The Diophantine equation $f(x)=g(y)$ for polynomials with simple rational roots

TL;DR

This paper investigates Diophantine equations of the form $f(x)=g(y)$ with rational roots, establishing conditions for infinitely many solutions and connecting these to Prouhet-Tarry-Escott tuples, with applications to products of consecutive integers.

Contribution

It provides necessary and sufficient conditions for the existence of infinitely many rational solutions with bounded denominators, linking these equations to Prouhet-Tarry-Escott tuples and related problems.

Findings

Conditions for infinite solutions are established.

Equations with simple rational roots relate to Prouhet-Tarry-Escott tuples.

Applications include problems on products of consecutive integers.

Abstract

In this paper we consider Diophantine equations of the form $f(x)=g(y)$ where $f$ has simple rational roots and $g$ has rational coefficients. We give strict conditions for the cases where the equation has infinitely many solutions in rationals with a bounded denominator. We give examples illustrating that the given conditions are necessary. It turns out that such equations with infinitely many solutions are strongly related to Prouhet-Tarry-Escott tuples. In the special, but important case when $g$ has only simple rational roots as well, we can give a simpler statement. Also we provide an application to equal products with terms belonging to blocks of consecutive integers of bounded length. The latter theorem is related to problems and results of Erd\H{o}s and Turk, and of Erd\H{o}s and Graham.