Comprehensive Gröbner systems in PBW algebras, Bernstein-Sato ideals and holonomic \(D\)-modules
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 \textit{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 \textit{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.
scientific article; zbMATH DE number 6881234
scientific article