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arXiv · 2605.27797

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

Toshinori Oaku
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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.

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BibTeXRIS

Toshinori Oaku. 2026-05-27. Indicial polynomials and $b$-functions of $D$-modules along arbitrary varieties and their computation. https://arxiv.org/abs/2605.27797

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