A polynomial-time algorithm for the Jacobson form for matrices of differential operators

We consider a ring R = K[∂; id, θ] of differential operators over a differential field (K, θ). This is a (left and right) principal ideal domain. Hence, if R is simple, then for every matrix M ∈ R n n there exist unimodular matrices S and T ∈ R n n and f ∈ R such that SMT = diag(1, . . . , 1, f, 0, . . . , 0). The proof of the existence goes back to Jacobson, Nakayama and Teichmüller. Therefore, this strong diagonal form is known as Jacobson form in Zerz (2007, Theorem 3.2) or Teichmüller-Nakayama normal form in Ilchmann and Mehrmann (2005, Theorem 2.1). In this paper, we present a polynomial time algorithm for computing a strong diagonal form (which we will later call Jacobson form) in the case where M has R-linearly independent rows. The method exploits the well-known existence of cyclic vectors for the module R/RM , and is applicable for all fields with “enough” constants and a “sufficiently” high degree over their constant field. We will achieve the form diag(1, . . . , 1, f) even for non-simple R.