Contribution effective dans le réel. (French) Zbl 0581.32016

Barlet, D.

Compos. Math. 56, 351-359 (1985).

Let \(f_{{\mathbb{R}}}: ({\mathbb{R}}^{n+1},0)\to ({\mathbb{R}}^+,0)\) be a germ of a real analytic function and \(f: ({\mathbb{C}}^{n+1},0)\to ({\mathbb{C}},0)\) its complexification. The author establishes a sufficient condition in order that the meromorphic continuation of the distribution \(\int_{X_{{\mathbb{R}}}}f^{\lambda}\square\) have effectively a pole of order at least k at a point of the form \(-u-\nu,\) where \(u\in {\mathbb{Q}}\cap [0,1]\) and \(\nu\in {\mathbb{N}}\). This theorem was suggested by its complex analogous. By means of some results of L. Łojasiewicz [Ann. Scuola Norm. Sup. Pisa, Sci. Fis. Mat., III. Ser. 18(1964), 449-474 (1965; Zbl 0128.171)], J. B. Poly [Bull. Soc. Math. France, Suppl., Mem. 38, 35-43 (1974; Zbl 0302.32010)] and M. Herrera in his joint paper with T. Bloom [Inventiones Math. 7, 275-296 (1968; Zbl 0175.373)] on semi-analytic sets, the author obtains, in an appendix, a strengthening of his theorem mentioned above.

Reviewer: P.Caraman

MSC:

32D15	Continuation of analytic objects in several complex variables
32B20	Semi-analytic sets, subanalytic sets, and generalizations
32A20	Meromorphic functions of several complex variables
32B10	Germs of analytic sets, local parametrization

Keywords:

germs of analytic functions; Milnor fiber; meromorphic continuation; distribution

Citations:

Zbl 0128.171; Zbl 0302.32010; Zbl 0175.373

References:

[1]	Contribution effective de la monodromie aux développments asymptotiques . Preprint Institut Elie Cartan Nancy janvier 83. A paraître aux Ann. Sc. ENS (84).
[2]	Contribution du cup-produit de la fibre de Milnor aux pôles de |f| 2\lambda . Preprint Institut Elie Cartan Nancy sept. 83 - à paraître aux Ann. de l’Institut Fourier. · Zbl 0525.32007
[3]	Monodromie et poles de \int X|f |2\lambda \square , preprint Institut Elie Cartan, Nancy, avril 1984.
[4]	T. Bloom et M. Herrera : De Rham cohomology of an analytic space , Invent. Math. 7 (1969) 275-296. · Zbl 0175.37301 · doi:10.1007/BF01425536
[5]	S. Łojasiewicz : Triangulation of semi-analytic sets , Ann. Sc. Norm. Sup. Pisa, Sc. Fis. Mat. ser. 3 vol 18 fasc. 4 (1964) 449-474. · Zbl 0128.17101
[6]	J. Milnor : Singular points of complex hypersurfaces , Ann. of Math. Studies 61, Princeton University Press (1968). · Zbl 0184.48405 · doi:10.1515/9781400881819
[7]	J.B. Poly : Sur l’homologie des courants à support dans un ensemble semi-analytique , Bull. Soc. Math. France, Memoire n^\circ 38 (1974) p. 35-43. · Zbl 0302.32010