Diophantine equations in separated variables and lacunary polynomials

Abstract

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 $f,g\in K[x]$, 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.

older paper (from 2015/2016)

References in corpus (1)

• On Ritt's polynomial decomposition theorems