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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

Citations:

Zbl 0525.32007
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References:

[1] D. BARLET , Développements asymptotiques des fonctions obtenues par intégration sur les fibres (Inv. Math., vol. 68, 1982 ). MR 84a:32021 | Zbl 0508.32003 · Zbl 0508.32003
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[3] M. KASHIWARA , B-functions and Holonomic Systems (Inv. Math., vol. 38-1, 1976 ). MR 55 #3309 | Zbl 0354.35082 · Zbl 0354.35082
[4] B. MALGRANGE , Polynôme de Bernstein-Sato , (Publication de l’IRMA de Strasbourg, vol. 28, RCP 25, 1980 ).
[5] B. MALGRANGE , Polynôme de Bernstein-Sato et cohomologie évanescente [Colloque d’Analyse et Topologie sur les espaces singuliers, Luminy, 1981 (à paraître)].
[6] J. MILNOR , Singular Points of Complex Hypersurfaces (Ann. of Math. Studies 61, Princeton, 1968 ). MR 39 #969 | Zbl 0184.48405 · Zbl 0184.48405
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