Algorithmic computation of local and de Rham cohomology in characteristic zero: An application of D-module theory to algebraic geometry and commutative algebra

This thesis deals with algorithmic algebraic geometry in characteristic zero. More specifically, it develops ideas that arise in the theory of linear differential operators and applies them to the construction of algorithms that compute cohomological functors and invariants associated to varieties or polynomials and modules.;The basic ideas were introduced by T. Oaku. They include the computation of Bernstein-Sato polynomials, restriction of holonomic modules to a hyperplane, localizations, and local cohomology at a hypersurface.;These results are extended in this thesis as follows. We give an algorithm to compute iterated local cohomology with arbitrary support of holonomic modules, thereby solving a longstanding problem in algorithmic algebraic geometry.;Then we develop an algorithm that computes the restriction of complexes of D-modules with holonomic cohomology to linear subspaces of affine space. This generalizes an algorithm by T. Oaku and N. Takayama for the module case. As an application we provide an algorithm to compute the de Rham cohomology of the complement of arbitrary subvarieties of affine space.;We also exhibit a method to localize a module at a hypersurface where the module is holonomic at the complement of the hypersurface, due to T. Oaku, N. Takayama and myself.