GrardLevelt membranes
We present an unexpected application of tropical convexity to the determination of invariants for linear systems of differential equations. We show that the classical GrardLevelt 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 BruhatTits building, which we call the GrardLevelt membrane. This provides a way to compute the true Poincar 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 Ardilas method for Bergman fans to the case of the tight-span of a valuated matroid.
tropical convexity meromorphic connections valuated matroids