Localizations at hyperplane arrangements: combinatorics and D-modules

We describe an algorithm deciding if the annihilating ideal of the meromorphic function 1 f , 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 due to `Alvarez-Montaner, Garc´ıa–L´opez and Zarzuela of the characteristic cycle of the D-module of meromorphic functions with respect to f.