Algebraic analysis of residue theory in several complex variables and algorithms

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Upon using the theory of holonomic D-modules and methods of computer algebra, we investigated algorithmic aspects of Grothendieck local residues. As applications, we derived and implemented diverse variety of algorithms.1. Algorithm for computing annihilating ideal in the Weyl algebra of a zero dimensional algebraic local cohomology are derived. Main ingredient of this derivation is the notion of holonomic D-modules.2. An algorithm that compute Grothendieck local residues is constructed. The resulting algorithm is efficient and available in use for generic case.3. Some improvement of the above algorithm are also studied.4. Solvability condition for an ordinary differential equation in a space of convergent power series and in a space of formal power series are investigated in the context of algebraic analysis. A necessary and sufficient condition for the solvability is described in terms of local residues. A regular singular system of ordinary differential equation which characterizes algebraic local cohomology solutions of the formal adjoint equation is introduced. The use of this system provides an effective method for computing formal solvability conditions.5. Algebraic local cohomology classes attached to a non quasi homogeneous isolated singularity are studies in the context of D-modules.