Computing the Bernstein Polynomial and the Krull-type Dimension of finitely generated $\boldsymbol{D}$-modules
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913578090496000 |
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| author | Prieto, Harry |
| author_facet | Prieto, Harry |
| contents | 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 Gröbner 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_09537 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Computing the Bernstein Polynomial and the Krull-type Dimension of finitely generated $\boldsymbol{D}$-modules Prieto, Harry Rings and Algebras 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 Gröbner 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. |
| title | Computing the Bernstein Polynomial and the Krull-type Dimension of finitely generated $\boldsymbol{D}$-modules |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2411.09537 |