Bernstein-Sato's polynomial for several analytic functions
The main theorem of this paper is an extension of the famous theorem of Kashiwara: any root of \(b\)-function of a polynomial of one single variable is a negative rational number, to the case of polynomials of several variables. Let \(X\) be a complex manifold, \({\mathcal D}_ X\) (resp. \({\mathcal O}_ X)\) the sheaf of differential operators (resp. holomorphic functions), \(x_ 0 \in X\), \(f_ 1, \dots,f_ l \in {\mathcal O}_{X,x_ 0}\), \(\zeta_ 1,\dots,\zeta_ l\) independent complex variables, and \({\mathcal D}_ X [\zeta_ 1,\dots, \zeta_ l] = {\mathcal D}_ X \bigotimes_ \mathbb{C} \mathbb{C} [\zeta_ 1,\dots,\zeta_ l]\). Then we have the following theorem: (1) For any \(\mu = (\mu_ 1,\dots, \mu_ l) \in \mathbb{Z}_{>0}\), there exist a differential operator \(P_ \mu (\zeta) \in {\mathcal D}_{X,x_ 0} [\zeta_ 1,\dots, \zeta_ l]\) and a nonzero polynomial \(b_ \mu (\zeta) \in \mathbb{C} [\zeta_ 1, \dots, \zeta_ l]\) such that \(P_ \mu (\zeta) f_ 1^{\zeta_ 1 + \mu_ 1} \dots f_ l^{\zeta_ l + \mu_ l} = b_ \mu (\zeta) f_ 1^{\zeta_ 1} \dots f_ l^{\zeta_ l}\). (2) Moreover, we can take \(b_ \mu (\zeta)\) so that \(b_ \mu (\zeta) = \prod_ i (\alpha_{i1} \zeta_ 1 + \cdots + \alpha_{il} \zeta_ l+a_ i)\), where \(\alpha_{ij} \in \mathbb{Z}_{>0}\), \(GCD(\alpha_{i1}, \dots,\alpha_{il})=1\) and \(a_ i \in \mathbb{Q}_{>0}\) for any \(i\).
scientific article; zbMATH DE number 558898