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An algorithm of computing \(b\)-functions. (English) Zbl 0893.32009
Let \(K\) be a field of characteristic 0. Let \(A_n(K): =K[x_1, \dots, x_n] \langle \partial_1, \dots, \partial_n \rangle\) be the Weyl algebra and let \(\widehat {\mathcal D}_n (K):= K[[x_1, \dots, x_n]] \langle\partial_1, \dots, \partial_n \rangle\), where \(\partial_i =\partial/ \partial x_i\). For a polynomial \(f(x)\in K[x_1, \dots, x_n]\), the minimal monic polynomial \(b(s)\) in \(s\) satisfying \[ P(s,x,\partial) f(x)^{s+1} =b(s)f(x)^s \] for some \(P(s,x,\partial) \in\widehat {\mathcal D}_n (K)[s]\) (resp. \(A_n(K))\) is called the \(b\)-function associated with \(f\) and denoted by \(b_f (s)\) (resp. \(\widetilde b_f(s))\). This notion was introduced by I. N. Bernstein and M. Sato [M. Sato and T. Shintani, Ann. Math., II. Ser. 100, 131-170 (1974; Zbl 0309.10014)], independently.
The author gives an algorithm to compute the \(b\)-function \(b_f\) and the operator \(P(s,x, \partial)\) above. He uses the Gröbner bases introduced by the author for left ideals of \(A_{n+1}(K)\) related to some filtration. To compute such bases he employs the homogenization technique.

MSC:
32C38 Sheaves of differential operators and their modules, \(D\)-modules
13N10 Commutative rings of differential operators and their modules
16S32 Rings of differential operators (associative algebraic aspects)
35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
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