of ${\mathcal{D}}_V$, the ring of algebraic differential operators on $V$ on $M[h^{-1}]$ by putting $\partial_{x_i}(g\cdot h^s):=\partial_{x_i}(g)\cdot h^s+g\cdot s \cdot h^{-1}\partial_{x_i}(h)h^s$. This action extends naturally to an ${\mathcal{D}}_V[s]$-action. Let $M$ the ${\mathcal{D}}_V[s]$-submodule of $M[h^{-1}]$ generated by $h^s$. Define an action of $t$ on $M[h^{-1}]$ by putting $t(g(s)\cdot h^s):=g(s+1)\cdot h\cdot h^{s+1}$. Then $b_h(s)$ is the minimal polynomial of the action of $s$ on the quotient $M/tM$.

This definition can be rephrased once more using the theory of $V$-filtrations on ${\mathcal{D}}$-modules. Without reviewing the details of the theory, we recall the following facts (see, e.g. [@MebkhMais section 4])

Let $X$ be any smooth algebraic variety, and $Y\subset X$ a smooth hypersurface defined by an ideal sheaf $I\subset{\mathcal{O}}_X$. We denote by $t\in{\mathcal{O}}_X$ a local generator of ${\mathcal{I}}$.

1.  Let ${\mathcal{D}}_X$ be the sheaf of algebraic differential operators, then define $$\begin{array}{rcl}
    V_k{\mathcal{D}}_X:=\left\{ P\in {\mathcal{D}}_X\,|\, P(I^j)\subset I^{j-k}\right\}
    \end{array}$$ For any left ${\mathcal{D}}_X$-module ${\mathcal{M}}$, a V-filtration on ${\mathcal{M}}$ is an increasing filtration $U_\bullet{\mathcal{M}}$ compatible with $V_\bullet{\mathcal{D}}_X$.

2.  A V-filtration $U_\bullet{\mathcal{M}}$ on a left ${\mathcal{D}}_X$-module ${\mathcal{M}}$ is good iff the *Rees*-module $\oplus z^k U_k{\mathcal{M}}$ is ${\mathcal{R}}_V{\mathcal{D}}_X$-coherent, where ${\mathcal{R}}_V{\mathcal{D}}_X:=\oplus_k z^k V_k{\mathcal{D}}_X$.

3.  A good V-filtration $U_\bullet{\mathcal{M}}$ is said to have a Bernstein polynomial iff there is a non-zero polynomial 