Multiple Poles at Negative Integers for ∫_<em>A</em><em>f^λ</em><font size="+1">☐</font> in the Case of an Almost Isolated Singularity

Abstract

We give a necessary and sufficient topological condition on A ∈ H_0({f ≠ 0},ℂ), for a real analytic germ f : (ℝ_n+1,0) → (ℝ,0), whose complexification has an isolated singularity relatively to the eigenvalue 1 of the monodromy, in order that the meromorphic continuation of ∫_A__fλ_☐ has a multiple pole at sufficiently “large” negative integers. We show that if such a multiple pole exists, it occurs already at λ = −(n + 1) with its maximal order which is computed topologically.