Improved bound for the $k$-variate Elekes--R\'onyai theorem

TL;DR

This paper improves the lower bounds on the size of the image of finite sets under multivariate polynomials, introducing a new notion of polynomial rank to generalize and strengthen previous results in combinatorial geometry.

Contribution

The paper introduces the concept of polynomial rank and establishes a new lower bound on the size of polynomial images, extending prior bounds for higher-rank polynomials.

Findings

New lower bound: |f(A_1,...,A_k)| = Ω(n^{(5r-4)/(2r) - ε}) for polynomials with rank r ≥ 3.

Introduces the notion of polynomial rank to analyze the structure of multivariate polynomials.

Application to bounding the number of distinct volumes spanned by points on the moment curve.

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 R\'onyai, 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)|=\Omega(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)|=\Omega\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 $\mathbb{R}^d$.