Non reduced plane curve singularities with \(b_{1}(F)=0\) and Bobadilla's question (Q1688206)

Let \(f:\mathbb{C}^2\to\mathbb{C}\) be a germ of a holomorphic function. It is proved, if the first Betti number of the Milnor fibre \(F\) of the plane curve singularity defined by \(f\) is zero, then \(f\) is equivalent to \(x^r\). This result is related to the Bobadilla conjectures [\textit{B. Hepler} and \textit{D. B. Massey}, Topology Appl. 217, 59--69 (2017; Zbl 1361.32010)]: Let \(\beta:=\dim H_1(F, F'), F'\) is the disjoint union of the transversal Milnor fibres \(F'_i\), one for each irreducible branch of the \(1\)--dimensional singular set. \(\beta=0\) implies that the singular set of \(f\) is a smooth line and \(f\) is equivalent to \(x^r\).