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,$\mathrm{d}\mathrm{e}\mathrm{g}$\ p,$\mathrm{d}\mathrm{e}\mathrm{g}$
q) $withintegers$ n,m \geq 2 $andnon-constantpolynomials$ 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 $p,q$.