in the last section, the logarithmic extension $G_0(\log\,D)$ of the family of Brieskorn lattices $G_0(*D)$ attached to $(f,h)$. We define the *logarithmic Brieskorn lattice* of $h$ to be the restriction $G_0(h):=i^*(G_0(\log\,D),\nabla)$, where $i:{{\mathds C}}\times\{0\}\hookrightarrow {{\mathds C}}\times T$.

Notice that it follows from lemma \[lemIndependenceLinearForm\] that $G_0(h)$ is independent of the choice of $f$ in $V^*\backslash D^*$, so that it makes sense to speak about *the* logarithmic Brieskorn lattice of $h$. The next result, which is an easy consequence of theorem \[theo:BernsteinEqualResidueGM\], can be considered as a variant of the corresponding classical statement of Malgrange ([@Mal6]) for the isolated singularity case.

\[theo:AdaptMalgrange\] Let $(G_0(h),\nabla)$ be the logarithmic Brieskorn lattice of a reductive linear free divisor $D$. Then $\nabla$ is regular singular at $\theta=0$. Consider the saturation $\widetilde{G}_0(h):=\sum_{k\geq 0} (\nabla_{\theta\partial_\theta})^kG_0(h)$, which has a logarithmic pole at $\theta=0$. Let $b_{\widetilde{G}_0(h)}(s)$ be the minimal polynomial of the residue endomorphism of $\nabla_\theta$ on $\widetilde{G}_0(h)$. Then $b_{\widetilde{G}_0(h)}(n(s+1)) = b_h(s)$.

The regularity follows easily from the particular form of the connection matrix . Namely, $G_0(h)$ is the Fourier-Laplace transformation of a regular ${{\mathds C}}[r]\langle \partial_r\rangle$-module, hence, its regularity is equivalent to the nilpotency of the polar part of the connection matrix, which is obviously the case here, 