# Diophantine equations in separated variables and lacunary polynomials

## TL;DR

This paper investigates Diophantine equations involving lacunary polynomials, focusing on their solutions and the structure of such polynomials under functional composition, advancing understanding of their algebraic properties.

## Contribution

The paper develops new results on the composition structure of lacunary polynomials and applies these to Diophantine equations of the form f(x)=g(y).

## Key findings

- Finiteness criterion for solutions in S-integers based on polynomial composition.
- New results on the functional decomposition of lacunary polynomials.
- Insights into the structure of lacunary polynomials with respect to solutions of Diophantine equations.

## 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.