Topics in Geometry
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.
KeywordsImplicit Function Theorem Tangent Cone Singular Locus Analytic Manifold Stein Manifold
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