Gröbner bases and logarithmicD -modules

Let be the ring of polynomials with complex coefficients and An the Weyl algebra of order nover . Elements in Anare linear differential operators with polynomial coefficients. For each polynomial f, the ring of rational functions with poles along f has a natural structure of a left An-module which is finitely generated by a classical result of I.N. Bernstein. A central problem in this context is how to find a finite presentation of M starting from the input f. In this paper we use Gröbner base theory in the non-commutative frame of the ring An to compare M to some other An-modules arising in Singularity Theory as the so-called logarithmic An-modules. We also show how the analytic case can be treated with computations in the Weyl algebra if the input data f is a polynomial.