The purpose of this paper is to give a reduction procedure for the construction of a Turrittin's canonical form associated with an invertible linear difference system. The nilpotent case is treated by methods of deformation of orbits under the adjoint representation of GL(n, C). We prove also a statement on uniqueness.

We define the notion of a Galois extension of a differential field; we work with differential algebras which are more easy than differential fields. This definition is geometric and does not use Weil's theorem found in Kolchin & Lang. We give the construction of the Picard-Vessiot extension associated to a differential equation; this construction is similar to the notion of a G-structure due to Bernard.

Abstract. We introduce and describe the characteristic class of a difference operator over the difference field (k((t)), τ ). Here k is an algebraically closed field of characteristic zero and τ is the k-linear automorphism of k((t)) defined by τ (t) = t/(1 + t). The approach is based on the characterization of simple difference operators in terms of their eigenvalues.Mathematics Subject Classifications (1991): 39A10, 39A70, 47B39.

Using the formal reduction by a method of deformation of orbits under the adjoint representation of GL(n, C), we have proved the existence and uniqueness (up to equivalence under GL(n, C)) of a formal canonical form of systems of singular linear difference equations. In this paper we study the stability of the irregular part of the canonical form under perturbation of the matrix coefficients.