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Comprehensive Gröbner systems in PBW algebras, Bernstein-Sato ideals and holonomic $$D$$-modules. (English) Zbl 1428.13047
Let $$f_1, \cdots, f_q$$ be elements in the polynomial ring $$K[x,u]=K[x_1, \cdots , x_n,u]$$ and $$A_n[s,u] = A_n \otimes_K K[s,u]$$ be the Weyl algebra over $$K[s,u]=K[s_1, \cdots, s_q]$$, where $$A_n$$ is the Weyl algebra over K. Here $$u$$ is as a parameter. The symbol $$f^s=f_1^{s_1} \cdots f_q^{s_q}$$ is a generator of the $$A_n[s,u]$$-modul $$A_n[s] f^s$$, where the action of differential operators on $$f^s$$ is the intuitive one. One is interested in the annihilator $${\operatorname Ann}(f^s)\subset A_n[s]$$ and also certain so-called Berstein-Sato ideals $\mathcal B_i =\{ b\in K[s,u] |b f^s \in A_n[s]\cdot f_if^s \}, \quad \mathcal B_\Sigma =\{ b\in K[s,u] | bf^s \in A_n[s]\cdot \sum_{i=1}^qf_if^s \}.$ A specialisation $$s$$ is a homomorphism $$s: K[x,u]\to K'[x]$$, wher $$K'$$ is a field, which then also induces a homomorphism $$s: A_n[s,u]\to A_n'[s]$$, where $$A_n'$$ is the Weyl algebra over $$K'$$. A finite set $$G$$ in $$K[x,u]$$ is a comprehenseve Gröbner basis if $$s(G)$$ is a Gröbner basis for each specialisation $$s$$. This notion was studied by V. Weispfenning [J. Symb. Comput. 14, No. 1, 1–29 (1992; Zbl 0784.13013)]. For the construction one decomposes $$\operatorname {Spec}K[u]$$ into constructible subsets, where one provides a comprehensive Gröbner basis for one such subset at a time. In the paper this notion is considered for the non-commutative ring $$A_n[s,u]$$ by extending algorithms by M. Kalkbrener [J. Symb. Comput. 24, No. 1, 51–58 (1997; Zbl 1054.13502)] in the commutative case to the ring $$A_n[s,u]$$. This is implemented to compute comprehensive Gröbner bases of $${\operatorname {Ann}}(f^s)$$ and also comprehensive Gröbner bases of $$\mathcal B_i$$ and $$\mathcal B_\Sigma$$. Some concrete examples are given and a link to used software is provided.
MSC:
 13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) 13D45 Local cohomology and commutative rings 13J05 Power series rings 14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials 32A27 Residues for several complex variables 32C37 Duality theorems for analytic spaces
Software:
Macaulay2; OpenXM; PGB; Plural; Risa/Asir; SINGULAR
Full Text:
References:
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