Pillai's conjecture for polynomials

In this paper we study the polynomial version of Pillai's conjecture on the exponential Diophantine equation \begin{equation*} p^n - q^m = f. \end{equation*} We prove that for any non-constant polynomial $f$ there are only finitely many vectors $(n,m,\mathrm{deg}\ p,\mathrm{deg}\ q)$ with integers $n,m \geq 2$ and non-constant polynomials $p,q$ such that Pillai's equation holds. Moreover, we will give some examples that there can still be infinitely many possibilities for the polynomials $p,q$.