Formal Stationary Phase for the Mellin Transform of a $\mathcal{D}$-Module

Abstract

Given a holonomic $\mathbb{C}[z,z^{-1}]⟨{\partial }_z⟩$-module $\mathbb{M}$, following Loeser and Sabbah (Comment. Math. Helv. $66$ (1991), 458–503), one can consider its Mellin transform, which is a difference system on the affine line over $\mathbb{C}$. In this note we prove a stationary phase formula, which shows that its formal behavior at infinity is determined by the local germs defined by $\mathbb{M}$ at its singular points.