G\'erard-Levelt membranes

TL;DR

This paper introduces a novel geometric approach using tropical convexity to compute invariants of linear differential systems, enabling efficient calculation of Poincaré and Katz ranks without complex transformations.

Contribution

It provides a geometric interpretation of the Gérard-Levelt procedure via tropical linear spaces, allowing direct computation of important invariants of meromorphic connections.

Findings

Provides a geometric understanding of the Gérard-Levelt lattice saturation

Develops an efficient algorithm for tropical projection in this context

Generalizes Ardila's method for Bergman fans to valuated matroids

Abstract

We present an unexpected application of tropical convexity to the determination of invariants for linear systems of differential equations. We show that the classical G\'erard-Levelt lattice saturation procedure can be geometrically understood in terms of a projection on the tropical linear space attached to a subset of the local affine Bruhat-Tits building, that we call the G\'erard-Levelt membrane. This provides a way to compute the true Poincar\'e rank, but also the Katz rank of a meromorphic connection without having to perform gauge transforms nor ramifications of the variable. We finally present an efficient algorithm to compute this tropical projection map, generalising Ardila's method for Bergman fans to the case of the tight-span of a valuated matroid.