Searcharxiv⌕ Search

arXiv subjects

Toshinori Oaku

Publications and source records attributed to Toshinori Oaku.

13 recordsLinked to original sources
arXiv2026

Indicial polynomials and $b$-functions of $D$-modules along arbitrary varieties and their computation

We define an indicial polynomial of a $D$-module along an arbitrary subvariety as a generalization of both the classical indicial polynomial for a single linear differential equation and the Bernstein-Sato polynomial of a variety defined by Budur-Mustata-Saito. An indicial polynomial is also a generalization of the $b$-function of a $D$-module along a submanifold and can be used in the computation of the $D$-module theoretic inverse image by the embedding instead of the $b$-function. We consider properties of indicial polynomials and relations with $b$-functions. An indicial polynomial may exist even if the $b$-function does not, and gives the set of the roots of the $b$-function if it exists. Computation of an indicial polynomial is easier than the $b$-function and naturally includes the case with parameters.

math.AG↗
arXiv2016

Localization, local cohomology, and the b-function of a D-module with respect to a polynomial

Given a $D$-module $M$ generated by a single element, and a polynomial $f$, one can construct several $D$-modules attached to $M$ and $f$ and can define the notion of the (generalized) $b$-function following M. Kashiwara. These modules are closely related to the localization and the local cohomology of $M$. We show that the $b$-function, if it exists, controls these modules and present general algorithms for computing these modules and the $b$-function (if it exists) without any further assumptions. For these algorithms, we reformulate the localization algorithm for $D$-modules.

math.AG↗
arXiv2016

Operational calculus for holonomic distributions in the framework of D-module theory

Let $f$ be a real polynomial of $x = (x_1,\dots,x_n)$ and $φ$ be a locally integrable function of $x$ which satisfies a holonomic system of linear differential equations. We study the distribution $f_+^λφ$ with a meromorphic parameter $λ$, especially its Laurent expansion and integration, from an algorithmic viewpoint in the framework of $D$-module theory.

math.CV↗
arXiv2015

Length and multiplicity of the local cohomology with support in a hyperplane arrangement

Let $R$ be the polynomial ring in $n$ variables with coefficients in a field $K$ of characteristic zero. Let $D_n$ be the $n$-th Weyl algebra over $K$. Suppose that $f \in R$ defines a hyperplane arrangement in the affine space $K^n$. Then the length and the multiplicity of the 1st local cohomology group $H^1_{(f)}(R)$ as left $D_n$-module coincide and are explicitly expressed in terms of the Poincaré polynomial or the Möbius function of the arrangement.

math.AG↗
arXiv2015

Annihilators of Laurent coefficients of the complex power for normal crossing singularity

Let $f$ be a real-valued real analytic function defined on an open set of $\mathbb{R}^n$. Then the complex power $f_+^λ$ is defined as a distribution with a holomorphic parameter $λ$. We determine the annihilator (in the ring of differential operators) of each coefficient of the principal part of the Laurent expansion of $f_+^λ$ about $λ=-1$ in case $f=0$ has a normal crossing singularity.

math.CV↗
arXiv2014

Properties of powers of functions satisfying second-order linear differential equations with applications to statistics

We derive properties of powers of a function satisfying a second-order linear differential equation. In particular we prove that the n-th power of the function satisfies an (n+1)-th order differential equation and give a simple method for obtaining the differential equation. Also we determine the exponents of the differential equation and derive a bound for the degree of the polynomials, which are coefficients in the differential equation. The bound corresponds to the order of differential equation satisfied by the n-fold convolution of the Fourier transform of the function. These results are applied to some probability density functions used in statistics.

math.CA↗
arXiv2012

An algorithm to compute the differential equations for the logarithm of a polynomial

We present an algorithm to compute the annihilator of (i.e., the linear differential equations for) the logarithm of a polynomial in the ring of differential operators with polynomial coefficients. The algorithm consists of differentiation with respect to the parameter s of the annihilator of f^s for a polynomial f and quotient computation. More generally, the annihilator of f^s(log f)^m for a complex number s and a positive integer m can be computed, which constitutes what is called a holonomic system in D-module theory. This enables us to compute a holonomic system for the integral of a function involving the logarithm of a polynomial by using integration algorithm for D-modules.

cs.SC↗
arXiv2011

Algorithms for integrals of holonomic functions over domains defined by polynomial inequalities

We present an algorithm for computing a holonomic system for a definite integral of a holonomic function over a domain defined by polynomial inequalities. If the integrand satisfies a holonomic difference-differential system including parameters, then a holonomic difference-differential system for the integral can also be computed. In the algorithm, holonomic distributions (generalized functions in the sense of L. Schwartz) are inevitably involved even if the integrand is a usual function.

cs.SC↗
arXiv2008

Algorithm for computing local Bernstein-Sato ideals

Given $p$ polynomials of $n$ variables over a field $k$ of characteristic 0 and a point $a \in k^n$, we propose an algorithm computing the local Bernstein-Sato ideal at $a$. Moreover with the same algorithm we compute a constructible stratification of $k^n$ such that the local Bernstein-Sato ideal is constant along each stratum. Finally, we present non-trivial examples computed with our algorithm.

math.AG↗
arXiv1999

A Localization Algorithm for $D$-modules

We present a method to compute the holonomic extension of a $D$-module from a Zariski open set in affine space to the whole space. A particular application is the localization of coherent $D$-modules which are holonomic on the complement of an affine variety.

math.AG↗
arXiv1998

An algorithm for de Rham cohomology groups of the complement of an affine variety via D-module computation

We give an algorithm to compute the following cohomology groups on $U = \C^n \setminus V(f)$ for any non-zero polynomial $f \in \Q[x_1, ..., x_n]$; 1. $H^k(U, \C_U)$, $\C_U$ is the constant sheaf on $U$ with stalk $\C$. 2. $H^k(U, \Vsc)$, $\Vsc$ is a locally constant sheaf of rank 1 on $U$. We also give partial results on computation of cohomology groups on $U$ for a locally constant sheaf of general rank and on computation of $H^k(\C^n \setminus Z, \C)$ where $Z$ is a general algebraic set. Our algorithm is based on computations of Gröbner bases in the ring of differential operators with polynomial coefficients.

math.AG↗