Équations fonctionnelles associées à des fonctions analytiques

Let $X$ be a  $X$, and denote by $F=f_1\dots f_p$ their product. Given a regular holonomic $\mathcal D_X$-module $\mathcal M$ and a section $m\in \mathcal M$, denote by $B(x,f_1,\dots ,f_p,m)$ the Bernstein–Sato ideal of $\mathbf C[s_1,\dots, s_p]$ consisting of polynomials $b(s_1,\dots ,s_p)$ such that there exists, in a neighborhood of $x\in F^{-1}(0)$, a differential operator $P(s_1,\dots,s_p)\in \mathcal D_X \otimes _{\mathbf C}\mathbf C[s_1,\dots , s_p]$ satisfying $P(s_1,\dots ,s_p)m f_1^{s_1+1}\dots f_p^{s_p+1} =b(s_1,\dots ,s_p)m f_1^{s_1}\dots f_p^{s_p}$. Claude Sabbah proved that this ideal is nonzero. One can associate to the characteristic variety of the $\mathcal D_X[s_1,\ldots ,s_p]$-module $\mathcal D_X[s_1,\ldots,s_p]m f_1^{s_1}\dots f_p^{s_p}$ a finite set ${\mathcal H}_{f,m}$ of hyperplanes in $\mathbf C^p$. We prove that there exists a Bernstein–Sato polynomial (i.e., a nonzero member of the Bernstein–Sato ideal) which is a product of one variable polynomials if and only if the set $\mathcal H_{f,m}$ is contained in the union of the coordinate hyperplanes. In the two variables case ($p=2$) we prove that there exist a Bernstein–Sato polynomial the higher degree form of which vanishes on and only on the set $\mathcal H_{f,m}$.