On generalized Dhombres equations with nonconstant polynomial solutions in the complex plane

We characterize functional equations of the form f (z f (z)) = f (z)^k+1, z ∈ C, with k ∈ N, as those generalized Dhombres equations f (z f (z))= φ (f (z)), z ∈ C, with given entire function φ, which have a nonconstant polynomial solution f.