Bernstein-Sato Polynomials and Spectral Numbers

Andréa G. Guimarães[1]; Abramo Hefez[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 (2007)

  • Volume: 57, Issue: 6, page 2031-2040
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Guimarães, Andréa G., and Hefez, Abramo. "Bernstein-Sato Polynomials and Spectral Numbers." Annales de l’institut Fourier 57.6 (2007): 2031-2040. <http://eudml.org/doc/10286>.

@article{Guimarães2007,
abstract = {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 $\mu $-constant family of germs of functions. An example will show that this set may sometimes consist of all common roots.},
affiliation = {Universidade Estadual do Rio de Janeiro IME R. São Francisco Xavier, 524, 6 o andar 20550-013 Rio de Janeiro, RJ (Brasil); Universidade Federal Fluminense Instituto de Matemática R. Mário Santos Braga, s/n 24020-140 Niterói, RJ (Brasil)},
author = {Guimarães, Andréa G., Hefez, Abramo},
journal = {Annales de l’institut Fourier},
keywords = {Bernstein polynomial; Spectral numbers; Gauss-Manin connection and Brieskorn lattice; spectral numbers},
language = {eng},
number = {6},
pages = {2031-2040},
publisher = {Association des Annales de l’institut Fourier},
title = {Bernstein-Sato Polynomials and Spectral Numbers},
url = {http://eudml.org/doc/10286},
volume = {57},
year = {2007},
}

TY - JOUR
AU - Guimarães, Andréa G.
AU - Hefez, Abramo
TI - Bernstein-Sato Polynomials and Spectral Numbers
JO - Annales de l’institut Fourier
PY - 2007
PB - Association des Annales de l’institut Fourier
VL - 57
IS - 6
SP - 2031
EP - 2040
AB - 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 $\mu $-constant family of germs of functions. An example will show that this set may sometimes consist of all common roots.
LA - eng
KW - Bernstein polynomial; Spectral numbers; Gauss-Manin connection and Brieskorn lattice; spectral numbers
UR - http://eudml.org/doc/10286
ER -

References

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  11. B. Malgrange, Le polynôme de Bernstein d’une singularité isolèe, Lecture Notes in Math. 459 (1975), 98-119, Springer-Verlag Zbl0308.32007
  12. F. Pham, Singularités des systèmes différentiels de Gauss-Manin, Progr. Math. 2 (1979), Birkhäuser Zbl0524.32015MR553954
  13. M. Saito, On the structure of Brieskorn lattice, Ann. Inst. Fourier (Grenoble) 39 (1989), 27-72 Zbl0644.32005MR1011977
  14. M. Saito, On b -function, spectrum and rational singularity, Math. Ann. 295 (1993), 51-74 Zbl0788.32025MR1198841
  15. J. H. M. Steenbrink, Mixed Hodge structure on the vanishing cohomology, Nordic Summer School Symposium in Math., Oslo (1976), 525-562 Zbl0373.14007MR485870
  16. A. N. Varchenko, The complex of a singularity does not change along the stratum μ -constant, Funct. Anal. Appl. 16 (1982), 1-9 Zbl0498.32010MR648803
  17. A. N. Varchenko, A. G. Khovanskii, Asymptotics of integrals over vanishing cycles and the Newton polyhedron, Soviet Math. Dokl. 32 (1985), 122-127 Zbl0595.32012
  18. T. Yano, On the theory of b -functions, RIMS, Kyoto Univ. 14 (1978), 11-202 Zbl0389.32005MR499664

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