Towards an Effective Version of a Theorem of Stafford

Simplifying a matrix of differential operators by elementary row and column operations is quite a fundamental task. If one considers matrices whose entries are ordinary timevarying differential operators with rational function coefficients (or more generally meromorphic functions) then a classical theorem tells you that by elementary row and column operations an analogue to the Smith form can be achieved (see, for example, Cohn, 1971, Chapter 8.1; Guralnick et al., 1988, the results going back to Jacobson, Nakayama and Teichmüller). In addition, by the simplicity of our ring of entries at most one diagonal entry can be different from 1 or 0. Thus time-varying differential matrices have a “simpler” diagonal form than constant ones. If one considers matrices whose entries are partial differential operators with rational function coefficients (now in finitely many variables x1, . . . , xn) then the ring of entries is still a simple one and one wonders what kind of simple form could be obtained by elementary row and column operations in this case. This question is obviously related to the following question: how many generators are necessary to generate a left or a right ideal in the ring of the entries? An astonishing theorem of J.T. Stafford from 1978 tells us that always two generators will be sufficient and even more: given three operators a, b, c then there must exist operators λ, μ s.t. a+λc and b + μc generate the same left ideal as a, b, and c. Thus the column [a, b, c] can be transformed to the column [a+λc, b+μc, 0] by four elementary row operations provided you are in possession of the according multiplicators. Stafford’s proof (see Stafford, 1978; Björk, 1979) is involved and does not indicate at every stage how one could determine in finitely many steps all the intermediate operators which are necessary to finally obtain two generators for an ideal. In what follows we will give in the main Section 3 a modified proof of this result which is (if at all) only slightly less involved, but which shows that Stafford’s approach can be made effective. To prove this we give an algorithmic procedure to find rather simply structured operators λ and μ which will do the job as described above. We will not give complexity bounds but it will