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

Abstract

Given a holonomic $\mathbb{C}\[z,z^{-1}]\langle \partial\_z\rangle$-module $\mathbb{M}$, following Loeser and Sabbah (Comment. Math. Helv. $\mathbf{66}$ (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.