2025/03/06 by Daniel Barlet, Barlet, Daniel · 1 citation
Mathematics · Engineering · #Approximation Theory and Sequence Spaces #Advanced Numerical Analysis Techniques #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2503.04383
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(2iπα) 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