##
**Diophantine approximation on linear algebraic groups. Transcendence properties of the exponential function in several variables.**
*(English)*
Zbl 0944.11024

Grundlehren der Mathematischen Wissenschaften. 326. Berlin: Springer. xxiii, 633 p. (2000).

This extensive monograph gives an excellent report on the present state of the art of some of the most important parts in modern transcendence theory. In the following, it will be tried to provide the reader with an idea of what he may find in this book.

The first four chapters (1: introduction and historical survey, 2: transcendence proofs in one variable, 3: heights of algebraic numbers, 4: criterion of Schneider-Lang) may serve as an introduction to the subject. For instance, the first three of these do not require much preliminary knowledge and include already complete proofs of a number of classical transcendence results. The first (of three) proof(s) of Baker’s theorem on linear independence of logarithms of algebraic numbers, in Ch. 4, follows an argument of Bertrand and Masser who derived it from the Schneider-Lang criterion concerning algebraic values of meromorphic functions on cartesian products.

Part II (linear independence of logarithms and measures) comprises chapters 5-7 on zero estimate (by D. Roy), linear independence of logarithms of algebraic numbers, homogeneous measures for linear independence. In particular, in Ch. 6 Schneider’s method is extended (as in Ch. 9 for the inhomogeneous case), and in Ch. 7 a relatively simple proof for a linear independence measure is explained.

Part III (multiplicities in higher dimension) contains multiplicity estimates (again by D. Roy), refined measures, Baker’s method as chapters 8-10 where in the last one Baker’s argument extending Gelfond’s solution of Hilbert’s seventh problem is explained.

A central result in this book, entitling Part IV, is the so-called linear subgroup theorem (LST) which occurs in two forms. The qualitative one (Ch. 11: points whose coordinates are logarithms of algebraic numbers) gives a lower bound for the dimension of the \({\mathbb C}\)-vector subspace of \({\mathbb C}^d\) spanned by points whose coordinates are either algebraic numbers or logarithms of such numbers. Finally, lower bounds for the rank of matrices (title of Ch. 12) are deduced from the LST.

The (last) Part V, simultaneous approximation of values of the exponential function in several variables, starts with Ch. 13: a quantitative version of the LST. This version, by no means a simple statement, includes a lot of diophantine estimates, as shown in Ch. 14: applications to diophantine approximation. Ch. 15 deals with algebraic independence (criteria, small and large transcendence degrees).

Altogether, the author’s emphasis is not only on the results, but also on the methods; this is why he gives sometimes several proofs of the same result. An original feature is certainly the systematic use of Laurent’s interpolation determinants instead of the classical construction of auxiliary functions via Thue-Siegel’s lemma.

What is excluded from the presentation? Not considered are elliptic curves, abelian varieties, and more generally nonlinear algebraic groups. Not discussed is Nesterenko’s breakthrough concerning the algebraic independence of \(\pi\) and \(e^\pi\); excluded are also elliptic, theta, and abelian functions as well as all kinds of non-archimedean considerations.

The reader having enough time and energy may learn from this carefully written book a great deal of modern transcendence theory from the very beginning. In this process, the many included exercises may be very helpful. Everybody interested in transcendence will certainly admire the author’s achievement to present such a clear and complete exposition of a topic growing so fast. The value of Waldschmidt’s new monograph for the further development of the subject cannot be overestimated.

The first four chapters (1: introduction and historical survey, 2: transcendence proofs in one variable, 3: heights of algebraic numbers, 4: criterion of Schneider-Lang) may serve as an introduction to the subject. For instance, the first three of these do not require much preliminary knowledge and include already complete proofs of a number of classical transcendence results. The first (of three) proof(s) of Baker’s theorem on linear independence of logarithms of algebraic numbers, in Ch. 4, follows an argument of Bertrand and Masser who derived it from the Schneider-Lang criterion concerning algebraic values of meromorphic functions on cartesian products.

Part II (linear independence of logarithms and measures) comprises chapters 5-7 on zero estimate (by D. Roy), linear independence of logarithms of algebraic numbers, homogeneous measures for linear independence. In particular, in Ch. 6 Schneider’s method is extended (as in Ch. 9 for the inhomogeneous case), and in Ch. 7 a relatively simple proof for a linear independence measure is explained.

Part III (multiplicities in higher dimension) contains multiplicity estimates (again by D. Roy), refined measures, Baker’s method as chapters 8-10 where in the last one Baker’s argument extending Gelfond’s solution of Hilbert’s seventh problem is explained.

A central result in this book, entitling Part IV, is the so-called linear subgroup theorem (LST) which occurs in two forms. The qualitative one (Ch. 11: points whose coordinates are logarithms of algebraic numbers) gives a lower bound for the dimension of the \({\mathbb C}\)-vector subspace of \({\mathbb C}^d\) spanned by points whose coordinates are either algebraic numbers or logarithms of such numbers. Finally, lower bounds for the rank of matrices (title of Ch. 12) are deduced from the LST.

The (last) Part V, simultaneous approximation of values of the exponential function in several variables, starts with Ch. 13: a quantitative version of the LST. This version, by no means a simple statement, includes a lot of diophantine estimates, as shown in Ch. 14: applications to diophantine approximation. Ch. 15 deals with algebraic independence (criteria, small and large transcendence degrees).

Altogether, the author’s emphasis is not only on the results, but also on the methods; this is why he gives sometimes several proofs of the same result. An original feature is certainly the systematic use of Laurent’s interpolation determinants instead of the classical construction of auxiliary functions via Thue-Siegel’s lemma.

What is excluded from the presentation? Not considered are elliptic curves, abelian varieties, and more generally nonlinear algebraic groups. Not discussed is Nesterenko’s breakthrough concerning the algebraic independence of \(\pi\) and \(e^\pi\); excluded are also elliptic, theta, and abelian functions as well as all kinds of non-archimedean considerations.

The reader having enough time and energy may learn from this carefully written book a great deal of modern transcendence theory from the very beginning. In this process, the many included exercises may be very helpful. Everybody interested in transcendence will certainly admire the author’s achievement to present such a clear and complete exposition of a topic growing so fast. The value of Waldschmidt’s new monograph for the further development of the subject cannot be overestimated.

Reviewer: P.Bundschuh (Köln)

### MSC:

11J81 | Transcendence (general theory) |

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

20G15 | Linear algebraic groups over arbitrary fields |

20G30 | Linear algebraic groups over global fields and their integers |