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.

翻译：我们定义了沿任意子簇的$D$-模的示性多项式，它同时推广了单个线性微分方程的经典示性多项式和由Budur-Mustata-Saito定义的簇的Bernstein-Sato多项式。示性多项式也是沿子流形的$D$-模的$b$-函数的一种推广，并可在嵌入的$D$-模理论逆像计算中替代$b$-函数。我们研究了示性多项式的性质及其与$b$-函数的关系。即使$b$-函数不存在，示性多项式也可能存在，且当$b$-函数存在时，它给出了$b$-函数根集。示性多项式的计算比$b$-函数更简单，且自然包含参数情形。