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.

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.

This work analyzes the optimal pico base station (BS) deployment in heterogeneous cellular networks (HetNet) with hotspots. Most of conventional works for HetNet focused on interference coordination and effective cell association methods, however, the problem of pico BS deployment (cell planning) for HetNet with hotspots has not been analyzed so much. In this paper, we extend the previously proposed optimization problem in terms of network parameters (spectrum splitting ratio and SINR bias value) to the optimal pico BS locations to maximize the system rate. Furthermore, the average user and outage user rates are evaluated numerically to show the effectiveness of the proposed optimization method. Numerical results show that the optimized pico BS locations can improve both the average and outage user rates in HetNet with hotspots.