Transcendental numbers.
(Трансцендентные числа.)

*(Russian)*Zbl 0629.10026
Moskva: “Nauka”. Glavnaya Redaktsiya Fiziko-Matematicheskoĭ Literatury. 448 p. R. 4.60 (1987).

The present book on transcendental numbers presents in a clear and beautiful way the method developed by Siegel, and later considerably improved by the author himself, for the consideration of the arithmetic properties of \(E\)-functions. The book also gives a detailed discussion on applications of the method for concrete \(E\)-functions and on the measures of transcendence and algebraic independence of the function values at algebraic points. Using this book one can study this topic of transcendental number theory well, and the book is also very useful for mathematicians working in this field, too.

The book begins with a chapter containing basic classical results on the approximation of real and algebraic numbers. The second chapter considers the values of the exponential function \(e^z\) at algebraic points and contains e.g. the proof of the Lindemann-Weierstrass theorem. In chapters 3 and 4 the author gives in a beautiful way in full generality the method of Siegel-Shidlovskiĭ on the transcendence and algebraic independence of the values of \(E\)-functions at algebraic points.

The following chapters 5–10 are devoted to the applications of the main theorems. First the E-function solutions of the first and second order linear differential equations are considered, including Bessel and Kummer functions. Chapter 7 gives some interesting applications to solutions of more general systems of differential equations. Chapter 8 introduces an arithmetical method for the investigation of algebraic independence of \(E\)-functions. The next chapter contains Nesterenko’s nice proof of an algebraic independence result for the solutions of a set of linear second order differential equations, implying e.g. Siegel’s most general statements on the values of Bessel functions. Chapter 10 considers \(E\)-function solutions of linear differential equations of prime order and is mainly based on works of Oleĭnikov and Salikhov concerning the irreducibility considerations of differential equations. Here we point to a recent paper of F. Beukers, W. D. Brownawell and G. Heckman [Ann. Math. (2) 127, No. 2, 279–308 (1988; Zbl 0652.10027)] on Siegel’s normality condition, which implies results analogous to some results of this chapter.

The rest of the book considers quantitative results for the values of \(E\)-functions. Measures of transcendence and algebraic independence in the case of \(IE\)-functions (\(I\) an imaginary quadratic field) are considered in chapter 11, and the general \(KE\)-function case (\(K\) an algebraic number field) is given in chapter 12. Chapter 13 considers the effectivity in the above measures. The book ends with a short discussion on some further results obtained by Siegel’s method.

The book begins with a chapter containing basic classical results on the approximation of real and algebraic numbers. The second chapter considers the values of the exponential function \(e^z\) at algebraic points and contains e.g. the proof of the Lindemann-Weierstrass theorem. In chapters 3 and 4 the author gives in a beautiful way in full generality the method of Siegel-Shidlovskiĭ on the transcendence and algebraic independence of the values of \(E\)-functions at algebraic points.

The following chapters 5–10 are devoted to the applications of the main theorems. First the E-function solutions of the first and second order linear differential equations are considered, including Bessel and Kummer functions. Chapter 7 gives some interesting applications to solutions of more general systems of differential equations. Chapter 8 introduces an arithmetical method for the investigation of algebraic independence of \(E\)-functions. The next chapter contains Nesterenko’s nice proof of an algebraic independence result for the solutions of a set of linear second order differential equations, implying e.g. Siegel’s most general statements on the values of Bessel functions. Chapter 10 considers \(E\)-function solutions of linear differential equations of prime order and is mainly based on works of Oleĭnikov and Salikhov concerning the irreducibility considerations of differential equations. Here we point to a recent paper of F. Beukers, W. D. Brownawell and G. Heckman [Ann. Math. (2) 127, No. 2, 279–308 (1988; Zbl 0652.10027)] on Siegel’s normality condition, which implies results analogous to some results of this chapter.

The rest of the book considers quantitative results for the values of \(E\)-functions. Measures of transcendence and algebraic independence in the case of \(IE\)-functions (\(I\) an imaginary quadratic field) are considered in chapter 11, and the general \(KE\)-function case (\(K\) an algebraic number field) is given in chapter 12. Chapter 13 considers the effectivity in the above measures. The book ends with a short discussion on some further results obtained by Siegel’s method.

Reviewer: Keijo Väänänen (Oulu)

##### MSC:

11J81 | Transcendence (general theory) |

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

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

11J91 | Transcendence theory of other special functions |