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The topology of the non-singular level set and the variation operator of a singularity

  • V. I. Arnold
  • S. M. Gusein-Zade
  • A. N. Varchenko
Chapter
Part of the Modern Birkhäuser Classics book series (MBC)

Abstract

Let f:\((\mathbb{C}^{n},0)\rightarrow(\mathbb{C},0)\) be a singularity, that is the germ ofa holomorphic function, with an isolated critical point at the origin. It follows from implicit function theorem that in a neighbourhood of the origin in the space \(\mathbb{C}^{n}\)the level setf-l \({f}^{-1}(\varepsilon)\)for for ε≠0 is a non-singular analytic manifold and the level set \({f}^{-1}(0)\) is a nonsingular manifold away from the origin. At the point \( 0 \,\epsilon \,\mathbb{C}^{n}\) the level set has a singular point.

Keywords

Variation Operator Topological Structure Intersection Matrix Homology Group Intersection Number 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • V. I. Arnold
  • S. M. Gusein-Zade
    • 1
  • A. N. Varchenko
    • 2
  1. 1.Department of Mechanics and MathematicsMoscow State UniversityMoscowRussia
  2. 2.Department of MathematicsThe University of North Carolina at Chapel HillChapel HillUSA

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