Constructible Sets in Real Geometry

Inhaltsverzeichnis

• 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.
• 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.