Formal stationary phase for the Mellin transform of a $\mathcal{D}$-module

Abstract

Given a holonomic $\mathbb{C}\left[z, z^{-1}\right]\left\langle\partial_z\right\rangle$-module $\mathbb{M}$, following Loeser and Sabbah (Comment. Math. Helv. 66 (1991), 458503), 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.