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Full Description
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.
Contents
I. A First Look at Semialgebraic Geometry.- 1. Real Closed Fields and Transfer Principles.- 2. What is Semialgebraic Geometry?.- 3. Real Spaces.- 4. Examples.- II. Real Algebra.- 1. The Real Spectrum of a Ring.- 2. Specializations, Zero Sets and Real Ideals.- 3. Real Valuations.- 4. Real Going-Up and Real Going-Down.- 5. Abstract Semialgebraic Functions.- 6. Cylindrical Decomposition.- 7. Real Strict Localization.- Notes.- III. Spaces of Signs.- 1. The Axioms.- 2. Forms.- 3. SAP-Spaces and Fans.- 4. Local Spaces of Signs.- 5. The Space of Signs of a Ring.- 6. Subspaces.- Notes.- IV. Spaces of Orderings.- 1. The Axioms Revisited.- 2. Basic Constructions.- 3. Spaces of Finite Type.- 4. Spaces of Finite Chain Length.- 5. Finite Type = Finite Chain Length.- 6. Local-Global Principles.- 7. Representation Theorem and Invariants.- Notes.- V. The Main Results.- 1. Stability Formulae.- 2. Complexity of Constructible Sets.- 3. Separation.- 4. Real Divisors.- 5. The Artin-Lang Property.- Notes.- VI. Spaces of Signs of Rings.- 1. Fans and Valuations.- 2. Field Extensions: Upper Bounds.- 3. Field Extensions: Lower Bounds.- 4. Algebras.- 5. Algebras Finitely Generated over Fields.- 6. Archimedean Rings.- 7. Coming Back to Geometry.- Notes.- VII. Real Algebra of Excellent Rings.- 1. Regular Homomorphisms.- 2. Excellent Rings.- 3. Extension of Orderings Under Completion.- 4. Curve Selection Lemma.- 5. Dimension, Valuations and Fans.- 6. Closures of Constructible Sets.- 7. Real Going-down for Regular Homomorphisms.- 8. Connected Components of Constructible Sets.- Notes.- VIII. Real Analytic Geometry.- 1. Semianalytic Sets.- 2. Semianalytic Set Germs.- 3. Cylindrical Decomposition of Germs.- 4. Rings of Global Analytic Functions.- 5. Hilbert's 17th Problem and Real Nullstellensatz.- 6.Minimal Generation of Global Semianalytic Sets.- 7. Topology of Global Semianalytic Sets.- 8. Germs at Compact Sets.- Notes.
