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**Mahler functions and transcendence.**
*(English)*
Zbl 0876.11034

Lecture Notes in Mathematics. 1631. Berlin: Springer. viii, 185 p. (1996).

Mahler’s method in transcendence makes a fascinating story. Mahler himself has described its beginnings in the study of the transcendence properties of the function \(f(z)= \sum z^{2^n}\) and the later development of the method into a machine for handling the algebraic independence of values of interesting functions such as \(\sum [n\omega] z^n\). (See, for example, the obituary [A. J. van der Poorten, J. Aust. Math. Soc., Ser. A 51, 343-380 (1991; Zbl 0738.01015)]). The key property of the first of these Mahler functions is the functional equation \(f(z)= f(z^2)+z\). The challenge to deal with functions satisfying simple functional equations of this type has prompted advances and applications in the analytic theory of recurrences (Kubota), differential algebra (Nishioka) and elimination theory (after Mendès-France observed the connection between Mahler functions and regular sequences generated by finite automata, giving new incentive to find strong theorems for algebraic independence with no unnecessary hypotheses).

The author has solved several of the old conjectures of Mahler and improved the general theory in a number of important respects. She has been able to correct errors and omissions in the work of others in this field on several occasions. She can fairly claim to be one of the leading exponents of the method and the related theory of differential algebra and elimination theory.

The notes on Mahler functions and transcendence bring together recent published work of the author and others. Chapter 1 illustrates the range of results and examples for functions of one variable, including recent examples of Becker. Chapter 2 gives the author’s proof of Masser’s vanishing theorem, which was one of the key breakthroughs in the general theory. Chapter 3 is her improvement of Kubota’s algebraic independence results. Chapter 4 contains the application of Nesterenko’s theory (although it still proves difficult to unravel the intricacies of Nesterenko’s method to the uninitiated). Chapter 5 describes some examples motivated by the study of regular sequences (including the proof of a very longstanding conjecture of Mahler). The notes cover the full range of the topic and the main results of chapters 2, 3 and 4 are of major significance.

The author has solved several of the old conjectures of Mahler and improved the general theory in a number of important respects. She has been able to correct errors and omissions in the work of others in this field on several occasions. She can fairly claim to be one of the leading exponents of the method and the related theory of differential algebra and elimination theory.

The notes on Mahler functions and transcendence bring together recent published work of the author and others. Chapter 1 illustrates the range of results and examples for functions of one variable, including recent examples of Becker. Chapter 2 gives the author’s proof of Masser’s vanishing theorem, which was one of the key breakthroughs in the general theory. Chapter 3 is her improvement of Kubota’s algebraic independence results. Chapter 4 contains the application of Nesterenko’s theory (although it still proves difficult to unravel the intricacies of Nesterenko’s method to the uninitiated). Chapter 5 describes some examples motivated by the study of regular sequences (including the proof of a very longstanding conjecture of Mahler). The notes cover the full range of the topic and the main results of chapters 2, 3 and 4 are of major significance.

Reviewer: J.H.Loxton (North Ryde)

### MSC:

11J81 | Transcendence (general theory) |

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

11B85 | Automata sequences |

11B83 | Special sequences and polynomials |

11J85 | Algebraic independence; Gel’fond’s method |