Computing Regular Meromorphic Differential Forms via Saito's Logarithmic Residues

Abstract

Logarithmic differential forms and logarithmic vector fields associated with a hypersurface with an isolated singularity are considered in the context of computational complex analysis. As applications, based on the concept of torsion differential forms due to A.G. Aleksandrov, regular meromorphic differential forms introduced by D. Barlet and M. Kersken, and Brieskorn formulae on Gauss-Manin connections are investigated. (i) A method is given to describe singular parts of regular meromorphic differential forms in terms of non-trivial logarithmic vector fields via Saito's logarithmic residues. The resulting algorithm is illustrated by using examples. (ii) A new link between Brieskorn formulae and logarithmic vector fields is discovered, and an expression that rewrites Brieskorn formulae in terms of non-trivial logarithmic vector fields is presented. A new effective method is described to compute non-trivial logarithmic vector fields, which are suitable for the computation of Gauss-Manin connections. Some examples are given for illustration.