Mahler Functions and Transcendence
This book is the first comprehensive treatise of the transcendence theory of Mahler functions and their values. Recently the theory has seen profound development and has found a diversity of applications. The book assumes a background in elementary field
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Kumiko Nishioka
Mahler Functions and Transcendence
Springer
Author Kumiko Nishioka Mathematics Keio University Hiyoshi Campus 4-1-1 Hiyoshi Kohokuku 233 Yokohama, JAPAN
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Nishioka, Kumiko: Mahler functions and transcendence / Kumiko Nishioka. Berlin; Heidelberg; New York; Barcelona; Budapest; Hong Kong; London; Milan; Paris ; Santa Clara; Singapore; Tokyo: Springer, 1996 (Lecture notes in mathematics ; 1631) ISBN 3-540-61472-9 NE:GT
Mathematics Subject Classification (1991): 1U81 ISSN 0075-8434 ISBN 3-540-61472-9 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1996 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the author SPIN: 10479803 46/3142-543210 - Printed on acid-free paper
Preface The present notes are based on the lectures the author gave at Keio University in 1989 and 1993. Recently the transcendence theory of Mahler functions has seen profound development and has found a diversity of applications. This volume is the first comprehensive treatise on the subject. The author hopes that it will be a source of further research. A Mahler function, for example, of one variable is a function which satisfies a functional equation under the transformation z --+ zd, where d is an integer greater than unity. The study of transcendence and algebraic independence of the values of those functions were started by Mahler's three papers in 1929, 1930. After a gap of about fifty years, it was again investigated by Kubota, Loxton, van der Poorten and the author. Especially Masser's vanishing theorem in 1982 gave a complete solution to a problem of Mahler which is important for the study of the values of Mahler functions of several variables. Next the present author applied elimination-theoretic method by Nesterenko and Philippon to Mahler functions to obtain a general algebraic independence result and a zero-order estimate. Amou , Becker and Topfer followed this approach. Very recently Barre-Sirieix, Diaz , Gramain and Phil
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