On the Diophantine equation G_n(x) = G_m(P(x)): Higher-order recurrences

Let K be a field of characteristic 0 and let (G_n(x)) _n=0^∞ be a linear recurring sequence of degree d in K[x] defined by the initial terms G₀, ..., G_d-1 ∈ K[x] and by the difference equation G_n+d(x) = A _d-1(x)G_n+d-1(x) + ... + A₀(x)G_n(x), for n ≥ 0, with A₀, ..., A_d-1 ∈ K [x]. Finally, let P(x) be an element of K[x]. In this paper we are giving fairly general conditions depending only on G₀,..., G_d-1 on P, and on A₀, ..., A_d-1 under which the Diophantine equation G _n(x) = G_m(P(x)) has only finitely many solutions (n, m) ∈ ℤ², n, m ≥ 0. Moreover, we are giving an upper bound for the number of solutions, which depends only on d. This paper is a continuation of the work of the authors on this equation in the case of second-order linear recurring sequences.