Filtration Relative, l'Id\'eal de Bernstein et ses pentes

Let $f_i: X \rightarrow {\bf C}$, for $i$ integer between $1$ and $p$, be analytic functions defined on a complex analytic variety $X$. Let us consider ${\cal D}_X$ the ring of linear differential operators and ${\cal D}_X [s_1, \ldots, s_p] = {\bf C}_X [s_1, \ldots, s_p] \otimes_ {\bf C} {\cal D}_X$. Let $m$ be a section of a holonomic ${\cal D}_X$-Module. We denote ${\cal B}(m, x_0, f_1, \ldots, f_p)$ the ideal of ${ \bf C} [s_1, \ldots, s_p]$ constituted by the polynomials $b$ satisfying in the neighborhood of $x_0 \in X$ : $$B (s_1, \ldots, s_p) m f_1^{ s_1} \ldots f_p ^{s_p} \in {\cal D}_X [s_1, \ldots, s_p] \, m f_1^{s_1 + 1} \ldots f_p^{s_p + 1} \; .$$ This ideal is called Bernstein's ideal. C. Sabbah shows the existence for every $x_0 \in X$ of a finite set ${\cal H}$ of linear forms with coefficients in ${\bf N}$, such that: $$\prod_{H \in {\cal H}} \prod_{i \in I_{\cal H}} (H (s_1, \ldots, s_p) + \alpha_{H , i}) \in {\cal B} (m, x_0, f_1, \ldots, f_p) \; ,$$ where $\alpha_{H,i}$ are complex numbers. The purpose of this article is to show in particular the existence of a minimal set ${\cal H}$. In addition, when $m$ is a section of a holonomic regular ${\cal D}_X$-Module, we will precise geometrically this set from the characteristic variety of ${\cal D}_X$-Module generated by $m$. We introduce and study especially the relative characteristic variety of the ${\cal D}_X [s_1, \ldots, s_p]$ - Modules related to our problem. This allows to specify the structure of the Bernstein's ideals.