Difference Algebra
Difference algebra grew out of the study of algebraic difference equations with coefficients from functional fields in much the same way as the classical algebraic geometry arose from the study of polynomial equations with numerical coefficients. The firs
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Algebra and Applications Volume 8 Managing Editor: Alain Verschoren University of Antwerp, Belgium Series Editors: Alice Fialowski Eötvös Loránd University, Hungary Eric Friedlander Northwestern University, USA John Greenlees Sheffield University, UK Gerhard Hiss Aachen University , Germany Ieke Moerdijk Utrecht University, The Netherlands Idun Reiten Norwegian University of Science and Technology, Norway Christoph Schweigert Hamburg University, Germany Mina Teicher Bar-llan University, Israel
Algebra an d Applications aims to publish well- written and carefully refereed monographs with up-to-date expositions of research in all fields of algebra, in cluding its classical impact on commutative and noncommutative algebraic and differential geometry, K-theor y and algebraic topology, and further applications in related do mains, such as number theory, homotopy and (co)homology theory through to discrete mathematics and mathematical physics. Particular emphasis will be put on state-of-the-art topics such as rings of differential operators, Lie algebras and super-algebras, group rings and algebras, Kac-Moody theory, arithmetic algebraic geometry, Hopf algebras and quantum groups, as well as their applications within mathematics and beyond. Books dedicated to computational aspects of these topics will also be welcome.
Alexander Levin
Difference Algebra
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Alexander Levin T he Catholic University of America Washington, D.C. USA
ISBN 978-1-4020-6946-8
e-ISBN 978-1-4020-6947-5
Library of Congress Control Number: 2008926109 c 2008 Springer Science+Business Media B.V. No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed on acid-free paper 9 8 7 6 5 4 3 2 1 springer.com
Preface Difference algebra as a separate area of mathematics was born in the 1930s when J. F. Ritt (1893 - 1951) developed the algebraic approach to the study of systems of difference equations over functional fields. In a series of papers published during the decade from 1929 to 1939, Ritt worked out the foundations of both differential and difference algebra, the theories of abstract algebraic structures with operators that reflect the algebraic properties of derivatives and shifts of arguments of analytic functions, respectively. One can say that differential and difference algebra grew out of the study of algebraic differential and difference equations with coefficients from functional fields in much the same way as the classical algebraic geometry arose from the study of polynomial equations with numerical coefficients. Ritt’s research in differential algebra was continued and extended by H. Raudenbuch, H. Levi, A. Seidenberg, A. Rosenfeld, P. Cassidy, J. Johnson, W. Keigher, W. Sit and many other ma
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