The topology of the non-singular level set and the variation operator of a singularity
Vladimir I. Arnold, Sabir Medzhidovich Gusein-Zade, Alexander N. Varchenko · Birkhäuser Boston eBooks · 2012
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.