\ddots & \vdots &\vdots \\
    0 &0 & \ldots &0& 0\\
    0&0& \ldots& -1&0
  \end{pmatrix},$$ $A_\infty=\operatorname{\textup{diag}}(\nu_1,\ldots,\nu_n)$ and $A'_\infty:=\operatorname{\textup{diag}}(0,1,\ldots,n-1)-A_\infty$. The constant $c\in {{\mathds C}}$ is defined by the equation $$f^n=-c\cdot h + \sum_{i=1}^{n-1}\xi_i(f)\cdot k_i,$$ where $\xi_i\in\Theta_{V/T}(\log\,D)$ and $k_i\in{\mathcal{O}}_V$ are homogenous polynomials of degree $n-1$. Here $$\omega_1=\iota_E\frac{\operatorname{\textup{vol}}}{h}=n\frac{\operatorname{\textup{vol}}}{dh},$$ where $\operatorname{\textup{vol}}=dx_1\wedge\ldots\wedge dx_n$. In particular, $(G(*D),\nabla)$ is flat. Moreover, $(\nu_1,\ldots,\nu_n)$ is the spectrum at ($\theta=$) infinity of the restriction of $G_0(\log\,D)$ to $t=0$.

The following obvious consequence will be used in the next section.

Consider the inclusion $j:\{1\}\times T\hookrightarrow {{\mathds C}}\times T$ and the restriction $(G_1(*D),\nabla):=j^*(G(*D),\nabla)$. This is a meromorphic bundle on $T$ with connection (the only pole being at $0\in T$), and can thus be seen as a coherent and holonomic left ${{{{\mathds C}}[t]\langle\partial_t\rangle}}$-module. Then we have an isomorphism of left ${{{{\mathds C}}[t]\langle\partial_t\rangle}}$-modules $$\varphi:{{{{\mathds C}}[t]\langle\partial_t\rangle}}/(b_{G_1(\log\,D)}(t\partial_t)+\frac{c}{n^n}\cdot t) \stackrel{\cong}{\longrightarrow} (G_1(*D),\nabla),$$ where $b_{G_1(\log\,D)}$ is the spectral polynomial of $G_1(\log\,D):=j^*(G(\log\,D),\nabla)$ at $0\in T$, i.e., $$b_{G_1(\log\,D)}(s):=\prod_{i=1}^n \left(s-\frac{i-1-\nu_i}{n}\right),$$ and where $\varphi(1)=\omega_1$. In particular, $\omega_1$ satisfies the functional equation $b_{G_1(\log\,D)}(t\partial_t)\omega_1=-c/n^n\cdot t\omega_1$ in $G_1(*D)$.

We note the following easy consequence from the definitions, which was not stated in [@dGMS].

\[lemIndependenceLinearForm\] Let $f_1,f_2\in V^*\backslash D^*$ be two generic linear forms. Denote by $(G_i(\log\,D),\nabla_i)$ the family of Brieskorn lattices attached to the pair $(f_i,h)$, 