Gérard–Levelt membranes

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 Gerard–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, which we call the Gerard–Levelt membrane. This provides a way to compute the true Poincare rank, but also the Katz rank of a meromorphic connection without having to perform either gauge transforms or 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.