Asymptotics for Pillai's problem with polynomials

Let $a_1(x)p_1(x)^n + \cdots + a_k(x)p_k(x)^n$ as well as $b_1(x)q_1(x)^m + \cdots + b_l(x) q_l(x)^m$ be two polynomial power sums where the complex polynomials $p_i(x)$ and $q_j(x)$ are all non-constant. Then in the present paper we will give an asymptotic for the number of pairs $(n,m) \in \mathbb{N}^2$ such that the degree of the sum of these two power sums is between $0$ and $d$ when $d$ goes to infinity.