Integral solutions of separated-variables equations

In my talk I shall explain how to explicitly characterize the finiteness of the set of integral solutions of a Diophantine equation with separated variables, i.e. an equation of the form $f(x)=g(y)$ with $f$, $g$ polynomials having integral coefficients, a result known as the Bilu–Tichy criterion. The proof uses the geometric description of the situation and ultimately follows as an application of Siegel's theorem. Afterwards, I shall discuss which additional information one can get if the equation has infinitely many solutions. These general statements will then be applied to concrete equations where Stirling numbers are involved. At the end of the talk, if time permits, I shall also mention some effective results for some (very) special cases.