## 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

### Keywords:

$${\mathcal D}$$-module; $$b$$-function
Full Text:

### References:

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