# Expanding polynomials on sets with few products

## TL;DR

This paper proves that for finite sets with small product sets, the image under any polynomial is nearly quadratic in size unless the polynomial has a specific monomial form, extending sum-product type results.

## Contribution

It establishes a lower bound on the size of polynomial images for sets with small product sets, characterizing the exceptional polynomials.

## Key findings

- Polynomial images are large for sets with small product sets
- The bound is tight up to the polynomial's form
- Identifies the structure of polynomials that do not expand

## Abstract

In this note, we prove that if $A$ is a finite set of real numbers such that $|AA| = K|A|$, then for every polynomial $f \in \mathbb{R}[x,y]$ we have that $|f(A,A)| = \Omega_{K,\operatorname{deg} f}(|A|^2)$, unless $f$ is of the form $f(x,y) = g(M(x,y))$ for some monomial $M$ and some univariate polynomial $g$.