×

zbMATH — the first resource for mathematics

Bernstein polynomials and spectral numbers for linear free divisors. (English) Zbl 1221.34237
By a classical result due to B. Malgrange, one can relate the eigenvalues of the monodromy acting on the Milnor fibre of an isolated hypersurface singularity to the roots of the Bernstein polynomial associated to the defining equation. In this article the author studies Bernstein polynomials of the defining equation \(h\) of an arbitrary reductive linear free divisor \(D\) and is able to prove a similar result in this case. First, he recalls some results concerning linear free divisors given in [I. D. Gregorio, D. Mond and C. Sevenheck, Compos. Math. 145, 5, 1305–1350 (2009; Zbl 1238.32022)] and Bernstein polynomials associated to functions and \(\mathcal{D}\)-modules. In the main result of the article, he compares the roots of the Bernstein polynomial associated to \(h\) to the roots of the Bernstein polynomial of a restriction of the logarithmic extension of the family of Gauß-Manin systems, which can be associated to a generic section of \(D\). The introduction of a suitable logarithmic Brieskorn lattice associated to \(h\) leads to an analouge of Malgrange’s result. Finally, he proves a symmetry property concerning the spectrum of this lattice at infinity, which is also similar to the case of an isolated hypersurface singularity, computes the Bernstein polynomials for some examples of linear free divisors and conjectures another symmetry property for the spectrum of the logarithmic Brieskorn lattice.

MSC:
34M35 Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms
11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
Software:
SINGULAR
PDF BibTeX XML Cite
Full Text: DOI EuDML arXiv
References:
[1] Bernstein, I. N., Analytic continuation of generalized functions with respect to a parameter, Functional Analysis and Its Applications, 6, 4, 26-40, (1972) · Zbl 0282.46038
[2] Björk, Jan-Erik, Analytic \({\mathcal{D}}\)-modules and applications, 247, (1993), Kluwer Academic Publishers Group, Dordrecht · Zbl 0805.32001
[3] Buchweitz, Ragnar-Olaf; Mond, David; Lossen, Christoph; Pfister, Gerhard, Linear free divisors and quiver representations, Singularities and computer algebra, 324, 41-77, (2006), Cambridge Univ. Press, Cambridge · Zbl 1101.14013
[4] Douai, Antoine, Examples of limits of Frobenius (type) structures: The singularity case, (2008)
[5] Douai, Antoine; Mann, Etienne, The small quantum cohomology of a weighted projective space, a mirror \(\mathcal{D}\)-module and their classical limits, (2009) · Zbl 1273.14112
[6] Douai, Antoine; Sabbah, Claude, Gauss-Manin systems, Brieskorn lattices and Frobenius structures. I, Ann. Inst. Fourier (Grenoble), 53, 4, 1055-1116, (2003) · Zbl 1079.32016
[7] Douai, Antoine; Sabbah, Claude, Frobenius manifolds, Gauss-Manin systems, Brieskorn lattices and Frobenius structures. II, 1-18, (2004), Vieweg, Wiesbaden · Zbl 1079.32017
[8] Granger, Michel; Mond, David; Nieto, Alicia; Schulze, Mathias, Linear free divisors and the global logarithmic comparison theorem., Ann. Inst. Fourier (Grenoble), 59, 1, 811-850, (2009) · Zbl 1163.32014
[9] Granger, Michel; Schulze, Mathias, On the symmetry of b-functions of linear free divisors., (2008) · Zbl 1202.14046
[10] Gregorio, Ignacio d.; Mond, David; Sevenheck, Christian, Linear free divisors and Frobenius manifolds, Compositio Mathematica, 145, 5, 1305-1350, (2009) · Zbl 1238.32022
[11] Gregorio, Ignacio d.; Sevenheck, Christian, Good bases for some linear free divisors associated to quiver representations
[12] Greuel, G.-M.; Pfister, G.; Schönemann, H., Singular 3.1.0 — A computer algebra system for polynomial computations, (2009) · Zbl 1344.13002
[13] Gyoja, Akihiko, Theory of prehomogeneous vector spaces without regularity condition, Publ. Res. Inst. Math. Sci., 27, 6, 861-922, (1991) · Zbl 0773.14025
[14] Hertling, Claus; Sevenheck, Christian, Nilpotent orbits of a generalization of Hodge structures., J. Reine Angew. Math., 609, 23-80, (2007) · Zbl 1136.32011
[15] Hertling, Claus; Stahlke, Colin, Bernstein polynomial and tjurina number, Geom. Dedicata, 75, 2, 137-176, (1999) · Zbl 0955.32022
[16] Iritani, Hiroshi, An integral structure in quantum cohomology and mirror symmetry for toric orbifolds, Adv. Math., 22, 3, 1016-1079, (2009) · Zbl 1190.14054
[17] Kashiwara, Masaki \(, B\)-functions and holonomic systems. rationality of roots of \(B\)-functions, Invent. Math., 38, 1, 33-53, (197677) · Zbl 0354.35082
[18] Maisonobe, Philippe; Mebkhout, Zoghman; Maisonobe, Philippe; Narváez Macarro, Luis, Éléments de la théorie des systèmes différentiels géométriques, 8, Le théorème de comparaison pour LES cycles évanescents, 311-389, (2004), Soc. Math. France, Paris · Zbl 1105.14017
[19] Maisonobe, Philippe; Narváez Macarro, Luis, Éléments de la théorie des systèmes différentiels géométriques, 8, (2004), Société Mathématique de France, Paris
[20] Malgrange, Bernard; Chazarain, J., Fourier integral operators and partial differential equations (Colloq. Internat., Univ. Nice, Nice, 1974), Le polynôme de Bernstein d’une singularité isolée, 98-119. Lecture Notes in Math., Vol. 459, (1975), Springer, Berlin · Zbl 0308.32007
[21] Mebkhout, Zoghman; Maisonobe, Philippe; Narváez Macarro, Luis, Éléments de la théorie des systèmes différentiels , géométriques, 8, Le théorème de positivité, le théorème de comparaison et le théorème d’existence de Riemann, 165-310, (2004), Soc. Math. France, Paris · Zbl 1082.32006
[22] Roucairol, Céline, Irregularity of an analogue of the Gauss-Manin systems, Bull. Soc. Math. France, 134, 2, 269-286, (2006) · Zbl 1122.32019
[23] Roucairol, Céline, The irregularity of the direct image of some \(\mathcal{D}\)-modules, Publ. Res. Inst. Math. Sci., 42, 4, 923-932, (2006) · Zbl 1132.32005
[24] Roucairol, Céline, Formal structure of direct image of holonomic \(\mathcal{D}\)-modules of exponential type, Manuscripta Math., 124, 3, 299-318, (2007) · Zbl 1140.32021
[25] Saito, Kyoji, Theory of logarithmic differential forms and logarithmic vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 27, 2, 265-291, (1980) · Zbl 0496.32007
[26] Sato, M.; Kimura, T., A classification of irreducible prehomogeneous vector spaces and their relative invariants, Nagoya Math. J., 65, 1-155, (1977) · Zbl 0321.14030
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.