Composite values of polynomial power sums

Let (Gn(x))n=0 be a d-th order linear recurrence sequence having polynomial characteristic roots, one of which has degree strictly greater than the others. Moreover, letm ≥ 2 be a given integer. We ask for n ∈ N such that the equation Gn(x) = g ◦ h is satisfied for a polynomial g ∈ C[x] with deg g = m and some polynomial h ∈ C[x]with degh > 1. We prove that for all but finitely many n these decompositions can be described in “finite terms” coming from a generic decomposition parameterized by an algebraic variety. All data in this description will be shown to be effectively computable.