Bernstein-Sato Polynomials and Spectral Numbers  [ Polynômes de Bernstein-Sato et nombres spectraux ]
Annales de l'Institut Fourier, Tome 57 (2007) no. 6, pp. 2031-2040.

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.

DOI : https://doi.org/10.5802/aif.2322
Classification : 32S40,  58K15,  58K65
Mots clés : Polynôme de Bernstein
@article{AIF_2007__57_6_2031_0,
     author = {Guimar\~aes, Andr\'ea G. and Hefez, Abramo},
     title = {Bernstein-Sato Polynomials and Spectral Numbers},
     journal = {Annales de l'Institut Fourier},
     pages = {2031--2040},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {57},
     number = {6},
     year = {2007},
     doi = {10.5802/aif.2322},
     mrnumber = {2377895},
     zbl = {1130.32013},
     language = {en},
     url = {archive.numdam.org/item/AIF_2007__57_6_2031_0/}
}
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/item/AIF_2007__57_6_2031_0/

[1] Bernstein, I. N. The analytic continuation of generalized functions with respect to a parameter, Funct. Anal. Appl., Volume 6 (1972), pp. 273-285 | Article | MR 320735 | Zbl 0282.46038

[2] Björk, J.-E. Rings of differential operators, North-Holland Mathematical Library, Volume 21, North-Holland Publishing Co., Amsterdam, 1979 | MR 549189 | Zbl 0499.13009

[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 | Article | Numdam | MR 1030839 | Zbl 0675.32008

[4] Brieskorn, E. Die Monodromie der isolierten Singularitäten von Hyperflächen, Manuscripta Math., Volume 2 (1970), pp. 103-161 | Article | MR 267607 | Zbl 0186.26101

[5] Cassous-Noguès, P. Racines de polynômes de Bernstein, Ann. Inst. Fourier (Grenoble), Volume 36 (1986) no. 4, pp. 1-30 | Article | Numdam | MR 867914 | Zbl 0597.32004

[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 0624.32006

[7] Hertling, C.; Stahlke, C. Bernstein polynomial and Tjurina number, Geom. Dedicata, Volume 75 (1999), pp. 137-176 | Article | MR 1686755 | Zbl 0955.32022

[8] Kashiwara, M. B-function and holonomic system, Invent. Math., Volume 38 (1976), pp. 33-53 | Article | MR 430304 | Zbl 0354.35082

[9] Kulikov, Valentine S. Mixed Hodge structures and singularities, Cambridge Tracts in Mathematics, Volume 132, Cambridge University Press, Cambridge, 1998 | MR 1621831 | Zbl 0902.14005

[10] Malgrange, B. Intègrales asymptotiques et monodromie, Ann. Sci. École Norm. Sup., Volume 7 (1974), pp. 405-430 | Numdam | MR 372243 | Zbl 0305.32008

[11] Malgrange, B. Le polynôme de Bernstein d’une singularité isolèe, Lecture Notes in Math., Volume 459, Springer-Verlag, 1975, pp. 98-119 | Zbl 0308.32007

[12] Pham, F. Singularités des systèmes différentiels de Gauss-Manin, Progr. Math. 2, Birkhäuser, 1979 | MR 553954 | Zbl 0524.32015

[13] Saito, M. On the structure of Brieskorn lattice, Ann. Inst. Fourier (Grenoble), Volume 39 (1989) no. 1, pp. 27-72 | Article | Numdam | MR 1011977 | Zbl 0644.32005

[14] Saito, M. On b-function, spectrum and rational singularity, Math. Ann., Volume 295 (1993), pp. 51-74 | Article | MR 1198841 | Zbl 0788.32025

[15] Steenbrink, J. H. M. Mixed Hodge structure on the vanishing cohomology, Nordic Summer School Symposium in Math., Oslo (1976), pp. 525-562 | MR 485870 | Zbl 0373.14007

[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 | Article | MR 648803 | Zbl 0498.32010

[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 0595.32012

[18] Yano, T. On the theory of b-functions, RIMS, Kyoto Univ., Volume 14 (1978), pp. 11-202 | MR 499664 | Zbl 0389.32005