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Contribution effective de la monodromie aux développements asymptotiques. (French) Zbl 0542.32003
Let \(f:({\mathbb{C}}^{n+1},0)\to({\mathbb{C}},0)\) be a germ of a non constant holomorphic function. Using the Bernstein-Sato polynomial of f it is easy to show that the distribution \(| f|^{2\lambda}\), defined near 0 for \(Re(\lambda)>0\) admits a meromorphic extension to \({\mathbb{C}}\). We prove that if the monodromy of f has a jordan block of size (k,k) for the eigenvalue \(e^{-i\pi r}\), the meromorphic extension of \(| f|^{2\lambda}\) has a pole of order at least k at \(-r/2-p\) for \(p\in {\mathbb{N}}\) big enough. Using Malgrange’s results linking monodromy and Bernstein-Sato polynomial, we can then deduce that each root of this polynomial produces a sequence of poles for the meromorphic extension of \(| f|^{2\lambda}.\)
Further results of this kind are obtained in the author’s following articles: ”Contribution du cup-produit de la fibre de Milnor aux pôles de \(| f|^{2\lambda}\)”, Ann. Inst. Fourier 34, No. 4, 75-107 (1984; Zbl 0525.32007); ”Monodromie et pôles du prolongement méromorphe de \(\int_{X}| f|^{2\lambda}\square\)”, Inst. E. Cartan, Univ. Nancy I (to appear).

MSC:
32S05 Local complex singularities
32B10 Germs of analytic sets, local parametrization
32C30 Integration on analytic sets and spaces, currents
32Sxx Complex singularities
32C35 Analytic sheaves and cohomology groups
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References:
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