Generalized Brieskorn Modules II: Higher Bernstein Polynomials and Multiple Poles
Daniel Barlet
Journal of Singularities
volume 29 (2026), 66-99
Received: 21 March 2025. In revised form: 18 January 2026
Abstract:
Our main result is to show that, if the p-th Bernstein polynomial of the (a,b)-module generated by a germ of a holomorphic volume form \omega in \Omega^{n+1}_0 in the (convergent) Brieskorn (a,b)-module associated to f, has a root -\alpha - N, there exists a pole of order at least p for the meromorphic extension of an analytic functional associated to \omega at some the point in -\alpha - N, under the hypothesis that f has an isolated singularity at the origin relative to the corresponding eigenvalue exp(2i\pi\alpha) of the monodromy. This implies the existence of at least p roots in -\alpha - N (counting multiplicities) for the usual reduced Bernstein polynomial of the germ of f at 0.
We also obtain in the case of an isolated singularity for f that the largest root -\alpha - m inside -\alpha - N of the reduced Bernstein polynomial of f produces a pole at the point \lambda = -\alpha - m for the meromorphic extension of the distribution \vert f\vert^{2\lambda}\bar f^{-h} for some h in N.
2020 Mathematical Subject Classification:
32S25; 32S40 ; 34E05
Key words and phrases:
Bernstein Polynomial; Higher Bernstein Polynomials; Generalized Brieskorn Modules; Asymptotic Expansions; Period-integral; Convergent $(a,b)$-Module; Hermitian Period; Geometric (convergent) $(a,b)$-Modules; (convergent) Fresco
Author(s) information:
Daniel Barlet
Institut Elie Cartan UMR 7502
Université de Lorraine, CNRS, INRIA et
Institut Universitaire de France
BP 239 - F - 54506 Vandoeuvre-lès-Nancy Cedex, France
email: daniel.barlet@univ-lorraine.fr