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Polynomials and Vanishing Cycles

Polynomials and Vanishing Cycles


Part of Cambridge Tracts in Mathematics

  • Date Published: May 2007
  • availability: Available
  • format: Hardback
  • isbn: 9780521829205

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About the Authors
  • The behaviour of vanishing cycles is the cornerstone for understanding the geometry and topology of families of hypersurfaces, usually regarded as singular fibrations. This self-contained tract proposes a systematic geometro-topological approach to vanishing cycles, especially those appearing in non-proper fibrations, such as the fibration defined by a polynomial function. Topics which have been the object of active research over the past 15 years, such as holomorphic germs, polynomial functions, and Lefschetz pencils on quasi-projective spaces, are here shown in a new light: conceived as aspects of a single theory with vanishing cycles at its core. Throughout the book the author presents the current state of the art. Transparent proofs are provided so that non-specialists can use this book as an introduction, but all researchers and graduate students working in differential and algebraic topology, algebraic geometry, and singularity theory will find this book of great use.

    • Self-contained and accessible tract proposing a systematic geometro-topological approach to vanishing cycles appearing in non-proper fibrations
    • The latest research on new topics such as topology of singularities of meromorphic functions and non-generic Lefschetz pencils is discussed in detail
    • Presents the current state of the art in this field
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    Reviews & endorsements

    '… written in a clear didactic style, with detailed explanations of all notions and results … the book reads fluently, it is accessible for graduate students while advanced researchers … can find here very interesting and useful material …' Zentralblatt MATH

    'Each chapter of the book is followed by exercises …' EMS Newsletter

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    Product details

    • Date Published: May 2007
    • format: Hardback
    • isbn: 9780521829205
    • length: 266 pages
    • dimensions: 235 x 160 x 25 mm
    • weight: 0.568kg
    • contains: 12 b/w illus. 3 tables 53 exercises
    • availability: Available
  • Table of Contents

    Part I. Singularities at Infinity of Polynomial Functions:
    1. Regularity conditions at infinity
    2. Detecting atypical values via singularities at infinity
    3. Local and global fibrations
    4. Families of complex polynomials
    5. Topology of a family and contact structures
    Part II. The Impact of Global Polar Varieties:
    6. Polar invariants and topology of affine varieties
    7. Relative polar curves and families of affine hypersurfaces
    8. Monodromy of polynomials
    Part III. Vanishing Cycles of Non-Generic Pencils:
    9. Topology of meromorphic functions
    10. Slicing by pencils of hypersurfaces
    11. Higher Zariski-Lefschetz theorems
    Appendix 1. Stratified singularities
    Appendix 2. Hints to exercises

  • Author

    Mihai Tibăr, Université de Lille
    Mihair Tibar is a Professor in the Mathematics Department at the Université des Sciences et Technologies de Lille, France.

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