Diophantine approximations and diophantine equations
type: Item
instance of: scholarly article
title: Diophantine approximations and diophantine equations
zbMATH Open document ID: 0754.11020
published in: Lecture Notes in Mathematics
publication date: 1992-09-18T00:00:00Z
review text: In 1909, A. Thue proved the following theorem. Let \(F(X,Y)\in\mathbb{Z}[X,Y]\) be an irreducible homogeneous polynomial of degree at least three. Then the diophantine equation \(F(X,Y)=m\) has only finitely many solutions \((x,y)\in\mathbb{Z}^ 2\). A curious property of his method is that, while it proves that the set of solutions is finite, it provides only an upper bound on the \textit{number} of solutions, not on their size. Thus, if this upper bound is not sharp, then one cannot prove that a given set of solutions actually contains all the solutions. Thus the method is described as being \textit{ineffective}. Over the years, this method has led to more general theorems, but in all of these generalizations the ineffectivity has remained. Morover, until recently most authors have been content to prove finiteness theorems in various contexts without mentioning any finite bounds on the number of solutions. Recently, however, Evertse, Silverman, Bombieri, Schmidt, Mueller, van der Poorten, and others have explicitly determined upper bounds for many of these theorems. That is a major theme of this book. This book may also be regarded as a general introduction to diophantine approximations. In that respect it is similar to \textit{W. M. Schmidt} [Diophantine approximation (Lect. Notes Math. 785) (Springer, Berlin 1980; Zbl 0421.10019)], but there is little overlap. The book starts with a general introduction to the geometry of numbers, Siegel's lemma, and heights; this includes the recent version of Siegel's lemma due to \textit{E. Bombieri} and \textit{J. Vaaler} [Invent. Math. 73, 11- 32; addendum 75, 377 (1983; Zbl 0533.10030)]. The second chapter discusses and proves Roth's theorem, assuming the generalized Dyson lemma of \textit{H. Esnault} and \textit{E. Viehweg} [Invent. Math. 78, 445-490 (1984; Zbl 0532.10020)]. (Recall that Roth's theorem is the assertion that for all \(\varepsilon>0\) and all \(\alpha\in\overline\mathbb{Q}\), the set of \(x/y\in\mathbb{Q}\) …
reviewed by: Paul Vojta
Mathematics Subject Classification ID: 11J25
Mathematics Subject Classification ID: 11J68
Mathematics Subject Classification ID: 11D57
Mathematics Subject Classification ID: 11-02
zbMATH DE Number: 52609
zbMATH Keywords: diophantine equations
zbMATH Keywords: diophantine approximations
zbMATH Keywords: Thue's method
zbMATH Keywords: explicit upper bounds
zbMATH Keywords: hyperelliptic equations
zbMATH Keywords: number of solutions
zbMATH Keywords: Roth's theorem
zbMATH Keywords: Thue equation
zbMATH Keywords: unit equation
zbMATH Keywords: superelliptic equations
zbMATH Keywords: Schmidt's Subspace Theorem
zbMATH Keywords: norm form equations
MaRDI profile type: Publication
Wikidata QID: Q29394607
DOI: 10.1007/BFB0098246
author: Wolfgang M. Schmidt
description: scientific article; zbMATH DE number 52609