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Functions on discriminants. (English) Zbl 0605.58011

In what follows k denotes the field of real or complex numbers. If \(D,0\subset k^ p\), 0 is the discriminant variety of the versal unfolding of an analytic function germ \(f: k^ n,0\to k,0\), it is of some interest to classify smooth functions h: D,0\(\to k,0.\)
In Commun. Pure Appl. Math. 29, 557-582 (1976; Zbl 0343.58003), V. I. Arnol’d classified such germs in the case when they are generic and f is a simple singularity of type \(A_ k\), \(D_ k\), \(E_ k\). Full details were, however, only given in the case of \(A_ k\). In this paper, using the basic vector fields of K. Saito [Invent. Math. 14, 123- 142 (1971; Zbl 0224.32011)] tangent to the discriminant D, we give another proof of Arnol’d’s results; the method of proof here is fairly unsophisticated. We then generalize these results to cover a wide collection of weighted homogeneous functions, which include the 14 weighted homogeneous unimodal germs, as well as the simple singularities. Furthermore we use these basic fields to prove that the discriminants of the simple elliptic singularities \(\tilde E_ k\), for \(k=6,7,8\), are topologically trivial along the modulus parameter. We also discuss the existence of stable germs on these discriminants. In the appendix we give a self-contained proof that Saito’s vector fields are tangent to the discriminant.

MSC:

58C25 Differentiable maps on manifolds
58K99 Theory of singularities and catastrophe theory
14B05 Singularities in algebraic geometry
32S05 Local complex singularities
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