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

Let $f_i:X\to \mathbf{C}$, for $i$ integer between $1$ and $p$, be analytic functions defined on a complex analytic variety $X$. Let us consider ${\mathcal{D}}_X$ 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$\mathrm{.}Let$ m $beasectionofaholonomic$ {\cal D}X $-Module\mathrm{.}Wedenote$ {\cal B}(m, x_0, f_1, \ldots, f_p) $theidealof$ { \bf C} [s_1, \ldots, s_p] $constitutedbythepolynomials$ b $satisfyingintheneighborhoodof$ 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 $\mathcal{H}$. In addition, when $m$ is a section of a holonomic regular ${\mathcal{D}}_X$-Module, we will precise geometrically this set from the characteristic variety of ${\cal D}_X$-Modulegeneratedby$m$. We introduce and study especially the relative characteristic variety of the ${\mathcal{D}}_X[s_1,\dots ,s_p]$ - Modules related to our problem. This allows to specify the structure of the Bernstein’s ideals.