Expanding Polynomials over the rationals

TL;DR

This paper characterizes non-expanding bivariate polynomials over the rationals, showing they must have specific additive or multiplicative forms, extending previous results over the reals.

Contribution

It extends the classification of non-expander polynomials from the reals to the rationals, identifying their specific algebraic structures.

Findings

Non-expander polynomials over rationals have additive or multiplicative forms.

Homogeneous non-expander polynomials are of the form c(x+ay)^α or c(xy)^α.

The result generalizes earlier work over the reals to the rationals.

Abstract

Let $F(x,y)$ be a polynomial over the rationals. We show that if $F$ is not an expander (over the rationals) then it has a special multiplicative or additive form. For example if $F$ is a homogeneous non-expander polynomial then $F(x,y)=c(x+ay)^\alpha$ or $F(x,y)=c(xy)^\alpha .$ This is an extension of an earlier result of Elekes and R\'onyai who described the structure of two-variate polynomials which are not expanders over the reals.