Improved bound for the $k$-variate Elekes--Rónyai theorem
Yaara Jahn, Orit E. Raz
TL;DR
The paper advances the Elekes–Rónyai program to $k$-variate polynomials by introducing the rank ${\rm rank}(f)$ and proving a new lower bound $|f(A_1,\ldots,A_k)|=\Omega\left(n^{\frac{5{\rm rank}(f)-4}{2{\rm rank}(f)}-\varepsilon}\right)$ for any $\varepsilon>0$, which improves previous results when ${\rm rank}(f)\ge3$. The authors develop an incidence-geometry framework, reducing general rank cases to a key rank-$k-1$ special case and leveraging the Sharir–Zahl incidence bounds for plane curves. They also fully characterize rank-1 polynomials and demonstrate an application to distinct $d$-volumes spanned by $(d+1)$-tuples on the moment curve in $\mathbb{R}^d$, illustrating the reach of their bounds. Overall, the work deepens the connection between algebraic structure and combinatorial expansion, with implications for higher-dimensional Erdős-type problems.
Abstract
Let $f\in \mathbb{R}[x_1,\ldots, x_k]$, for $k\ge 2$. For any finite sets $A_1,\ldots, A_k\subset \mathbb{R}$, consider the set $$ f(A_1,\ldots, A_k):=\{f(a_1,\ldots, a_k)\mid (a_1,\cdots,a_k)\in A_1\times\cdots \times A_k\}, $$ that is, the image of $A_1\times \cdots\times A_k$ under $f$. Extending a theorem of Elekes and Rónyai, which deals with the case $k=2$, and a result of Raz, Sharir, and De Zeeuw, dealing with the case $k=3$, it was proved Raz and Shem Tov, that for every choice of finite $A_1,\ldots, A_k\subset \mathbb{R}$, each of size $n$, one has \begin{equation}\label{RSbound} |f(A_1,\ldots,A_k)|=Ω(n^{3/2}), \end{equation} unless $f$ has some degenerate special form. In this paper, we introduce the notion of a {\it rank} of a $k$-variate polynomial $f$, denoted as ${\rm rank}(f)$. Letting $r={\rm rank}(f)$, we prove that \begin{equation} |f(A_1,\ldots,A_k)|=Ω\left(n^{\frac{5r-4}{2r}-\varepsilon}\right), \end{equation} for every $\varepsilon>0$, where the constant of proportionality depends on $\varepsilon$ and on ${\rm deg}(f)$. This improves the previous lower bound, for polynomials $f$ for which ${\rm rank}(f)\ge 3$. We present an application of our main result, to lower bound the number of distinct $d$-volumes spanned by $(d+1)$-tuples of points lying on the moment curve in $\mathbb{R}^d$.