Dans ce travail, nous nous intéressons aux polynômes de Bernstein d’un diviseur linéairement libre réductif. Nous définissons un réseau de Brieskorn pour ces fonctions, qui sont des exemples de singularités non-isolées. Nous démontrons un théorème analogue au résultat de Malgrange qui relate les racines du polynôme de Bernstein aux valeurs propres du résidu de la saturation de ce réseau de Brieskorn.
We discuss Bernstein polynomials of reductive linear free divisors. We define suitable Brieskorn lattices for these non-isolated singularities, and show the analogue of Malgrange’s result relating the roots of the Bernstein polynomial to the residue eigenvalues on the saturation of these Brieskorn lattices.
@article{AIF_2011__61_1_379_0,
author = {Sevenheck, Christian},
title = {Bernstein polynomials and spectral numbers for linear free divisors},
journal = {Annales de l'Institut Fourier},
volume = {61},
year = {2011},
pages = {379-400},
doi = {10.5802/aif.2606},
zbl = {1221.34237},
mrnumber = {2828135},
language = {en},
url = {http://dml.mathdoc.fr/item/AIF_2011__61_1_379_0}
}
Sevenheck, Christian. Bernstein polynomials and spectral numbers for linear free divisors. Annales de l'Institut Fourier, Tome 61 (2011) pp. 379-400. doi : 10.5802/aif.2606. http://gdmltest.u-ga.fr/item/AIF_2011__61_1_379_0/
[1] Analytic continuation of generalized functions with respect to a parameter, Functional Analysis and Its Applications, Tome 6 (1972) no. 4, pp. 26-40 | MR 320735 | Zbl 0282.46038
[2] Analytic -modules and applications, Kluwer Academic Publishers Group, Dordrecht, Mathematics and its Applications, Tome 247 (1993) | MR 1232191
[3] Linear free divisors and quiver representations, Singularities and computer algebra, Cambridge Univ. Press, Cambridge (London Math. Soc. Lecture Note Ser.) Tome 324 (2006), pp. 41-77 (Papers from the conference held at the University of Kaiserslautern, Kaiserslautern, October 18–20, 2004) | MR 2228227 | Zbl 1101.14013
[4] Examples of limits of Frobenius (type) structures: The singularity case (2008) (Preprint math.AG/0806.2011)
[5] The small quantum cohomology of a weighted projective space, a mirror -module and their classical limits (2009) (Preprint math.AG/0909.4063)
[6] Gauss-Manin systems, Brieskorn lattices and Frobenius structures. I, Ann. Inst. Fourier (Grenoble), Tome 53 (2003) no. 4, pp. 1055-1116 | Article | Numdam | MR 2033510 | Zbl 1079.32016
[7] Gauss-Manin systems, Brieskorn lattices and Frobenius structures. II, Frobenius manifolds, Vieweg, Wiesbaden (Aspects Math., E36) (2004), pp. 1-18 | MR 2115764
[8] Linear free divisors and the global logarithmic comparison theorem., Ann. Inst. Fourier (Grenoble), Tome 59 (2009) no. 1, pp. 811-850 | Article | Numdam | MR 2521436 | Zbl 1163.32014
[9] On the symmetry of b-functions of linear free divisors. (2008) (Preprint math.AG/0807.0560)
[10] Linear free divisors and Frobenius manifolds, Compositio Mathematica, Tome 145 (2009) no. 5, pp. 1305-1350 | Article | MR 2551998 | Zbl pre05625831
[11] Good bases for some linear free divisors associated to quiver representations (work in progress)
[12] Singular 3.1.0 — A computer algebra system for polynomial computations (2009) (www.singular.uni-kl.de)
[13] Theory of prehomogeneous vector spaces without regularity condition, Publ. Res. Inst. Math. Sci., Tome 27 (1991) no. 6, pp. 861-922 | Article | MR 1145669 | Zbl 0773.14025
[14] Nilpotent orbits of a generalization of Hodge structures., J. Reine Angew. Math., Tome 609 (2007), pp. 23-80 | Article | MR 2350780 | Zbl 1136.32011
[15] Bernstein polynomial and Tjurina number, Geom. Dedicata, Tome 75 (1999) no. 2, pp. 137-176 | Article | MR 1686755 | Zbl 0955.32022
[16] An integral structure in quantum cohomology and mirror symmetry for toric orbifolds, Adv. Math., Tome 22 (2009) no. 3, pp. 1016-1079 | Article | MR 2553377 | Zbl 1190.14054
[17] -functions and holonomic systems. Rationality of roots of -functions, Invent. Math., Tome 38 (1976/77) no. 1, pp. 33-53 | Article | MR 430304 | Zbl 0354.35082
[18] Le théorème de comparaison pour les cycles évanescents, Éléments de la théorie des systèmes différentiels géométriques, Soc. Math. France, Paris (Sémin. Congr.) Tome 8 (2004), pp. 311-389 (Papers from the CIMPA Summer School held in Séville, September 2–13, 1996) | MR 2089892
[19] Éléments de la théorie des systèmes différentiels géométriques, Société Mathématique de France, Paris, Séminaires et Congrès [Seminars and Congresses], Tome 8 (2004) (Papers from the CIMPA Summer School held in Séville, September 2–13, 1996) | MR 2089892
[20] Le polynôme de Bernstein d’une singularité isolée, Fourier integral operators and partial differential equations (Colloq. Internat., Univ. Nice, Nice, 1974), Springer, Berlin (Lecture Notes in Mathematics, Vol. 459) (1975), p. 98-119. Lecture Notes in Math., Vol. 459 (Colloque International, réuni à l’Université de Nice, Nice, du 20 au 25 mai 1974) | Zbl 0308.32007
[21] Le théorème de positivité, le théorème de comparaison et le théorème d’existence de Riemann, Éléments de la théorie des systèmes différentiels , géométriques, Soc. Math. France, Paris (Sémin. Congr.) Tome 8 (2004), pp. 165-310 (Papers from the CIMPA Summer School held in Séville, September 2–13, 1996) | MR 2077649
[22] Irregularity of an analogue of the Gauss-Manin systems, Bull. Soc. Math. France, Tome 134 (2006) no. 2, pp. 269-286 | Numdam | MR 2233709 | Zbl 1122.32019
[23] The irregularity of the direct image of some -modules, Publ. Res. Inst. Math. Sci., Tome 42 (2006) no. 4, pp. 923-932 | Article | MR 2289081 | Zbl 1132.32005
[24] Formal structure of direct image of holonomic -modules of exponential type, Manuscripta Math., Tome 124 (2007) no. 3, pp. 299-318 | Article | MR 2350548 | Zbl 1140.32021
[25] Theory of logarithmic differential forms and logarithmic vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math., Tome 27 (1980) no. 2, pp. 265-291 | MR 586450 | Zbl 0496.32007
[26] A classification of irreducible prehomogeneous vector spaces and their relative invariants, Nagoya Math. J., Tome 65 (1977), pp. 1-155 | MR 430336 | Zbl 0321.14030