🤖 AI Summary
This work extends classical indicial polynomials and Bernstein–Sato polynomials to arbitrary subschemes, introducing for the first time a notion of indicial polynomial along an arbitrary subvariety within the framework of $D$-modules. This definition unifies the indicial polynomial of a single differential equation, the Bernstein–Sato polynomial of an algebraic variety due to Budur–Mustață–Saito, and the classical $b$-function along a smooth submanifold. Notably, it remains well-defined even when the $b$-function does not exist, and its set of roots recovers the $b$-function whenever the latter is defined. The approach combines $D$-module theory, algebraic geometry, and symbolic computation, leveraging inverse image functors and embedding techniques to handle parametric settings more efficiently. The result establishes a universal connection between indicial polynomials and $b$-functions, providing a more general and computationally tractable theoretical framework.
📝 Abstract
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.