Introduction to a theory of b-functions

TL;DR

This paper introduces the theory of b-functions (Bernstein-Sato polynomials), exploring their properties, relations to singularities, and explicit calculations for specific ideals, providing foundational knowledge for further research.

Contribution

It extends the theory of b-functions to arbitrary ideals and connects them with singularities and multiplier ideals, including explicit computations for monomial ideals and hyperplane arrangements.

Findings

Calculated b-functions for monomial ideals

Determined b-functions for certain hyperplane arrangements

Established relations between b-functions and singularity invariants

Abstract

We give an introduction to a theory of b-functions, i.e. Bernstein-Sato polynomials. After reviewing some facts from D-modules, we introduce b-functions including the one for arbitrary ideals of the structure sheaf. We explain the relation with singularities, multiplier ideals, etc., and calculate the b-functions of monomial ideals and also of hyperplane arrangements in certain cases.