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Algebraic Independence
 
Yu. V. Nesterenko Moscow State University, Moscow, Russia
A publication of Tata Institute of Fundamental Research
Algebraic Independence
Softcover ISBN:  978-81-7319-984-4
Product Code:  TIFR/14
List Price: $60.00
AMS Member Price: $48.00
Please note AMS points can not be used for this product
Algebraic Independence
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Algebraic Independence
Yu. V. Nesterenko Moscow State University, Moscow, Russia
A publication of Tata Institute of Fundamental Research
Softcover ISBN:  978-81-7319-984-4
Product Code:  TIFR/14
List Price: $60.00
AMS Member Price: $48.00
Please note AMS points can not be used for this product
  • Book Details
     
     
    Tata Institute of Fundamental Research Publications
    Volume: 142008; 157 pp
    MSC: Primary 11;

    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.

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

    A publication of 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.

  • Requests
     
     
    Review Copy – for publishers of book reviews
    Accessibility – to request an alternate format of an AMS title
Volume: 142008; 157 pp
MSC: Primary 11;

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.

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

A publication of 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.

Review Copy – for publishers of book reviews
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.