Generalized Brieskorn Modules II: Higher Bernstein Polynomials and Multiple Poles

Abstract

Our main result is to show that the existence of a root in. --$α$--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$π$$α$) of the monodromy, produces poles of order at least p for themeromorphic extension of the (conjugate) analytic functional given by polar partsat points--$α$--N for N well chosen integer. This result is new, even forp= 1. As a corollary, this implies that, in the case of an isolated singularity for f,the existence of a root in. --$α$--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 $α$ thebiggest root --$α$--m. of the reduced Bernstein polynomial of f in --$α$--N producesa pole at--$α$--m for the meromorphic extension of the associated distribution