arXiv subjects

Katsusuke Nabeshima

Publications and source records attributed to Katsusuke Nabeshima.

5 recordsLinked to original sources

arXiv2021

Computing Regular Meromorphic Differential Forms via Saito's Logarithmic Residues

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.

math.AG↗

arXiv2021

Methods for computing $b$-functions associated with $μ$-constant deformations -- Case of inner modality 2 --

New methods for computing parametric local $b$-functions are introduced for $μ$-constant deformations of semi-weighted homogeneous singularities. The keys of the methods are comprehensive Gröbner systems in Poincaré-Birkhoff-Witt algebra and holonomic ${\mathcal D}$-modules. It is shown that the use of semi-weighted homogeneity reduces the computational complexity of $b$-functions associated with $μ$-constant deformations. In the case of inner modality 2, local $b$-functions associated with $μ$-constant deformations are obtained by the resulting method and given the list of parametric local $b$-functions.

math.AG↗

arXiv2020

An effective method for computing Grothendieck point residue mappings

Grothendieck point residue is considered in the context of computational complex analysis. A new effective method is proposed for computing Grothendieck point residues mappings and residues. Basic ideas of our approach are the use of Grothendieck local duality and a transformation law for local cohomology classes. A new tool is devised for efficiency to solve the extended ideal membership problems in local rings. The resulting algorithms are described with an example to illustrate them. An extension of the proposed method to parametric cases is also discussed as an application.

cs.SC↗

arXiv2019

Testing zero-dimensionality of varieties at a point

Effective methods are introduced for testing zero-dimensionality of varieties at a point. The motivation of this paper is to compute and analyze deformations of isolated hypersurface singularities. As an application, methods for computing local dimensions are also described. For the case where a given ideal contains parameters, the proposed algorithms can output in particular a decomposition of a parameter space into strata according to the local dimension at a point of the associated varieties. The key of the proposed algorithms is the use of the notion of comprehensive Gröbner systems.

cs.SC↗

arXiv2015

Algebraic Local Cohomology with Parameters and Parametric Standard Bases for Zero-Dimensional Ideals

A computation method of algebraic local cohomology with parameters, associated with zero-dimensional ideal with parameter, is introduced. This computation method gives us in particular a decomposition of the parameter space depending on the structure of algebraic local cohomology classes. This decomposition informs us several properties of input ideals and the output of our algorithm completely describes the multiplicity structure of input ideals. An efficient algorithm for computing a parametric standard basis of a given zero-dimensional ideal, with respect to an arbitrary local term order, is also described as an application of the computation method. The algorithm can always output "reduced" standard basis of a given zero-dimensional ideal, even if the zero-dimensional ideal has parameters.

cs.SC↗