Composite values of polynomial power sums

Abstract

Let (G n (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, let m≥2 be a given integer. We ask for n∈ℕ such that the equation G n (x)=g∘h is satisfied for a polynomial g∈ℂ[x] with degg=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.