scholarly journals A residual duality over Gorenstein rings with application to logarithmic differential forms

2018 ◽  
Author(s):  
Mathias Schulze ◽  
Laura Tozzo
Keyword(s):  
Differential Forms ◽  
Gorenstein Rings ◽  
Logarithmic Differential Forms

scholarly journals Computing Regular Meromorphic Differential Forms via Saito's Logarithmic Residues

2021 ◽  
Author(s):  
Shinichi Tajima ◽  
◽  
Katsusuke Nabeshima ◽  
Keyword(s):  
Vector Fields ◽  
Complex Analysis ◽  
Differential Forms ◽  
Isolated Singularity ◽  
Logarithmic Differential Forms ◽  
Meromorphic Differential

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.

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