Diophantine approximations and diophantine equations.

*(English)*Zbl 0754.11020
Lecture Notes in Mathematics. 1467. Berlin etc.: Springer-Verlag. viii, 217 p. (1991).

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 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 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 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 E. Bombieri and 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 H. Esnault and 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 \(x/y\in\mathbb{Q}\) with \(x\in\mathbb{Z}\), \(y\in\mathbb{N}\), and \(| x/y- \alpha|<y^{-2-\varepsilon}\) is finite; this has also been generalized to number fields.) In addition to discussing bounds on the number of good approximations in Roth’s theorem, he also gives a “moving targets” generalization, where the algebraic numbers \(\alpha\) may vary in a bounded way with respect to \(y\).

Chapter III deals with the Thue equation discussed above. This includes proving bounds on the number of solutions, as well as sharper bounds if \(F(X,Y)\) has at most three terms. Chapter IV deals with the unit equation \(u+v=1\), where \(u\) and \(v\) are units in a subring of \(\overline\mathbb{Q}\) of finite type over \(\mathbb{Z}\). This equation has only finitely many solutions; in turn this implies finiteness results for hyperelliptic and superelliptic equations \(y^ n=f(x)\) \((n=2,\deg f\geq 3\) or \(n\geq 3\), \(\deg f\geq 2)\). Finally, Chapter V describes higher dimensional results, all of which are corollaries of Schmidt’s Subspace Theorem. This includes more general unit equations \(u_ 1+\cdots+u_ n=1\), but the primary goal of the chapter is proving a theorem (Theorem 3B) on norm form equations.

The author is careful not to require a lot of knowledge on the part of the reader. Many of the more technical proofs have been omitted, and whatever specialized knowledge is required (e.g., use of elliptic curves in Chapter IV) is described in detail.

There is one error, however, concerning Dyson’s lemma on page 46. In the inequality for the \(m=2\) case, \(k'-1\) should be \(k'-2\). Also Viola proved the same Dyson lemma as Bombieri; the variant that Schmidt mentions was an intermediate step in his proof.

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 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 E. Bombieri and 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 H. Esnault and 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 \(x/y\in\mathbb{Q}\) with \(x\in\mathbb{Z}\), \(y\in\mathbb{N}\), and \(| x/y- \alpha|<y^{-2-\varepsilon}\) is finite; this has also been generalized to number fields.) In addition to discussing bounds on the number of good approximations in Roth’s theorem, he also gives a “moving targets” generalization, where the algebraic numbers \(\alpha\) may vary in a bounded way with respect to \(y\).

Chapter III deals with the Thue equation discussed above. This includes proving bounds on the number of solutions, as well as sharper bounds if \(F(X,Y)\) has at most three terms. Chapter IV deals with the unit equation \(u+v=1\), where \(u\) and \(v\) are units in a subring of \(\overline\mathbb{Q}\) of finite type over \(\mathbb{Z}\). This equation has only finitely many solutions; in turn this implies finiteness results for hyperelliptic and superelliptic equations \(y^ n=f(x)\) \((n=2,\deg f\geq 3\) or \(n\geq 3\), \(\deg f\geq 2)\). Finally, Chapter V describes higher dimensional results, all of which are corollaries of Schmidt’s Subspace Theorem. This includes more general unit equations \(u_ 1+\cdots+u_ n=1\), but the primary goal of the chapter is proving a theorem (Theorem 3B) on norm form equations.

The author is careful not to require a lot of knowledge on the part of the reader. Many of the more technical proofs have been omitted, and whatever specialized knowledge is required (e.g., use of elliptic curves in Chapter IV) is described in detail.

There is one error, however, concerning Dyson’s lemma on page 46. In the inequality for the \(m=2\) case, \(k'-1\) should be \(k'-2\). Also Viola proved the same Dyson lemma as Bombieri; the variant that Schmidt mentions was an intermediate step in his proof.

Reviewer: P.Vojta (Berkeley)

##### MSC:

11J25 | Diophantine inequalities |

11J68 | Approximation to algebraic numbers |

11D57 | Multiplicative and norm form equations |

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |