Residue Currents and Bezout Identities
A very primitive form of this monograph has existed for about two and a half years in the form of handwritten notes of a course that Alain Y ger gave at the University of Maryland. The objective, all along, has been to present a coherent picture of the al
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Series Editors J. Oesterle A. Weinstein
Carlos A. Berenstein RogerGay Alekos Vidras AlainYger
Residue Currents and Bezout Identities
Springer Basel AG
Authors: Carlos A. Berenstein Mathematics Department & Institute of Systems Research University of Maryland College Park, MD 20742 USA
Alekos Vidras Research Institute for Mathematical Sciences Kyoto University 606 Kyoto Japan
Roger Gay and Alain Yger Centre de Recherche en Mathematiques Universite de Bordeaux 1 33405 Talence (Cedex) France
A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA Deutsche Bibliothek Cataloging-in-Publication Data Residue curreots aod bezout identities / Carlos A. Berenstein ... - Basel ; Boston ; Berlin : Birkhăuser, 1993 (Progress in mathematics ; VoI. 114) ISBN 978-3-0348-9680-1 ISBN 978-3-0348-8560-7 (eBook)
DOI 10.1007/978-3-0348-8560-7 NE: Berenstein, Carlos A.; GT This work is subject to copyright. Ali rights are reserved, whether the whole or part of the material is concemed, specifically the rights of translation, reprinting, re-use of illustrations, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 1993 Springer Basel AG Originally published by Birkhăuser Verlag Basel in 1993 Softcover reprint ofthe hardcover Ist edition 1993 Camera-ready copy prepared by the authors Printed on acid-free paper produced of chlorine-free pulp ISBN 978-3-0348-9680-1 987654321
To our families, for their infinite patience.
In memoriam Miguel Herrera
Table of Contents Introduction.........................................................
ix
1. Residue Currents in one Dimension. Different Approaches 1. Residue attached to a holomorphic function ..................... 2. Some other approaches to the residue current ................... 3. Some variants of the classical Pompeiu formula . . . . . . . . . . . . . . . . . . 4. Some applications of Pompeiu's formulas. Local results. . . . . . . . . . 5. Some applications of Pompeiu's formulas. Global results ........ References for Chapter 1 ...............................................
1 4 8 10 14 20
2. Integral Formulas in Several Variables 1. Chains and cochains, homology and cohomology 2. Cauchy's formula for test functions ............................. 3. Weighted Bochner-Martinelli formulas .......................... 4. Weighted Andreotti-Norguet formulas........................... 5. Applications to systems of algebraic equations. . . . . . . . . . . . . . . . . . . References for Chapter 2 ...............................................
21 23 27 38 41 46
3. Residue Currents and Analytic Continuation 1. Leray iterated residues .......................................... 2. Multiplication of principal values and residue currents ........... 3. The Dolbeault complex and the Grothendieck residue ........... 4. Residue currents ................................................ 5. The local duality theorem.. . .. . . . . . . . . . .
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