Localization at hyperplane arrangements: Combinatorics and D-modules

We describe an algorithm deciding if the annihilating ideal of the meromorphic function , where f=0 defines an arrangement of hyperplanes, is generated by linear differential operators of order 1. The algorithm is based on the comparison of two characteristic cycles and uses a combinatorial description of the characteristic cycle of the D-module of meromorphic functions with respect to f, due to Àlvarez Montaner, García López and Zarzuela.