Generalized Brieskorn Modules II: Higher Bernstein Polynomials and Multiple Poles

Our main result is to show that the existence of a root in. --$\alpha$--Nfor the p-th Bernstein polynomial of the (a,b)-module generated by a holomorphicform in the (convergent) Brieskorn (a,b)-module associated to f, under the hypothesis that f has an isolated singularity at the origin relative to the eigenvalue exp(2i$\pi$$\alpha$) of the monodromy, produces poles of order at least p for themeromorphic extension of the (conjugate) analytic functional given by polar partsat points--$\alpha$--N for N well chosen integer. This result is new, even forp= 1. As a corollary, this implies that, in the case of an isolated singularity for f,the existence of a root in. --$\alpha$--N for the p-th Bernstein polynomial of the (a,b)-module generated by a holomorphic form implies the existence of at leastp roots (counting multiplicities) for the usual reduced Bernstein polynomial of thegerm of f at the origin.In the case of an isolated singularity for f, we obtain that for each $\alpha$ thebiggest root --$\alpha$--m. of the reduced Bernstein polynomial of f in --$\alpha$--N producesa pole at--$\alpha$--m for the meromorphic extension of the associated distribution