**Tata Institute of Fundamental Research Publications**

Volume: 14;
2008;
157 pp;
Softcover

MSC: Primary 11;
**Print ISBN: 978-81-7319-984-4
Product Code: TIFR/14**

List Price: $40.00

Individual Member Price: $32.00

# Algebraic Independence

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*Yu. V. Nesterenko*

A publication of the Tata Institute of Fundamental Research

This book is an expanded version of the notes of a course of lectures given by at the Tata Institute of Fundamental Research in 1998. It deals with several important results and methods in transcendental number theory.

First, the classical result of Lindemann–Weierstrass and its applications are dealt with. Subsequently, Siegel's theory of \(E\)-functions is developed systematically, culminating in Shidlovskii's theorem on the algebraic independence of the values of the \(E\)-functions satisfying a system of differential equations at certain algebraic values. Proof of the Gelfond–Schneider Theorem is given based on the method of interpolation determinants introduced in 1992 by M. Laurent.

The author's famous result in 1996 on the algebraic independence of the values of the Ramanujan functions is the main theme of the reminder of the book. After deriving several beautiful consequences of his result, the author develops the algebraic material necessary for the proof. The two important technical tools in the proof are Philippon's criterion for algebraic independence and zero bound for Ramanujan functions. The proofs of these are covered in detail.

The author also presents a direct method, without using any criterion for algebraic independence as that of Philippon, by which one can obtain lower bounds for transcendence degree of finitely generated field \(\mathbb Q(\omega_1,\ldots,\omega_m)\). This is a contribution towards Schanuel's conjecture.

The book is self-contained and the proofs are clear and
lucid. A brief history of the topics is also given. Some sections
intersect with Chapters 3 and 10 of *Introduction to Algebraic
Independence Theory*, Lecture Notes in Mathematics, Springer,
1752, edited by Yu. V. Nesterenko and P. Philippon.

A publication of the Tata Institute of Fundamental Research. Distributed worldwide except in India, Bangladesh, Bhutan, Maldavis, Nepal, Pakistan, and Sri Lanka.

Narosa Publishing House for the Tata Institute of Fundamental Research. Distributed worldwide except in India, Bangladesh, Bhutan, Maldavis, Nepal, Pakistan, and Sri Lanka.

#### Table of Contents

# Table of Contents

## Algebraic Independence

#### Readership

Graduate students and research mathematicians interested in number theory.