Papers by Shinichi Tajima
RIMS Kokyuroku Bessatsu, Nov 1, 2014
Algebraic local cohomology classes attached to Lê cycles of isolated line singularities are consi... more Algebraic local cohomology classes attached to Lê cycles of isolated line singularities are considered. A simple method that uses algebraic local cohomology classes to guess the microlocal \ ma t h r m{ b }-functions associated with line singularities is described. The correctness of the method is shown by using the index theorem of regular singular ordinary differential equations and the notion of vertical monodromy on the stratum of singular locus of a line singularity.
arXiv (Cornell University), Mar 29, 2019
Effective methods are introduced for testing zero-dimensionality of varieties at a point. The mot... more 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.
Mathematics in Computer Science, Jun 5, 2020
The torsion module of Kähler differential forms is considered in the context of symbolic computat... more The torsion module of Kähler differential forms is considered in the context of symbolic computation. Relations between logarithmic differential forms and logarithmic vector fields are investigated. As an application, an effective method is proposed for computing torsion differential forms associated with a hypersurface with an isolated singularity. The main ingredients of the proposed method are logarithmic vector fields and local cohomology.
数理解析研究所講究録別冊 = RIMS Kokyuroku Bessatsu, Jun 1, 2019
A hypersurface with a smooth 2-dimensional singular locus is considered in the context of Computa... more A hypersurface with a smooth 2-dimensional singular locus is considered in the context of Computational Algebraic Analysis. The holonomic D-module associated with each root of the reduced b-function is computed. Local cohomology solutions to the holonomic D-module are explicitly computed.
Advanced studies in pure mathematics, Oct 4, 2018
Journal of Symbolic Computation, Nov 1, 2018
A computation method of comprehensive Gröbner systems is introduced in Poincaré-Birkhoff-Witt (PB... more A computation method of comprehensive Gröbner systems is introduced in Poincaré-Birkhoff-Witt (PBW) algebras and applications to Bernstein-Sato ideals, holonomic D-modules for parametric cases are considered. The proposed method provides in particular holonomic D-modules associated with primary components of Bernstein-Sato ideals. Furthermore, with our implementation, effective methods are illustrated for computing b-functions of μ-constant deformations of hypersurface isolated singularities by utilizing holonomic D-modules. High versatility of the proposed method is also illustrated by several examples.
arXiv (Cornell University), Nov 18, 2020
Grothendieck point residue is considered in the context of computational complex analysis. A new ... more 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.
arXiv (Cornell University), Nov 19, 2018
Grothendieck local residue is considered in the context of symbolic computation. Based on the the... more Grothendieck local residue is considered in the context of symbolic computation. Based on the theory of holonomic D-modules, an effective method is proposed for computing Grothendieck local residues. The key is the notion of Noether operator associated to a local cohomology class. The resulting algorithm and an implementation are described with illustrations.
arXiv (Cornell University), Sep 11, 2022
An effective exact method is proposed for computing generalized eigenspaces of a matrix of intege... more An effective exact method is proposed for computing generalized eigenspaces of a matrix of integers or rational numbers. Keys of our approach are the use of minimal annihilating polynomials and the concept of the Jordan-Krylov basis. A new method, called Jordan-Krylov elimination, is introduced to design an algorithm for computing Jordan-Krylov basis. The resulting algorithm outputs generalized eigenspaces as a form of Jordan chains. Notably, in the output, components of generalized eigenvectors are expressed as polynomials in the associated eigenvalue as a variable.
RIMS Kokyuroku, Oct 1, 2009
RIMS Kokyuroku, Aug 1, 1991
Japan Journal of Industrial and Applied Mathematics, Jul 14, 2022
Lecture Notes in Computer Science, 2020
A generalization of integral dependence relations in a ring of convergent power series is studied... more A generalization of integral dependence relations in a ring of convergent power series is studied in the context of symbolic computation. Based on the theory of Grothendieck local duality on residues, an effective algorithm is introduced for computing generalized integral dependence relations. It is shown that, with the aid of local cohomology, generalized integral dependence relations in the ring of convergent power series can be computed in a polynomial ring. An extension of the proposed method to parametric cases is also discussed.
Kyushu Journal of Mathematics, 2000
Mathematics in Computer Science, Jan 14, 2020
Grothendieck local residue is considered in the context of symbolic computation. Based on the the... more Grothendieck local residue is considered in the context of symbolic computation. Based on the theory of holonomic D-modules, an effective method is proposed for computing Grothendieck local residues. The key is the notion of Noether operator associated to a local cohomology class. The resulting algorithm and an implementation are described with illustrations.
Publications of The Research Institute for Mathematical Sciences, 2005
The purpose of this paper is to study hypersurface isolated singularities by using partial differ... more The purpose of this paper is to study hypersurface isolated singularities by using partial differential operators based on D-modules theory. Algebraic local cohomology classes supported at a singular point that constitute the dual space of the Milnor algebra are considered. It is shown that an isolated singularity is quasi-homogeneous if and only if an algebraic local cohomology class generating the dual space can be characterized as a solution of a holonomic system of first order partial differential equations.
arXiv (Cornell University), Jan 5, 2021
New methods for computing parametric local b-functions are introduced for µ-constant deformations... more 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 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.
Springer eBooks, 2020
Logarithmic vector fields along an isolated complete intersection singularity (ICIS) are consider... more Logarithmic vector fields along an isolated complete intersection singularity (ICIS) are considered in the context of computational complex analysis. Based on the theory of local polar varieties, an effective method is introduced for computing a set of generators, over a local ring, of the modules of germs of logarithmic vector fields. Underlying ideas of our approach are the use of a parametric version of the concept of local cohomology and the Matlis duality. The algorithms are designed to output a set of representatives of logarithmic vector fields which is suitable to study their complex analytic properties. Some examples are given to illustrate the resulting algorithms.
Publications of The Research Institute for Mathematical Sciences, 2012
Algebraic local cohomology classes and holonomic systems attached to non-quasihomogeneous isolate... more Algebraic local cohomology classes and holonomic systems attached to non-quasihomogeneous isolated unimodal singularities are considered in the context of algebraic analysis. Holonomic systems and their algebraic local cohomology solution spaces attached to a unimodal singularity are studied in a constructive manner. The holonomic system constructed from linear partial differential operators of order at most two that annihilate the given algebraic local cohomology is proven to be simple for the case of nonquasihomogeneous unimodal singularities.
Uploads
Papers by Shinichi Tajima