To quote the author: “The theory of diophantine equations, having the same deep history as that of human culture as a whole, has from time to time undergone its ups and downs.”
The purpose of this outstanding monograph is to show that in the last 15 years this subject has undergone a period of explosive growth, at least in the theory of diophantine equations with two integer unknowns. Starting from Baker’s results on linear forms in the logarithms of algebraic numbers, the author develops a systematic theory of all general types of diophantine equations and considers solutions in rational integers, \(S\)-integers and algebraic integers of a fixed field. This makes it possible to obtain a wider arsenal of arithmetic facts than would be possible if the author had considered only the classical cases. This also allows the analysis of several types of equations in several unknowns (e.g., norm form equations).
In the course of analysis the author’s main focus is on the influence of the basic parameters of the associated algebraic number field on the construction of upper bounds for the solutions of the diophantine equation. Here one soon discovers the influence of the regulator on these bounds (and, through the Brauer-Siegel theorem, the class number). This idea is used to construct a large number of algebraic number fields with large class number. The author also shows how further work in this direction would lead to a substantial improvement in the bounds for the solutions of the diophantine equation.
Let us now give a chapter summary. In Chapter 1 the author gives us a bird’s-eye view of all the basic early results on the subject. First we find an outline of a proof of Runge’s Theorem, the forerunner of all theorems on the finiteness of the number of solutions of certain diophantine equations. Then he presents a summary of Liouville’s inequality, the Thue-Siegel-Roth theorem, Skolem’s \(p\)-adic method and Hilbert’s Seventh Problem with Baker’s generalizations. This chapter gives an excellent introduction to these subjects for anyone wishing to learn the basics of transcendental number theory without getting lost in a myriad of technical details.
In Chapter 2 the author first develops some basic lemmas on the size and height of algebraic numbers and then proves two key lemmas on units and regulators which he uses throughout the rest of the work. He also gives a summary of the Brauer-Siegel theorem and the \(p\)-adic complex variable theory necessary for Chapter 3.
Chapter 3 is devoted to a development of the Baker-Coates theory of linear forms in the logarithms of algebraic numbers. Most of this chapter is devoted to the \(p\)-adic case, with only a brief summary in Section 8 being given to the real case. Here it must be emphasized that the results given for the real case are those of Baker (1977) and are not the best known today. M. Waldschmidt [Acta Arith. 37, 257–283 (1980; Zbl 0357.10017)] has obtained a substantial improvement of Baker’s work and van der Poorten, Loxton and Waldschmidt have improved Waldschmidt’s earlier result still more. Regrettably, this latter work has never been published.
Chapters 4 and 5 are devoted to an analysis of one of the central problems of diophantine analysis, the representation of numbers by binary forms (Thue’s equation). In Chapter 4 the author returns to the connection between the magnitude of the solutions of Thue’s equation and rational approximations to algebraic numbers. The analysis is contrary to Thue’s in that the author gets bounds for the approximations as a corollary of the bounds derived for the solutions of the equation. In this way the author obtains an effective strengthening of Liouville’s inequality. Special attention is paid to showing how the regulator, height and size of the associated algebraic number field influence the size of solutions of Thue’s equation. In Chapter 5 the author develops the theory of solutions of Thue’s equation in \(S\)-integers and uses this to obtain rational approximations to algebraic numbers in different non-Archimedean metrics. He then studies the solution of Thue’s equation in rational numbers whose denominators contain a fixed set of primes. The chapter concludes with a study of approximations of algebraic numbers by algebraic numbers of a fixed field.
Chapters 6 and 7 are devoted to a study of the equation \(f(x)=Ay^m\), \(m\ge 2\), where \(f(x)\) is a polynomial having at least 3 simple roots if \(m=2\) and at least 2 simple roots if \(m\ge 3\). Here we find bounds derived for Mordell’s equation \(y^2 - k = x^3\), the general equation \(f(x)=Ay^m\), where \(m\) is both fixed and variable, and Catalan’s equation \(x^u - y^v =1\).
In Chapter 8 the author uses the Brauer-Siegel theorem and the preceding results on the influence of the regulator of the associated algebraic number field to study the more general problem of the size of the ideal class group. He shows that algebraic number fields with “small” regulator (“large” class number) occur quite often and, in a certain sense, constitute a majority. He also shows how further work in this area would lead to substantial improvements in the bounds for solutions of Thue’s equation.
In Chapter 9, the concluding chapter, the author studies some effective variants of Hilbert on irreducibility of polynomials and uses these to study abelian points on algebraic curves.
All in all, this is a truly excellent work, very diligently done, and certainly worthy of translation into many languages.