Complex analysis of residues currents and computational algebraic analysis

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Hypersurface isolated singularities and associated residues currents are considered in the context of algebraic analysis.・An efficient algorithm that computes bases of a dual vector space of a Milnor algebra associated to a singular point has been constructed.・A new method for computing standard bases of a zero-dimensional ideal in a power series ring has been proposed. The key ingredient in this approach is the concept of algebraic local cohomology and the Grothendieck local duality.・An algorithm for construction holonomic D-modules attached to hypersurface isolated singularities has been derived and the structure of these holonomic D-modules have been investigated.・An algorithmic method for computing homological indices of holomorphic vector fields has been proposed.