Extensions of the theorem of Julia-Ritt for polynomials

In 1922 Julia characterized the pairs of rational functions over the field of complex numbers which satisfy f(g) = g(f). The next year Ritt used his theory on decomposition of rational functions to give an alternative proof for a slightly refined form of Julia’s theorem. Several authors have extended the Julia-Ritt theorem for polynomials to wider classes of fields. In the lecture I shall report on joint work with Lajos Hajdu resulting into extensions of the Julia-Ritt for polynomials to equations of the form f(g(h)) = h(g(f)) and f(g) = g(h). Our proofs are based on a paper by Zieve and Mueller (2008).