Logarithmic Jacobian ideals, quasi-ordinary hypersurfaces and equisingularity

Let $(S, 0) \subset (\mathbb C^{d+1},0)$ be an irreducible germ of hypersurface. The germ $(S,0)$ is quasi-ordinary if $(S,0)$ has a finite projection to $(\mathbb C^d,0)$ which is unramified outside the coordinate hyperplanes. This implies that the normalization of $S$ is a toric singularity. One also has a monomial variety associated to $S$, which is a toric singularity with the same normalization, and with possibly higher embedding dimension. Since $(S,0)$ is quasi-ordinary, the extension of the Jacobian ideal of $S$ to the local ring of its normalization is a monomial ideal. We describe this monomial ideal by comparing it with the logarithmic Jacobian ideals of $S$ and of its associated monomial variety and we give some applications.