An algorithm for de Rham cohomology groups of the complement of an affine variety via D-module computation

Abstract

We give an algorithm to compute the following cohomology groups on U= C n⧹V(f) for any non-zero polynomial f∈ Q [x 1,…,x n] : 1. H k(U, C U), C U is the constant sheaf on U with stalk C . 2. H k(U, V), V is a locally constant sheaf of rank 1 on U. We also give partial results on computation of cohomology groups on U for a locally constant sheaf of general rank and on computation of H k( C n⧹Z, C ) where Z is a general algebraic set. Our algorithm is based on computations of Gröbner bases in the ring of differential operators with polynomial coefficients.