Grothendieck point residues are studied in the context of symbolic computation. Based on the theory of holonomic D-modules associated with local cohomology, we introduce new effective algorithms for computing Grothendieck residue mappings and residues. The strategies are to use holonomic systems of first order linear partial differential operators for the relevant local cohomology classes and to avoid syzygy computations in deriving these operators.

Grothendieck point residue is considered in the context of symbolic computation. Based on the theory of holonomic D-modules associated to a local cohomology class, a new effective method is given for computing Grothendieck point residue mappings. A basic strategy of our approach is the use of holonomic systems of first order linear partial differential equations. The resulting algorithm is easy to implement and can also be used to compute Grothendieck point residues in an effective manner.

We provide a concrete computational algorithm for computing the standard basis for a mixed module proposed by Gatermann and Hosten [1]. We extend it to parametric standard system for a mixed module and provide an algorithm to compute it. We demonstrate our algorithm by applying it to classification of map-germs relative to A in which complicated moduli structures appear.

An algorithm for computing comprehensive Gröbner systems (CGS) is introduced in rings of linear partial differential operators. Their applications to b-functions are considered. The resulting algorithm designed for a wide use of computing comprehensive Gröbner systems can be used to compute all the roots of b-functions and relevant holonomic D-modules. Furthermore, with our implementation, effective methods are illustrated for computing holonomic D-modules associated with hypersurface singularities. It is shown that the proposed algorithm is full of versatility.

Logarithmic vector fields associated with parametric semi-quasihomogeneous hypersurface isolated singularities are considered in the context of symbolic computation. A new algorithm for computing the logarithmic vector fields is introduced. The keys of this approach are the concept of a polar variety and parametric local cohomology systems. The resulting algorithm also provides a decomposition of the parameter space depending on the structure of the logarithmic vector fields.

A new algorithm is given for computing parametric local cohomology classes associated with semi-quasihomogeneous singularities. The essential point of the proposed algorithm involves Poincaré polynomials and weighted degrees. The proposed algorithm gives a suitable decomposition of the parameter space depending on the structure of the parametric local cohomology classes. As an application, an algorithm for computing parametric standard bases of zero-dimensional ideals, is given. These algorithms work for non-parametric cases, too.

We introduce a new algorithm for computing comprehensive Gröbner systems.There exists the Suzuki-Sato algorithm for computing comprehensive Gröbner systems. The Suzuki-Sato algorithm often creates overmuch cells of the parameter space for comprehensive Gröbner systems. Therefore the computation becomes heavy. However, by using inequations ("not equal zero"), we can obtain different cells. In many cases, this number of cells of parameter space is smaller than that of Suzuki-Sato's. Therefore, our new algorithm is more efficient than Suzuki-Sato's one, and outputs a nice comprehensive Gröbner system. Our new algorithm has been implemented in the computer algebra system Risa/Asir We compare the runtime of our implementation with the Suzuki-Sato algorithm and find our algorithm superior in many cases.