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On microlocal \(b\)-function. (English) Zbl 0810.32004
Let \(X\) be a complex manifold and let \(f\) be a germ of holomorphic functions at \(x \in X\). A \(b\)-function \(b_ f(s)\) is defined to be a monic generator of the ideal consisting of polynomials \(b(s)\) satisfying the relation \(b(s)f^ s = Pf^{s+1}\) for \(P \in {\mathcal D}_{X,x} [s]\), in \({\mathcal O}_{X,x} [f^{-1}] [s]f^ s\). Here, \({\mathcal O}_{X,x} [f^{-1}] [s]f^ s\) is the stalk of the sheaf of holomorphic functions on \(X\) at \(x\) and \({\mathcal D}_{X,x}\) is the stalk of the sheaf of differential operators on \(X\) at \(x\). The author introduces another “\(b\)-function” \(\widetilde b_ f (s)\) and shows that \((s+1) \widetilde b_ f (s) = b_ f(s)\). Let \(R_ f\) be the roots of \(\widetilde b_ f (s)\), \(\alpha_ f = \min R_ f\) and \(m_ \alpha (f)\) the multiplicity of \(\alpha \in R_ f\). Then we prove \(R_ f \subset [\alpha_ f, n - \alpha_ f]\) and \(m_ \alpha (f) \leq n - \alpha_ f - \alpha + 1(\leq n - 2 \alpha_ f + 1)\). The Thom-Sebastiani type theorem for \(b\)-functions is valid under a certain hypothesis.
Reviewer: M.Muro (Yanagido)

MSC:
32B05 Analytic algebras and generalizations, preparation theorems
32C38 Sheaves of differential operators and their modules, \(D\)-modules
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
32B10 Germs of analytic sets, local parametrization
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