Topics in Geometry

  • Steven G. Krantz
  • Harold R. Parks
Part of the Birkhäuser Advanced Texts book series (BAT)


Suppose F(x, y), (x, y) ∈ ℝ n × ℝ, is real analytic in a neighborhood of the origin, is not identically zero, and satisfies F(0,0) =0. To study the locus of the equation F(x, y) = 0 near the origin, we would apply the implicit function theorem if possible, but when the linear term in the Taylor series for F vanishes, then the use of the implicit function theorem is not possible. Instead, the tool that can be used is the Weierstrass preparation theorem.


Implicit Function Theorem Tangent Cone Singular Locus Analytic Manifold Stein Manifold 
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Copyright information

© Springer Science+Business Media New York 2002

Authors and Affiliations

  • Steven G. Krantz
    • 1
  • Harold R. Parks
    • 2
  1. 1.Department of MathematicsWashington UniversitySt. LouisUSA
  2. 2.Department of MathematicsOregon State UniversityCorvallisUSA

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