ to arbitrary $d \\ge 3$.

Abstract

We prove the following statement. Let $f\in\mathbb{R}[x_1,\ldots,x_d]$, for some $d\ge 3$, and assume that $f$ depends non-trivially in each of $x_1,\ldots,x_d$. Then one of the following holds. (i) For every finite sets $A_1,\ldots,A_d\subset \mathbb{R}$, each of size $n$, we have $$|f(A_1\times\ldots\times A_d)|=\Omega(n^{3/2}), $$ with constant of proportionality that depends on ${\rm deg} f$. (ii) $f$ is of one of the forms \begin{align*} f(x_1,\ldots, x_d)&=h(p_1(x_1)+\cdots+p_d(x_d))~~\text{or}\\ f(x_1,\ldots, x_d)&=h(p_1(x_1)\cdot\ldots\cdot p_d(x_d)), \end{align*} for some univariate real polynomials $h(x)$, $p_1(x),\ldots,p_d(x)$. This generalizes the results from [ER00,RSS, RSdZ], which treat the cases $d=2$ and $d=3$.