A bound on the multiplicity of horizontal sections for a connection on a Riemann surface

We address the question of bounding the multiplicity of the solutions of a linear differential system, setting the problem in invariant terms. A meromorphic connection is considered on a holomorphic vector bundle over a compact Riemann surface. We produce an upper bound on the order of vanishing of an arbitrary horizontal section, which depends only on global data, provided the connection has only regular singularities or the underlying monodromy is irreducible.