Diophantine equations in separated variables and lacunary polynomials

We study Diophantine equations of type f(x)=g(y), where f and g are lacunary polynomials. According to a well known finiteness criterion, for a number field K and nonconstant f,g∈K[x], the equation f(x)=g(y) has infinitely many solutions in S-integers x,y only if f and g are representable as a functional composition of lower degree polynomials in a certain prescribed way. The behaviour of lacunary polynomials with respect to functional composition is a topic of independent interest, and has been studied by several authors. In this paper we utilize known results and develop some new results on the latter topic.