Boundary singularities and symmetric boundary singularities (Q1896968)

Let \(f : (\mathbb{C}^{n + 1}, 0) \to (\mathbb{C},0)\) be a holomorphic function germ and \(f_0\) be the restriction of \(f\) to a hyperplane \(H\), containing the origin. Assume \(f\) and \(f_0\) have isolated critical points at 0. Then the couple \((f,H)\) is called the boundary singularity. Further, let \(z_0, \ldots, z_n\) be coordinates in \(\mathbb{C}^{n + 1}\), \(H = \{z_0 = 0\}\) and \(\overline f(z_0, z_1, \ldots, z_n) = f(z^2_0, z_1, \ldots, z_n)\). Then \((\overline f,H)\) is called the symmetric boundary singularity. The author analyses relationships between the basic topological objects associated with \((f,H)\) and \((\overline f,H)\) such as the homology groups of the Milnor fibres, distinguished bases, intersection forms, Dynkin diagrams, etc. This enables her to give a clear interpretation of her results in Bull. Soc. Math. France 118, No. 4, 451-486 (1990; Zbl 0726.58010) in terms of an earlier work by \textit{V. I. Arnol'd} [Russ. Math. Surv. 33, 99-116 (1978); translation from Usp. Mat. Nauk 33, No. 5, 91-105 (1978; Zbl 0408.58009)] and vice versa.