Diophantine equations in separated variables and polynomial power sums

TL;DR

This paper investigates Diophantine equations where polynomials are power sums, proving that under certain conditions, only finitely many rational solutions with bounded denominators exist unless in trivial cases.

Contribution

It applies Bilu and Tichy's finiteness criterion to polynomial power sum equations, establishing conditions for the finiteness of rational solutions.

Findings

Infinitely many rational solutions are only possible in trivial cases.

Under certain assumptions, solutions with bounded denominators are finite.

The approach uses a finiteness criterion by Bilu and Tichy.

Abstract

We consider Diophantine equations of the shape $ f(x) = g(y) $, where the polynomials $ f $ and $ g $ are elements of power sums. Using a finiteness criterion of Bilu and Tichy, we will prove that under suitable assumptions infinitely many rational solutions $ (x,y) $ with a bounded denominator are only possible in trivial cases.