The analytic standard fan of a image-module

In this paper, we associate with any monogeneous module over the ring image of germs of linear differential operators at the origin of Cn, with holomorphic coefficients, a combinatorial object which we call the standard fan of this image-module (see Section 6 for a precise geometric description of this object). The main tool of the proof is the homogenization technique and a convergent division theorem in the homogenization ring image This last result is the key tool to an extension to the analytic image-module case of our results in the algebraic case of the Weyl algebra (see Assi et al., J. Pure Appl. Algebra, 150 (1) (2000) 27–39.