# 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.

#### Readership

Graduate students and research mathematicians interested in number theory.