Diophantine equations in separated variables and lacunary polynomials

Abstract

We study Diophantine equations of type [Formula: see text], where [Formula: see text] and [Formula: see text] are lacunary polynomials. According to a well-known finiteness criterion, for a number field [Formula: see text] and nonconstant [Formula: see text], the equation [Formula: see text] has infinitely many solutions in [Formula: see text]-integers [Formula: see text] only if [Formula: see text] and [Formula: see text] 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.