Diophantine equations in separated variables
type: Item
instance of: scholarly article
title: Diophantine equations in separated variables
zbMATH Open document ID: 1399.11105
author: Robert F. Tichy
author: Dijana Kreso
published in: Periodica Mathematica Hungarica
publication date: 2018-04-27T00:00:00Z
review text: Let \(K\) be a number field, a finite set \(S\) of places of \(K\) that contains all Archimedean places and the ring \(O_S\) of \(S\)-integers of \(K\), and \(f, g\in K[x]\). The study of solutions in \(O_S\) of Diophantine equations of type \(f(x)= g(y)\) have been of long-standing interest to number theorists and several important results have been obtained \textit{D. Kreso} and \textit{R. F. Tichy} [Grazer Math. Ber. 363, 143--170 (2015; Zbl 1395.11054)]. In this paper, some standard methods of Galois theory for maps between curves are used to give simple and unifying proof of most of these results. For a polynomial \(f\), the roots of the derivative \(f^{\prime}\) are called critical points, and the values of \(f\) at critical points are called critical values. If for critical points \(b_i\) 's of \(f\), one has \(f(b_i) \neq f(b_j)\) when \(b_i \neq b_j\), then \(f\) is said to have all distinct critical values. In this paper it is proved that if \( \deg f \geq 3\), \(\deg g \geq 3\), \(f\) and \(g\) both have at least two distinct critical points and all distinct critical values, then the equation \(f(x) = g(y)\) has infinitely many solutions \(x\), \(y\) with a bounded \(O_S\)-denominator (i.e. there is a nonzero \(\delta\in O_S\) such \(\delta x, \delta y \in O_S\)) if and only if \(f(x) = g(\mu(x))\) for some linear \(\mu \in K[x]\). Furthermore, it is proved that if both \(f\) and \(g\) have at least two distinct critical points and equal critical values at at most two distinct critical points, and their derivatives do not satisfy some condition, then the equation \(f (x) = g(y)\) has only finitely many solutions with a bounded \(O_S\)-denominator, unless either \((\deg f, \deg g) = (3, 5)\), or \(f\) is indecomposable and \(g(x) = f (\nu(x))\) for some quadratic \(\nu\in K[x]\). Moreover, some interesting corollaries have been given.
Mathematics Subject Classification ID: 11D61
zbMATH DE Number: 6865418
zbMATH Keywords: Diophantine equations
zbMATH Keywords: monodromy group
zbMATH Keywords: permutation groups
zbMATH Keywords: polynomial decomposition
reviewed by: Dimitros Poulakis
MaRDI profile type: Publication
full work available at URL: s10998-017-0195-y
OpenAlex ID: W2741035571
Wikidata QID: Q59607794
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DOI: 10.1007/S10998-017-0195-Y
description: scientific article; zbMATH DE number 6865418