Abstract:
We study the solutions of irregular A-hypergeometric systems that are constructed from Gröbner degenerations with respect to generic positive weight vectors. These are formal logarithmic Puiseux series that belong to explicitly described Nilsson rings, and are therefore called (formal) Nilsson series. When the weight vector is a perturbation of (1, . . . , 1), these series converge and provide a basis for the (multivalued) holomorphic hypergeometric functions in a specific open subset of Cn. Our results are more explicit when the parameters are generic or when the solutions studied are logarithm-free. We also give an alternative proof of a result of Schulze and Walther that inhomogeneous A-hypergeometric systems have irregular singularities. © European Mathematical Society.
Registro:
Documento: |
Artículo
|
Título: | Nilsson solutions for irregular A-hypergeometric systems |
Autor: | Dickenstein, A.; Martínez, F.N.; Matusevich, L.F. |
Filiación: | Departamento de Matemática, FCEN, Universidad de Buenos Aires, C1428EGA Buenos Aires, Argentina Department of Mathematics, Texas A and M University, College Station, TX 77843-3368, United States
|
Palabras clave: | A-hypergeometric functions; Formal Nilsson series; Gröbner degenerations in the Weyl algebra; Irregular holonomic D-modules |
Año: | 2012
|
Volumen: | 28
|
Número: | 3
|
Página de inicio: | 723
|
Página de fin: | 758
|
DOI: |
http://dx.doi.org/10.4171/rmi/689 |
Título revista: | Revista Matematica Iberoamericana
|
Título revista abreviado: | Rev. Mat. Iberoam.
|
ISSN: | 02132230
|
Registro: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_02132230_v28_n3_p723_Dickenstein |
Referencias:
- Adolphson, A., Hypergeometric functions and rings generated by monomials (1994) Duke Math. J., 73 (2), pp. 269-290
- Assi, A., How to calculate the slopes of a D-module (1996) Compositio Mathematica, 104 (2), pp. 107-123
- Assi, A., Castro-Jiménez, F.J., Granger, J.M., The Gröbner fan of an An-module (2000) J. Pure Appl. Algebra, 150 (1), pp. 27-39
- Berkesch, C., (2010) Euler-Koszul Methods in Algebra and Geometry, , PhD thesis, Purdue University
- Björk, J.-E., (1979) Rings of Differential Operators, , North-Holland Mathematical Library, 21. North-Holland, Amsterdam-New York
- Castro-Jiménez, F.J., Fernández-Fernández, M.C., Gevrey solutions of the irregular hypergeometric system associated with an affine monomial curve (2011) Trans. Amer. Math. Soc., 363 (2), pp. 923-948
- Castro-Jimenez, F.J., Takayama, N., Singularities of the hypergeometric system associated with a monomial curve (2003) Transactions of the American Mathematical Society, 355 (9), pp. 3761-3775
- Cattani, E., Dickenstein, A., Rodríguez Villegas, F., The structure of bivariate rational hypergeometric functions (2011) Int. Math. Res. Not. IMRN, 2011 (11), pp. 2496-2533
- Cope, F.T., Formal solutions of irregular linear differential equations. Part i (1934) Amer. J. Math, 56 (1-4), pp. 411-437
- Fernández-Fernández, M.C., Irregular hypergeometric D-modules (2010) Adv. Math., 224 (5), pp. 1735-1764
- Gel'Fand, I.M., Graev, M.I., Zelevinsky, A.V., Holonomic systems of equations and series of hypergeometric type (1987) Dokl. Akad. Nauk SSSR, 295 (1), pp. 14-19
- (1988) Soviet Math. Dokl., 36 (1), pp. 5-10. , translation
- Gel'Fand, I.M., Kapranov, M.M., Zelevinsky, A.V., (1994) Discriminants, Resultants, and Multidimensional Determinants, , Mathematics: Theory & Applications. Birkhäuser, Boston, MA
- Gel'Fand, I.M., Zelevinsky, A.V., Kapranov, M.M., Hypergeometric functions and toric varieties (1989) Funktsional. Anal. i Prilozhen., 23 (2), pp. 12-26
- (1989) Funct. Anal. Appl., 23 (2), pp. 94-106. , translation
- Hartillo Hermoso, M.I., Slopes of hypergeometric systems of codimension one (2003) Proceedings of the International Conference on Algebraic Geometry and Singularities (Sevilla, 2001)". Rev. Mat. Iberoamericana, 19 (2), pp. 455-466
- Hartillo-Hermoso, M.I., Irregular hypergeometric systems associated with a singular monomial curve (2005) Transactions of the American Mathematical Society, 357 (11), pp. 4633-4646. , DOI 10.1090/S0002-9947-04-03614-1, PII S0002994704036141
- Hotta, R., Equivariant D-modules (1991) Proceedings of ICPAM Spring School in Wuhan, , arXiv:math/9805021v1
- Laurent, Y., Polygône de Newton et b-fonctions pour les modules microdifférentiels. (1987) Ann. Sci. École Norm. Sup. (4), 20 (3), pp. 391-441
- Laurent, Y., Mebkhout, Z., Pentes algébriques et pentes analytiques d'un D-module (1999) Ann. Sci. École Norm. Sup. (4), 32 (1), pp. 39-69
- Matusevich, L.F., Miller, E., Walther, U., Homological methods for hypergeometric families (2005) Journal of the American Mathematical Society, 18 (4), pp. 919-941. , DOI 10.1090/S0894-0347-05-00488-1, PII S0894034705004881
- Mebkhout, Z., Le théorème de comparaison entre cohomologies de de Rham d'une variété algébrique complexe et le théorème d'existence de Riemann. (1989) Inst. Hautes Études Sci. Publ. Math., 69, pp. 47-89
- Mora, T., Robbiano, L., The Gröbner fan of an ideal (1988) J. Symbolic Comput., 6 (2-3), pp. 183-208
- Ohara, K., Takayama, N., Holonomic rank of A -hypergeometric differentialdifference equations (2009) J. Pure Appl. Algebra, 213 (8), pp. 1536-1544
- Saito, M., Logarithm-free A-hypergeometric series (2002) Duke Math. J., 115 (1), pp. 53-73
- Saito, M., Sturmfels, B., Takayama, N., (2000) Gröbner Deformations of Hypergeometric Differential Equations, , Algorithms and Computation in Mathematics, 6. Springer-Verlag, Berlin
- Schulze, M., Walther, U., Irregularity of hypergeometric systems via slopes along coordinate subspaces (2008) Duke Math. J., 142 (3), pp. 465-509
- Sturmfels, B., (1996) Gröbner Bases and Convex Polytopes, , University Lecture Series, 8. American Mathematical Society, Providence, RI
- Sturmfels, B., Trung, N.V., Vogel, W., Bounds on degrees of projective schemes (1995) Math. Ann., 302 (3), pp. 417-432
Citas:
---------- APA ----------
Dickenstein, A., Martínez, F.N. & Matusevich, L.F.
(2012)
. Nilsson solutions for irregular A-hypergeometric systems. Revista Matematica Iberoamericana, 28(3), 723-758.
http://dx.doi.org/10.4171/rmi/689---------- CHICAGO ----------
Dickenstein, A., Martínez, F.N., Matusevich, L.F.
"Nilsson solutions for irregular A-hypergeometric systems"
. Revista Matematica Iberoamericana 28, no. 3
(2012) : 723-758.
http://dx.doi.org/10.4171/rmi/689---------- MLA ----------
Dickenstein, A., Martínez, F.N., Matusevich, L.F.
"Nilsson solutions for irregular A-hypergeometric systems"
. Revista Matematica Iberoamericana, vol. 28, no. 3, 2012, pp. 723-758.
http://dx.doi.org/10.4171/rmi/689---------- VANCOUVER ----------
Dickenstein, A., Martínez, F.N., Matusevich, L.F. Nilsson solutions for irregular A-hypergeometric systems. Rev. Mat. Iberoam. 2012;28(3):723-758.
http://dx.doi.org/10.4171/rmi/689