Paco Castro: Annihilating ideals of rational functions

Abstract:  We consider the Weyl algebra A_n whose elements are linear 
differential operators with coefficients in the polynomial ring R of complex 
polynomials in n variables. For a given non-zero polynomial f, the ring 
of rational functions R_f is finitely generated considered as a module 
over A_n (this main result is due to J. Bernstein). Finding a system of 
generators of such a module is a computationally difficult task and it 
is related to the computation of the annihilating ideal of the rational 
function 1/f. This annihilating ideal can be computed by using 
algorithms of T. Oaku and N. Takayama. These algorithms use Groebner 
basis computations in the Weyl algebra A_n.
In this talk we will describe the role of logarithmic D-modules in the 
"logarithmic comparison conjecture", an open problem concerning both the 
annihilating ideal of 1/f and the comparison between the cohomology of 
the meromorphic and the logarithmic de Rham complexes associated to the 
polynomial f.