Constructible Sets in Real Geometry

This book presents a systematic and unified report on the minimal description of constructible sets. It starts at a very basic level (almost undergraduate) and leads up to state-of-the-art results, many of which are published in book form for the very first time. The book contains numerous examples, 63 figures and each chapter ends with a section containing historical notes. The authors tried to keep the presentation as self-contained as it can possibly be.

The plan to write this book was laid out in April 1987 at Oberwolfach, dur- ing the Conference "Reelle Algebraische Geometrie." Afterwards we met at various conferences and seminars in Luminy, Madrid, Munster, Oberwolfach, Segovia, Soesterberg, Trento and La Turballe. We would like to thank the or- ganizers and the institutions which supported these meetings. With pleasure we remember the special year on Real Algebraic Geometry and Quadratic Forms (Ragsquad) in Berkeley 1990/91 where an essential part of this book was written. Thanks to T.Y. Lam and R. Robson. We are indepted to our Departments: Universidad Complutense de Madrid and WestfiiJische Wilhelms-Universitiit Munster, as well as the D.A.A.D. and the D.G.I.C.y'T .. It is not possible to mention here all colleagues and friends who showed permanent interest in the project. They encouraged us to continue and com- plete this work. In particular, we are obliged to Jacek Bochnak, Mike Buchner, Michel Coste and Claus Scheiderer for proofreading and helpful suggestions. While the work was still in progress, Manfred Knebusch used parts of our manuscript for a course on Real Algebraic Geometry. His experience con- vinced us that it was worth pursuing a fully abstract approach. We were also in permanent contact with Murray Marshall, and the reader will recognize the mutual influence of ideas. We are also indebted to Professor Reinhold Remmert and to the Springer- Verlag for publishing the book in this series.

This ERGEBNISSE volume deals with the theory of the minimal description of real constructible sets. It gives a comprehensive state-of-the-art report with many results published for the first time in book form. The authors tried to keep the presentation as self-contained a possible. The book contains numerous examples, historical notes and 63 figures.

Contents: A First Look at Semialgebraic Geometry.- Real Algebra.- Spaces of Signs.- Spaces of Orderings.- The Main Results.- Spaces of Signs of Rings.- Real Algebra of Excellent Rings.- Real Analytic Geometry.