Bernstein-Sato Polynomials and Spectral Numbers

Guimarães, Andréa G.; Hefez, Abramo

doi:10.5802/aif.2322

Bernstein-Sato Polynomials and Spectral Numbers
[Polynômes de Bernstein-Sato et nombres spectraux]

Guimarães, Andréa G.¹ ; Hefez, Abramo ²

¹ Universidade Estadual do Rio de Janeiro IME R. São Francisco Xavier, 524, 6 o andar 20550-013 Rio de Janeiro, RJ (Brasil)
² Universidade Federal Fluminense Instituto de Matemática R. Mário Santos Braga, s/n 24020-140 Niterói, RJ (Brasil)

Annales de l'Institut Fourier, Tome 57 (2007) no. 6, pp. 2031-2040.

Résumé
Abstract

Dans cet article nous décrivons un ensemble de racines du polynôme de Bernstein-Sato associées à un germe de fonction analytique à plusieurs variables complexes, avec un point critique isolé à l’origine, qui peuvent être obtenues en connaissant seulement les nombres spectraux du germe. Ceci nous donnera aussi un ensemble de racines communes aux polynômes de Bernstein-Sato associées aux membres d’une famille à $μ$ -constant de germes de fonctions. Un exemple nous montrera que cet ensemble peut parfois donner toutes les racines communes.

In this paper we will describe a set of roots of the Bernstein-Sato polynomial associated to a germ of complex analytic function in several variables, with an isolated critical point at the origin, that may be obtained by only knowing the spectral numbers of the germ. This will also give us a set of common roots of the Bernstein-Sato polynomials associated to the members of a $μ$ -constant family of germs of functions. An example will show that this set may sometimes consist of all common roots.

MR Zbl

DOI : 10.5802/aif.2322

Classification : 32S40, 58K15, 58K65
Keywords: Bernstein polynomial, Spectral numbers, Gauss-Manin connection and Brieskorn lattice
Mot clés : Polynôme de Bernstein

Affiliations des auteurs :

Guimarães, Andréa G. ¹ ; Hefez, Abramo ²

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     title = {Bernstein-Sato {Polynomials} and {Spectral} {Numbers}},
     journal = {Annales de l'Institut Fourier},
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Guimarães, Andréa G.; Hefez, Abramo. Bernstein-Sato Polynomials and Spectral Numbers. Annales de l'Institut Fourier, Tome 57 (2007) no. 6, pp. 2031-2040. doi : 10.5802/aif.2322. http://archive.numdam.org/articles/10.5802/aif.2322/

Bibliographie
Cité par

[1] Bernstein, I. N. The analytic continuation of generalized functions with respect to a parameter, Funct. Anal. Appl., Volume 6 (1972), pp. 273-285 | DOI | MR | Zbl

[2] Björk, J.-E. Rings of differential operators, North-Holland Mathematical Library, 21, North-Holland Publishing Co., Amsterdam, 1979 | MR | Zbl

[3] Briançon, J.; Granger, M.; Maisonobe, Ph.; Miniconi, M. Algorithme de calcul du polynôme de Bernstein: cas non dégénéré, Ann. Inst. Fourier (Grenoble), Volume 39 (1989) no. 3, pp. 553-610 | DOI | Numdam | MR | Zbl

[4] Brieskorn, E. Die Monodromie der isolierten Singularitäten von Hyperflächen, Manuscripta Math., Volume 2 (1970), pp. 103-161 | DOI | MR | Zbl

[5] Cassous-Noguès, P. Racines de polynômes de Bernstein, Ann. Inst. Fourier (Grenoble), Volume 36 (1986) no. 4, pp. 1-30 | DOI | Numdam | MR | Zbl

[6] Cassous-Noguès, P. Comportement du polynômes de Bernstein lors d’une déformation à $μ$ -constant de $x^{a} + y^{b}$ avec $(a, b) = 1$ , Compositio Math., Volume 63 (1987), pp. 291-313 | Numdam | Zbl

[7] Hertling, C.; Stahlke, C. Bernstein polynomial and Tjurina number, Geom. Dedicata, Volume 75 (1999), pp. 137-176 | DOI | MR | Zbl

[8] Kashiwara, M. B-function and holonomic system, Invent. Math., Volume 38 (1976), pp. 33-53 | DOI | MR | Zbl

[9] Kulikov, Valentine S. Mixed Hodge structures and singularities, Cambridge Tracts in Mathematics, 132, Cambridge University Press, Cambridge, 1998 | MR | Zbl

[10] Malgrange, B. Intègrales asymptotiques et monodromie, Ann. Sci. École Norm. Sup., Volume 7 (1974), pp. 405-430 | Numdam | MR | Zbl

[11] Malgrange, B. Le polynôme de Bernstein d’une singularité isolèe, Lecture Notes in Math., Volume 459, Springer-Verlag, 1975, pp. 98-119 | Zbl

[12] Pham, F. Singularités des systèmes différentiels de Gauss-Manin, Progr. Math. 2, Birkhäuser, 1979 | MR | Zbl

[13] Saito, M. On the structure of Brieskorn lattice, Ann. Inst. Fourier (Grenoble), Volume 39 (1989) no. 1, pp. 27-72 | DOI | Numdam | MR | Zbl

[14] Saito, M. On $b$ -function, spectrum and rational singularity, Math. Ann., Volume 295 (1993), pp. 51-74 | DOI | MR | Zbl

[15] Steenbrink, J. H. M. Mixed Hodge structure on the vanishing cohomology, Nordic Summer School Symposium in Math., Oslo (1976), pp. 525-562 | MR | Zbl

[16] Varchenko, A. N. The complex of a singularity does not change along the stratum $μ$ -constant, Funct. Anal. Appl., Volume 16 (1982), pp. 1-9 | DOI | MR | Zbl

[17] Varchenko, A. N.; Khovanskii, A. G. Asymptotics of integrals over vanishing cycles and the Newton polyhedron, Soviet Math. Dokl., Volume 32 (1985), pp. 122-127 | Zbl

[18] Yano, T. On the theory of $b$ -functions, RIMS, Kyoto Univ., Volume 14 (1978), pp. 11-202 | MR | Zbl

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