Expanding Polynomials over 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.