Z}}$, $U_k {\mathcal{N}}:=V_k{\mathcal{D}}_{T\times V}\delta(t-h)$. This $V$-filtration admits a Bernstein polynomial (namely, a Bernstein polynomial for the section $\delta(t-h)$), which is exactly the polynomial $b_h(s)$. We denote, as in [@Mal6], by ${\mathcal{M}}$ the $V_0{\mathcal{D}}_{T\times V}$-module $U_0 {\mathcal{N}}$.

2.  The direct image $(i_h)_+{\mathcal{O}}_V(*D)$ is the localization of both ${\mathcal{N}}$ and ${\mathcal{M}}$ along $t=0$, and is thus denoted by ${\mathcal{M}}[t^{-1}]$. As ${\mathcal{N}}$ has no $t$-torsion, we have an exact sequence $$0\longrightarrow {\mathcal{N}}\longrightarrow {\mathcal{M}}[t^{-1}] \longrightarrow C \longrightarrow 0.$$ where $C$ is a ${\mathcal{D}}_{T\times V}$-module. A Bernstein polynomial for a local section $m\in {\mathcal{N}}$ is also a Bernstein polynomial for $m$, seen as a local section in ${\mathcal{M}}[t^{-1}]$.

We can now state and prove the main result of this paper.

\[theo:BernsteinEqualResidueGM\] Let $D=h^{-1}(0)$ be reductive linear free divisor and $f\in V^*$ be generic. Consider the family of Gauß-Manin systems $G(*D)$, the logarithmic extension $G(\log\,D)$ and the restrictions $G_1(\log\,D)\subset G_1(*D)$ from above. Then we have that $b_h(s)=b_{G_1(\log\,D)}(s+1)$ (recall that $b_{G_1(\log\,D)}(s)$ is the spectral polynomial of $G_1(\log\,D))$.

In order to prove this result, we start with a preliminary lemma.

Let ${\mathcal{M}}[t^{-1}]:=(i_h)_+{\mathcal{O}}_V(*D)$ as above. Consider the twisted module $(i_h)_+{\mathcal{O}}_V(*D)e^{-f}$. Then the section $\delta(t-h)e^{-f}\in(i_h)_+{\mathcal{O}}_V(*D)e^{-f}$ admits $b_h(s)$ as a Bernstein polynomial, with associated functional equation $$\label{eq:twistedBernstein}
\left(t\cdot h^*(\partial_{x_i}+a_i)-b_h(-\partial_t t)\right)\delta(t-h)e^{-f}=0,$$ where 