Lectures in real geometry

Lectures in real geometry / editor Fabrizio Broglia.

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Bibliographic Details
Other Authors: Broglia, Fabrizio, 1948-
Format: eBook
Language:English
Published: Berlin ; New York : Walter de Gruyter, 1996.
Series:De Gruyter expositions in mathematics ; 23.
Subjects:
Geometry, Analytic.
Geometry, Algebraic.
MATHEMATICS > Geometry > Algebraic.
Geometry, Algebraic
Geometry, Analytic
Online Access:Click for online access
Table of Contents:
  • Foreword
  • Introduction
  • Basic algorithms in real algebraic geometry and their complexity: from Sturmâ€?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 inequalities5.2. Thomâ€?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
  • Â1. Introduction
  • Â2. Nash functions
  • Â3. Approximation Theorem
  • Â4. Nash manifolds
  • Â5. Sheaf theory of Nash function germs
  • Â6. Nash groups
  • References
  • Approximation theorems in real analytic and algebraic geometry
  • IntroductionI. 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 curvesIntroduction
  • 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. Riemannâ€?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 Comessatti2.6. 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

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