be any function, then there is a polynomial $B\in{{\mathds C}}[s]$ and a differential operator $P(x_i,\partial_{x_i},s)\in{\mathcal{D}}_V[s]$ such that $$P(x_i,\partial_{x_i},s) h^{s+1} = B(s) h^s$$ All polynomials $B(s)\in{{\mathds C}}[s]$ having this property form an ideal in ${{\mathds C}}[s]$, and we denote by $b_h(s)$ the unitary generator of this ideal. $b_h(s)$ is called the Bernstein polynomial of $h$.

If $h$ defines a linear free divisor, then the theory of pre-homogenous vector spaces shows that the functional equation defining $b_h(s)$ is of a particular type.

\[theo:BernsteinPolLinFree\] Let $D=h^{-1}(0)$ be a reductive linear free divisor, then the operator $P$ appearing in Bernstein’s functional equation is given by $P:=h^*(\partial_{x_1},\ldots,\partial_{x_1})$ (remember that $h^*(\underline{y})=\overline{h(\overline{\underline{y}})}$, where $x_i$ are the unitary coordinates and $y_i$ are their duals). In particular, it is an element of ${{\mathds C}}\langle\partial_{x_1},\ldots,\partial_{x_n}\rangle$. Moreover, the degree of $b_h(s)$ is equal to $n$ and the roots of $b_h(s)$ are contained in the open interval $(-2,0)$ and are symmetric around $-1$. In particular, $-1$ is the only integer root.

The following classical reformulation of the definition of the Bernstein polynomial will be useful in the sequel.

Consider the ring ${{\mathds C}}[s,h^{-1}]$, and denote by $M[h^{-1}]:={{\mathds C}}[s,h^{-1}] h^s$ the rank one ${{\mathds C}}[s,h^{-1}]$-module generated by the symbol $h^s$. Define an action 