Expanding Polynomials over the rationals

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 $F(x,y)=c(x+ay)^α$ or $F(x,y)=c(xy)^α.$ This is an extension of an earlier result of Elekes and Rónyai who described the structure of two-variate polynomials which are not expanders over the reals.

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