Computing Regular Meromorphic Differential Forms via Saito's Logarithmic Residues

TL;DR

This paper develops algorithms to compute regular meromorphic differential forms and logarithmic vector fields for hypersurfaces with isolated singularities, linking them to Gauss-Manin connections and Brieskorn formulae.

Contribution

It introduces a novel method to describe singular parts of meromorphic forms using Saito's residues and links Brieskorn formulae with logarithmic vector fields.

Findings

Algorithm for singular parts of meromorphic forms using logarithmic residues

New expression of Brieskorn formulae in terms of logarithmic vector fields

Effective method for computing logarithmic vector fields for Gauss-Manin connections

Abstract

Logarithmic differential forms and logarithmic vector fields associated to 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.