by putting $t=0$ in $A_0$. Now the saturation of $G_0(h)$ is easy to calculate: We put $\widetilde{\omega}_i:=\theta^{1-i}\omega_i$, then $G(\log\,D)=
\oplus_{i=1}^n{{\mathds C}}[\theta,\theta^{-1},t]\widetilde{\omega}_i$, but $G_0(\log\,D)\subsetneq\oplus_{i=1}^n{{\mathds C}}[\theta,t]\widetilde{\omega}_i$. It is evident that $\widetilde{G}_0(h)=\oplus_{i=1}^n{{\mathds C}}[\theta]\widetilde{\omega}_i$, in particular, this module is invariant under $\theta\nabla_\theta$, i.e., logarithmic at $\theta=0$. We have $(\theta\partial_\theta)\underline{\widetilde{\omega}}=\underline{\widetilde{\omega}}
\cdot(\widetilde{A}_0+\operatorname{\textup{diag}}(\{1-i+\nu_i\}_{i=1,\ldots,n})$, where $\widetilde{A}_0:=(A_0)_{|t=0}$. We see by theorem \[theo:BernsteinEqualResidueGM\] that the residue eigenvalues of $\nabla_\theta$ at $\theta=0$ are the roots of the Bernstein polynomial of $h$ after dividing by $n$ and shift by $-1$, and moreover that the residue endomorphism is regular (i.e., its minimal and characteristic polynomial coincide), as it has a cyclic generator. This proves the theorem.

**Remark:** One may ask what the meaning of the rescaling by $n$ occurring in $b_{\widetilde{G}_0(h)}(n(s+1))$ is. The same kind of twist occurs in [@dGMS proposition 4.5i(v)], where it is performed on the base, i.e., where the pull-back $u^*(G(*D))$ with $u:{{\mathds C}}^2\rightarrow{{\mathds C}}\times T$, $(\theta,t')\mapsto(\theta,(t')^n)$ is considered, and where it is shown that after this pull-back, the resulting bundle has the “rescaling property”, i.e., that it is invariant under $\nabla_{\theta\partial_\theta - t'\partial_{t'}}$.

The following easy consequence is a somewhat reverse argumentation compared to Malgrange’s result, where the rationality of the roots of the Bernstein polynomial was deduced from 