Galois Theory of Linear Differential Equations
Linear differential equations form the central topic of this volume, Galois theory being the unifying theme. A large number of aspects are presented: algebraic theory especially differential Galois theory, formal theory, classification, algorithms to deci
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Series editors A. Chenciner S.S. Chern B. Eckmann P. de la Harpe F. Hirzebruch N. Hitchin L. H¨ormander M.-A. Knus A. Kupiainen G. Lebeau M. Ratner D. Serre Ya. G. Sinai N.J.A. Sloane B. Totaro A. Vershik M. Waldschmidt
Editor-in-Chief M. Berger
J. Coates
S.R.S. Varadhan
Springer-Verlag Berlin Heidelberg GmbH
Marius van der Put Michael F. Singer
Galois Theory of Linear Differential Equations
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Marius van der Put University of Groningen Department of Mathematics 9700 AV Groningen The Netherlands e-mail: [email protected] Michael F. Singer Department of Mathematics North Carolina State University Raleigh. Ne 27695-8205 USA e-mail: [email protected] Cataloging-in-Publication Data applied for A catalog record for this book is available from the Library of Congress. Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliotheklists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de
Mathematics Subject Qassification (2000): 12H05. 12HlO. 12H25. 13NlO. 14F10. 34A25. 34A30. 34A55. 34M. 34E. 34-04> 34M99 ISSN 0072-7830 ISBN 978-3-642-62916-7
ISBN 978-3-642-55750-7 (eBook)
DOI 10.1007/978-3-642-55750-7 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. reuse of illustrations, recitation. broadcasting. reproduction on microfilm 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
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Springer-Verlag Berlin Heidelberg
2003
OriginaIly published by Springer-Verlag Berlin Heidelberg New York in 2003
Softcover reprint of the hardcover lS1: edition 2003 The use of general descriptive names. registered names, trademarks, etc. in this publication does nat imply, even in the absence of a specific statement, that such names are exempt from the relevant pratective laws and regulations and therefore free for general use. Cover design: design & production GmbH, Heidelberg Printed on acid-free paper
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Preface
This book is an introduction to the algebraic, algorithmic, and analytic aspects of the Galois theory of homogeneous linear differential equations. Although the Galois theory has its origins in the 19th Century and was put on a firm footing by Kolchin in the middle of the 20th Century, it has experienced a burst of activity in the last 30 years. In this book we present many of the recent results and new approaches to this classical field. We have attempted to make this subject accessible to anyone with a background in algebra and analysis at the level of a first-year graduate student. Our hope is that this book will prepare and entice the reader to delve fur
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