A characterization of linear polynomials (Q2581378)

The characterization is the following: linear polynomials over a number field \(k\) are the only non-constant polynomials \(f\) for which the equation \(f(x)=g(y)\) has a \(k\)-rational solution for each \(g\in k[X]\). The proof of this result is elementary. The author considers also possible conditions on the respective degrees \(m\) of \(f\) and \(n\) of \(g\), both with integer coefficients, when \(m>1\), in order that there exists such a polynomial \(g\) of degree \(n\) for which the previous equation \(f(x)=g(y)\) has no rational solution. He considers several questions related to this problem.