Multiple poles at negative integers for \({\int_A}f^\lambda\square\) in the case of an almost isolated singularity
Let \(f: ({\mathbb R}^{n+1},0) \to ({\mathbb R},0)\) be a real analytic germ, and \(A\in H^0(\{f\neq 0\},{\mathbb C})\) a complex linear combination of connected components of \(\{f\neq 0\}\). The aim of the paper is to give a necessary and sufficient topological condition in order that the meromorphic extension of the holomorphic current \(\lambda \to \int_A f^\lambda\square\) has a multiple pole at sufficiently ``large'' negative integers. This is achieved under the assumption that the complexification \(f_{\mathbb C}: ({\mathbb C}^{n+1},0) \to ({\mathbb C},0)\) has an isolated singularity for the eigenvalue 1, i.e. the monodromy of \(f\) in nearby points does not have 1 as eigenvalue (on reduced cohomology of the local Milnor fibre).
scientific article; zbMATH DE number 1437021