Computing the Bernstein Polynomial and the Krull-type Dimension of finitely generated $\boldsymbol{D}$-modules

We establish the existence of the Bernstein polynomial in one indeterminate $t$, and provide a method for its explicit computation. The Bernstein polynomial is associated with finitely generated modules over the Weyl algebra, known as $D$-modules, and is notoriously difficult to compute directly. Our approach is constructive, offering a systematic method to compute the Bernstein polynomial and its associated invariants explicitly. We begin by introducing the Weyl algebra as a ring of operators and stating some of its main properties, followed by considering the class of numerical polynomials. We then develop a generalization of the theory of Gr\"obner bases specifically for $D$-modules and use it to compute the Bernstein polynomial and its invariants. As an application of the properties of the Bernstein polynomial, we develop the concept of the Krull-type dimension for $D$-modules, which sheds light on the structure of these modules.