Lectures in Real Geometry
1st Edition
Published on 10. October 2011
XIV, 268 pages
978-3-11-081111-7 (ISBN)
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Fabrizio Broglia
Lectures in Real Geometry
Book
10/1996
1st Edition
De Gruyter
€179.95
Shipment within 7-9 days
Fabrizio Broglia
Lectures in Real Geometry
Book
01/1996
1st Edition
De Gruyter
€259.00
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Content
- Intro
- Foreword
- Introduction
- Basic algorithms in real algebraic geometry and their complexity: from Sturm's theorem to the existential theory of reals
- 1. Introduction
- 2. Real closed fields
- 2.1. Definition and first examples of real closed fields
- 2.2. Cauchy index and real root counting
- 3. Real root counting
- 3.1. Sylvester sequence
- 3.2. Subresultants and remainders
- 3.3. Sylvester-Habicht sequence
- 3.4. Quadratic forms, Hankel matrices and real roots
- 3.5. Summary and discussion
- 4. Complexity of algorithms
- 5. Sign determinations
- 5.1. Simultaneous inequalities
- 5.2. Thom's lemma and its consequences
- 6. Existential theory of reals
- 6.1. Solving multivariate polynomial systems
- 6.2. Some real algebraic geometry
- 6.3. Finding points on hypersurfaces
- 6.4. Finding non empty sign conditions
- References
- Nash functions and manifolds
- §1. Introduction
- §2. Nash functions
- §3. Approximation Theorem
- §4. Nash manifolds
- §5. Sheaf theory of Nash function germs
- §6. Nash groups
- References
- Approximation theorems in real analytic and algebraic geometry
- Introduction
- I. The analytic case
- 1. The Whitney topology for sections of a sheaf
- 2. A Whitney approximation theorem
- 3. Approximation for sections of a sheaf
- 4. Approximation for sheaf homomorphisms
- II. The algebraic case
- 5. Preliminaries on real algebraic varieties
- 6. A- and B-coherent sheaves
- 7. The approximation theorems in the algebraic case
- III. Algebraic and analytic bundles
- 8. Duality theory
- 9. Strongly algebraic vector bundles
- 10. Approximation for sections of vector bundles
- References
- Real abelian varieties and real algebraic curves
- Introduction
- 1. Generalities on complex tori
- 1.1. Complex tori
- 1.2. Homology and cohomology of tori
- 1.3. Morphisms of complex tori
- 1.4. The Albanese and the Picard variety
- 1.5. Line bundles on complex tori
- 1.6. Polarizations
- 1.7. Riemann's bilinear relations and moduli spaces
- 2. Real structures
- 2.1. Definition of real structures
- 2.2. Real models
- 2.3. The action of conjugation on functions and forms
- 2.4. The action of conjugation on cohomology
- 2.5. A theorem of Comessatti
- 2.6. Group cohomology
- 2.7. The action of conjugation on the Albanese variety and the Picard group
- 2.8. Period matrices in pseudonormal form and the Albanese map
- 3. Real abelian varieties
- 3.1. Real structures on complex tori
- 3.2. Equivalence classes for real structures on complex tori
- 3.3. Line bundles on complex tori with a real structure
- 3.4. Riemann bilinear relations for principally polarized real varieties
- 3.5. Moduli spaces of principally polarized real abelian varieties
- 3.6. Real theta functions
- 4. Applications to real curves
- 4.1. The Jacobian of a real curve
- 4.2. Real theta-characteristics
- 4.3. Examples
- 4.4. Moduli spaces and the theorem of Torelli
- 4.5. Singular curves
- References
- Appendix
- Mario Raimondo's contributions to real geometry
- Mario Raimondo's contributions to computer algebra
