Constructible sets in real geometry

InhaltsverzeichnisI. 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.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.V. The Main Results.1. Stability Formulae.2. Complexity of Constructible Sets.3. Separation.4. Real Divisors.5. The Artin-Lang Property.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.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.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.