Grobner bases and logarithmic D-modules

Let C[x]= C[x(1),...x(n)] be the ring of polynomials with complex coefficients and A(n) the Weyl algebra of order it over C. Elements in A(n) are linear differential operators with polynomial coefficients. For each polynomial f, the ring M = C[x](f) of rational functions with poles along f has a natural structure of a left A(n)-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 Grobner base theory in the non-commutative frame of the ring A(n) to compare M to some other A(n)-modules arising in Singularity Theory as the so-called logarithmic A(n)-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. (c) 2005 Elsevier Ltd. All rights reserved.