global sections and considering again the isomorphism from lemma \[lem:CompDirectImage\], 4., we obtain $$H^0(V, \Omega^n_V\otimes_{{\mathcal{D}}_V}({\mathcal{O}}_V(*D)e^{-f}[\partial_t])) \cong G_1(*D)$$ Notice that the section $\operatorname{\textup{vol}}\otimes\delta(t-h)e^{-f}$ is mapped to the section $\omega_1/n=\operatorname{\textup{vol}}/dh$ under this isomorphism. From the equation $\left(t\cdot h^*(\partial_{x_i}+a_i)-b_h(-\partial_t t)\right)\delta(t-h)e^{-f}=0$ in ${\mathcal{O}}_V(*D)e^{-f}[\partial_t]$ (equation ) we deduce that the element $\operatorname{\textup{vol}}\otimes \left(t\cdot h^*(\partial_{x_i}+a_i)-b_h(-\partial_t t)\right)\delta(t-h)e^{-f}$ is zero in $h_*(\Omega^n_V\otimes_{{\mathcal{D}}_V}({\mathcal{O}}_V(*D)e^{-f}[\partial_t]))$. Hence $$t\cdot (h^*(\partial_{x_i}+a_i)(\operatorname{\textup{vol}}) )\otimes \delta(t-h)e^{-f} = b_h(-\partial_t t)(\operatorname{\textup{vol}}\otimes \delta(t-h)e^{-f})$$ holds in $h_*\left(\Omega^n_V\otimes_{{\mathcal{D}}_V}({\mathcal{O}}_V(*D)e^{-f}[\partial_t])\right)$, where the operator $h^*(\partial_{x_i}+a_i)$ acts on $\operatorname{\textup{vol}}$ by the right ${\mathcal{D}}_V$-action on $\Omega^n_V$. Now develop the polynomial $h^*(y_i+a_i)$ as $h^*(y_i+a_i) = \sum_{1 \leq |I|\leq n} a_I y^I + h^*(a_i)$, then $$h^*(\partial_{x_i}+a_i) = \sum_{1\leq|I|\leq n} a_I \partial_{x_1}^{i_1} \ldots \partial_{x_n}^{i_n} + h^*(a_i)$$ and the action $h^*(\partial_{x_i}+a_i)(\operatorname{\textup{vol}})$ is given by $$\sum_{1\leq|I|\leq n} a_I \left(\underbrace{\operatorname{\textup{Lie}}_{\partial_{x_1}}\cdots\operatorname{\textup{Lie}}_{\partial_{x_1}}}_{i_1} \cdots
\underbrace{\operatorname{\textup{Lie}}_{\partial_{x_n}}\cdots\operatorname{\textup{Lie}}_{\partial_{x_n}}}_{i_n} \right)(\operatorname{\textup{vol}})+ h^*(a_i)\cdot\operatorname{\textup{vol}}$$ But obviously $\operatorname{\textup{Lie}}_{\partial_{x_i}}\operatorname{\textup{vol}}=0$ for any $i\in\{1,\ldots,n\}$, so that finally we see that the section $\operatorname{\textup{vol}}\otimes\delta(t-h)e^{-f}$ of $h_*\left(\Omega^n_V\otimes_{{\mathcal{D}}_V}({\mathcal{O}}_V(*D)e^{-f}[\partial_t])\right)$ is annihilated by $h^*(a_i)\cdot t-b_h(-\partial_t t)$. It follows that $b_h(-\partial_t t)$ sends $U_0 G(*D) = V_0{{{{\mathds C}}[t]\langle\partial_t\rangle}}\omega_1$ into $U_{-1}G(*D)$, hence, we have $b_{G_1(\log\,D)}(s+1)|b_h(s)$. Now the theorem follows as both $b_h$ and $b_{G_1(\log\,D)}$ are of degree $n$.

Consequences and Examples {#sec:ConsqExamples}
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Let $D$ be a reductive linear free divisor with defining equation $h\in{\mathcal{O}}_V$ and $f\in V^*$ a generic linear form. Consider, as 