On the Mellin transform of a D-module

TL;DR

This paper establishes a formal stationary phase formula for the Mellin transform of holonomic D-modules on a 1-dimensional torus, linking its behavior at infinity to local data at singular points.

Contribution

It introduces a formal stationary phase formula that relates the asymptotic behavior of the Mellin transform to local germs of the D-module at singularities.

Findings

Proves a formal stationary phase formula for Mellin transforms of D-modules.

Shows the behavior at infinity is determined by local germs at singular points.

Connects global Mellin transform behavior with local data.

Abstract

Given a holonomic D-module M on a 1-dimensional torus, one can consider its Mellin transform, which is a difference system on the affine line. In this note we prove a formal stationary phase formula, which shows that the formal behavior at infinity of the Mellin transform of M is determined by the local germs defined by M at its singular points.