Linear systems with nilpotent leading term

Abstract

For systems of meromorphic linear ODE with nilpotent leading term, there exist certain analytic transformations simplifying the coefficient matrix in a certain sense. At first glance, these transformation appear to be formal in the sense that the power series representation generally has radius of convergence equal to zero. Here we show that this is not the case—indeed, we show how one can explicitly obtain the transformation in terms of the coefficient matrix of the system without solving any differential equations. When the Jordan form of the leading term is a single nilpotent block, this construction is equivalent to finding a cyclic vector of a certain form.