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

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 Erdős and Turk, and of Erdős and Graham.

On polynomials with only rational roots

“…They cover norm forms that are crucial in Schmidt's Subspace Theorem [20], and index forms and discriminant forms, see Evertse and Győry [12]. Many papers on Diophantine equations deal with polynomials in $${\mathbb {Z}}[x]$$ with only rational roots themselves, see, for example, Section 2 of Hajdu and Tijdeman [15].…”