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 is an element of 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 behavior 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 on the latter topic, and develop new ones, in relation to Diophantine applications.