Augmentation of singularities of smooth mappings. (English) Zbl 1054.58029
Let \(K\) be the field \(\mathbb R\) (resp. \(\mathbb C\)) of real (resp. complex) numbers. Suppose \(h: (K^n,S) \longrightarrow (K^p,0)\) is a smooth multi-germ with a stable 1-parameter unfolding \(H, H(x,\lambda) = (H_{\lambda}(x),\lambda)\), and \(g:(K^n,0) \longrightarrow (K,0)\) a smooth function.
It is proved that \(\text{cod}(h)\tau(g) \leq \text{cod}(A_{H,g}(h))\) with equality if \(g\) is quasihomogeneous. Here \(A_{H,g}\) is the augmentation of \(h\) by \(H\) and \(g, A_{H,g}(x,z) = (h_{g(z)}(x),z)\). Moreover it is proved, if a map-germ has a 1-parameter stable unfolding which is \(\mathcal A\)-equivalent to the \(q\)-fold trivial unfolding of a stable map, then it is an augmentation of a map by a function of q variables.
It is proved that \(\text{cod}(h)\tau(g) \leq \text{cod}(A_{H,g}(h))\) with equality if \(g\) is quasihomogeneous. Here \(A_{H,g}\) is the augmentation of \(h\) by \(H\) and \(g, A_{H,g}(x,z) = (h_{g(z)}(x),z)\). Moreover it is proved, if a map-germ has a 1-parameter stable unfolding which is \(\mathcal A\)-equivalent to the \(q\)-fold trivial unfolding of a stable map, then it is an augmentation of a map by a function of q variables.
Reviewer: Gerhard Pfister (Kaiserslautern)
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