Real Algebraic Geometry Proceedings of the Conference held in Rennes
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Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, Zurich F. Takens, Groningen
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M. Coste L. Mahe
M.-F. Roy (Eds.)
Real Algebraic Geometry Proceedings of the Conference held in Rennes, France, June 24-28, 1991
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Editors Michel Coste Louis Mahe Marie-Francoise Roy IRMAR (CNRS, U.R.A. 305) Universite de Rennes I Campus Beaulieu F-35402 Rennes Cedex, France
Mathematics Subject Classification (1991): Primary: 14P Secondary: 12015, 32B, 32C, 58A35, 68Q40 ISBN 3-540-55992-2 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-55992-2 Springer-Verlag New York Berlin Heidelberg 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 1992 Printed in Germany
Typesetting: Camera-ready by author/editor 46/3140-543210 - Printed on acid-free paper
PREFACE
The meeting on "Real Algebraic Geometry" was held in La Turballe, on the seashore not far from Rennes, from June 24 to 28, 1991. It took place ten years after the first meeting on "Geornetrie Algebrique Reelle et Formes Quadratiques" (*). These Proceedings contain survey papers on some of the developements of real algebraic geometry in the last ten years, and also contributions by the participants. Every paper has been submitted to a referee, and we want to thank all of them for their collaboration. The meeting, and the collaboration between the european teams which made it possible, received support from the Universite de Rennes 1, the GDR MathernatiquesInformatique (CNRS), and the programs Reseau Europeen de Laboratoires, Acces, Alliance, Actions Integrees FrancoEspagnoles. We would like also to thank SpringerVerlag for publishing this volume, and to express our gratitude to Ms. Yvette Brunel, for her precious help for the secretary of the meeting. Michel Coste Louis MaM MarieFrancoise Roy
(*) Lecture Notes in Mathematics 959, Springer (1982)
TABLE OF CONTENTS
Survey papers Knebusch M.: Semialgebraic topology in the last ten years
....
1
Parimala R.: Algebraic and topological invariants of real algebraic varieties Polotovskii G.M.: On the classification of decomposing plane algebraic curves
37 52
Scheiderer C.: Real algebra and its applications to geometry in the last ten years: some major developments and results
75
Shustin E.I.: Topology of real plane algebraic curves
97
Silhol R.: Moduli problems in real algebraic geometry
110
Other papers Akbulut S., King H.:
