Indicial polynomials and $b$-functions of $D$-modules along arbitrary varieties and their computation
This work provides a new theoretical tool for D-module theory that simplifies computation and extends applicability to cases where b-functions are not available, benefiting researchers in algebraic analysis and singularity theory.
The authors define indicial polynomials for D-modules along arbitrary subvarieties, generalizing classical indicial polynomials and Bernstein-Sato polynomials. They show that indicial polynomials can exist when b-functions do not, and that they are easier to compute while still providing the roots of the b-function when it exists.
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.