The formal theory of differential operators with К = k((x)) and т = x-f-have been studied by Turrittin [Tur55], Levelt [Lev75], Moser [Mos60], Hi-lali [HW87], Babbitt-Varadarajan [BV83] and many others. Several invariants related to the nature of the singularity of the differential operator have been denned, e.g., the invariant of Moser (cf. [Mos60]) and Hilali (cf. [HW87]), the order of the singularity of Katz (cf. [Del70, II. 1.10]), the irregularity index of Malgrange (cf. [Mal74]), the p-invariants of Gerard and Levelt (cf. [GL73]), the Newton-Puiseux polygon and the Newton-Puiseux polynomials (cf. [Die87]). These invariants and their connection with the characteristic class are described in section 3.7 and section 3.8. However these invariants give only partial information about the differential operator. For instance they don't give enough information to decide whether a differential operator is simple or whether two differential operators are isomorphic.
If the differential operator D: V —*■ V has a regular singularity then it is well-known that there exists а К -basis v = (vlt..., vn) of V such that С = M(D, v) is a constant matrix and that the monodromy matrix exp(27riC) determines the isomorphism class of the differential operator, cf. [Man65]. For semisimple differential operators D: V —> V and D':V —» V with a regular singularity this implies the following. Suppose M(D, v) — С and M(D',«') = C" for some К -bases v of V and v' of V with С and C" constant matrices. Then the К\т\-modules V and V are isomorphic if and only if det(A/ - C) = det(A/ - С ), where the equivalence relation = on A; [A] is denned as the transitive closure of
