Distinguished bases of non-simple singularities (Q5946889)

Let \(f: (\mathbb C^n,0) \to (\mathbb C,0)\) be one of the simple singularities \(A_k, D_k, E_6, E_7, E_8\) and \(n\) odd. \textit{S. M. Gusein-Zade} [Funct. Anal. Appl. 14, 307--308 (1980); translation from Funkts. Anal. Prilozh. 14, No. 4, 73--74 (1980; Zbl 0484.32002)] proved that for any vanishing cycle \(\Delta\) and any distinguished basis \(\Delta_1, \dots, \Delta_\mu\) there exists a sequence of elementary substitutions, turning it into a distinguished basis \(\Delta^\prime_1, \dots, \Delta^\prime_\mu\) with \(\Delta_1^\prime = \pm \Delta\). The author proves that this is wrong if \(f\) defines a non--simple singularity.