Constructible Sets in Real Geometry

Carlos Andradas, Ludwig Bröcker, Jesus M. Ruiz (Autoren)

Buch | Softcover

IX, 270 Seiten

2011
Springer Berlin (Verlag)
9783642800269 (ISBN)

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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.

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.

Erscheint lt. Verlag	14.12.2011
Reihe/Serie	Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics
Zusatzinfo	IX, 270 p.
Verlagsort	Berlin
Sprache	englisch
Maße	155 x 235 mm
Gewicht	434 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
Schlagworte	Algebraic Geometry • constructible set • Dimension • Divisor • Grad • konstruierbare Mengen • local-global principles • real spectrum • Reelle Spektren • Space of signs • Stabilitätsindex • stability index
ISBN-13	9783642800269 / 9783642800269
Zustand	Neuware