Parametrization of integral values of polynomials

We will recall a recent result about the classification of those polynomial in one variable with rational coefficients whose image over the integer is equal to the image of an integer coefficients polynomial in possibly many variables. These set is polynomially generated over the integers by a family of polynomials whose denominator is $2$ and they have a symmetry with respect to a particular axis. We will also give a description of the linear factors of the bivariate separated polynomial $f(X)-f(Y)$ over a number field $K$, which we need to formulate a conjecture for a generalization of the previous result over a generic number field