# B-functions and holonomic systems

@article{Kashiwara1976BfunctionsAH, title={B-functions and holonomic systems}, author={Masaki Kashiwara}, journal={Inventiones mathematicae}, year={1976}, volume={38}, pages={33-53} }

A b-function of an analytic function f(x) is, by definition, a gcnerator of the ideal formed by the polynomials b(s) satisfying P(s, x, Dx) f (x)" + 1 = b(s) f(x) ~ for some differential operator P(s, x, Dx) which is a polynomial on s. Professor M.Sato introduced the notions of "a-function", "b-function" and "'c-function" for relative invariants on prehomogeneous vector spaces, when he studied the fourier transforms and ~-functions associated with them (see [10, 12]). He defined, in the same…

## 370 Citations

On the holonomic systems of linear differential equations, II

- Mathematics
- 1978

In this paper we shall study the restriction of holonomic systems of differential equations. Let X be a complex manifold and Y a submanifold, and let (9 x and D x be the sheaf of the holomorphic…

On the Holonomic System of f s and ^-Function

- Mathematics
- 2006

M. Sato initiated a theory of prehomogenous vector spaces in 1961 and defined a-, band c-functions. The general theory of ^-function has extensively been developed since 1972, and an amount of…

Onb-function, spectrum and rational singularity

- Mathematics
- 1993

by duality [18], because R~Tr,CCy, = 0 for i > 0 by [6, 31] (this follows also from [13, 21, 23]) where rc is assumed projective. Here COy denotes the dualizing sheaf (i.e., the dualizing complex…

Multiplier Ideals, B-function, and Spectrum of a Hypersurface Singularity

- Mathematics
- 2007

We prove that certain roots of the Bernstein-Sato polynomial (i.e. b-function) are jumping coefficients up to a sign, showing a partial converse of a theorem of L. Ein, R. Lazarsfeld, K.E. Smith, and…

Brian\c{c}on-Skoda exponents and the maximal root of reduced Bernstein-Sato polynomials

- Mathematics
- 2021

. For a holomorphic function f on a complex manifold X , the Brian¸con-Skoda exponent e BS ( f ) is the smallest integer k with f k ∈ ( ∂f ) (replacing X with a neighborhood of f − 1 (0)), where ( ∂f…

Multiplier ideals, $b$-function, and spectrum of a hypersurface singularity

- MathematicsCompositio Mathematica
- 2007

We prove that certain roots of the Bernstein–Sato polynomial (i.e. $b$-function) are jumping coefficients up to a sign, showing a partial converse of a theorem of L. Ein, R. Lazarsfeld, K. E. Smith,…

ON THE POLES OF p-ADIC COMPLEX POWERS AND THE b- FUNCTIONS OF PREHOMOGENEOUS VECTOR SPACES

- Mathematics
- 1990

Introduction. In the real case, it is well known that there exists an intimate relation between poles of complex powers of polynomial functions and the roots of the b-functions (cf. [B]). Many…

Inverse image of D-modules and quasi-b-functions

- Mathematics
- 2008

The usual b-function of a holonomic \( \mathcal{D} \)-module is associated to the Euler vector field but the elementary case of a ramification map shows that this Euler vector field is not preserved…

Algorithms for b-Functions, Restrictions, and Algebraic Local CohomologyGroups of D-Modules

- Mathematics
- 2011

Our purpose is to present algorithms for computing some invariants and functors attached to algebraic D-modules by using Grobner bases for ̈ differential operators. Let K be an algebraically closed…

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Le polynome de Bernstein d'une singularit6 isolee. Lecture notes in Math

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The theory of b functions. Mater's these presented to Kyoto University (1975) (here we find many interesting examples of b-functions)

- The theory of b functions. Mater's these presented to Kyoto University (1975) (here we find many interesting examples of b-functions)