Criteria of isolated weighted homogeneous hypersurface singularities using Logarithmic vector fields

We prove a conjecture of da Silva Machado and Seade that characterizes weighted homogeneous isolated hypersurface singularities through the existence of a logarithmic vector field transverse to the link. For a reduced isolated hypersurface germ $(D,0)$ in $\C^{n+1}$ with $n\ge2$, or with $n=1$ and $D$ irreducible, we prove that weighted homogeneity is equivalent to the existence, in suitable coordinates, of a logarithmic vector field everywhere transverse in the real-Euclidean sense to all small links. We also prove the equivalent formulation that $(D,0)$ admits an ambient holomorphic vector field tangent to $D$ that has a non-degenerate isolated singularity at $0$. We further show that the transversality condition must be read after allowing a coordinate change: there exists a weighted homogeneous germ admitting no logarithmic field transverse to the standard round links in certain linear coordinates. The main result of this paper was obtained by the Rethlas system.