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Transactions of the American Mathematical Society

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Singularities of the hypergeometric system associated with a monomial curve


Authors: Francisco Jesús Castro-Jiménez and Nobuki Takayama
Journal: Trans. Amer. Math. Soc. 355 (2003), 3761-3775
MSC (2000): Primary 32C38, 13N10; Secondary 13P10, 14F10, 14M25
DOI: https://doi.org/10.1090/S0002-9947-03-03300-2
Published electronically: May 29, 2003
MathSciNet review: 1990172
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Abstract: We compute, using $\mathcal{D}$-module restrictions, the slopes of the irregular hypergeometric system associated with a monomial curve. We also study rational solutions and reducibility of such systems.


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  • 1. Adolphson, A., Hypergeometric functions and rings generated by monomials, Duke Mathematical Journal 73 (1994), 269-290. MR 96c:33020
  • 2. Assi, A., Castro-Jiménez, F. J., and Granger, J. M., How to calculate the slopes of a ${{\mathcal D} }$-module, Compositio Math. 104 (1996), no. 2, 107-123. MR 98i:32010
  • 3. Assi, A., Castro-Jiménez, F. J., and Granger, M.,
    The Gröbner fan of an ${A}\sb n$-module, J. Pure Appl. Algebra 150 (2000), 1, 27-39. MR 2001j:16036
  • 4. Beukers, F., Brownawell, W. D., and Heckman, G., Siegel normality, Ann. of Math. (2) 127 (1988), no. 2, 279-308. MR 90e:11106
  • 5. Cattani, E., D'Andrea, C., and Dickenstein, A., The ${\mathcal A}$-hypergeometric system associated with a monomial curve, Duke Mathematical J. 99 (1999), 179-207. MR 2001f:33018
  • 6. Gel'fand, I. M., Zelevinski{\u{\i}}\kern.15em, A. V., and Kapranov, M. M., Hypergeometric functions and toric varieties, Funktsional. Anal. i Prilozhen., 23 (1989), 2, 12-26; English transl., Funct. Anal. Appl. 23 (1989), no. 2, 94-106. MR 90m:22025
  • 7. Grayson, D. and Stillman, M., Macaulay2, a software system for research in algebraic geometry, available at http://www.math.uiuc.edu/Macaulay2.
  • 8. Hotta, R., Equivariant $D$-modules. Preprint math.RT/9805021.
  • 9. Laurent, Y. Théorie de la deuxième microlocalisation dans le domaine complexe, Progress in Math. 53, Birkhäuser, 1985. MR 86k:58113
  • 10. Laurent, Y., Polygone de Newton et $b$-fonctions pour les modules microdifférentiels, Ann. Sci. École Norm. Sup. (4) 20 (1987), no. 3, 391-441. MR 89k:58282
  • 11. Laurent, Y. and Mebkhout, Z., Pentes algébriques et pentes analytiques d'un ${{\mathcal D} }$-module, Ann. Sci. École Norm. Sup. (4) 32 (1999), no. 1, 39-69. MR 2001b:32015
  • 12. Laurent, Y. and Mebkhout, Z., Image inverse d'un ${{\mathcal D} }$-module et polygone de Newton, Compositio Math. 131 (2002), no. 1, 97-119.
  • 13. Leykin, A. and Tsai, H., D-module package for Macaulay 2. http://www.math.cornell. edu/htsai
  • 14. Mebkhout, Z., Le théorème de positivité de l'irrégularité pour les ${{\mathcal D} }_X$-modules, The Grothendieck Festschrift, Vol. III, 83-132, Progr. Math., 88, Birkhäuser, Boston, MA, 1990. MR 92j:32031
  • 15. Oaku, T., Algorithms for $b$-functions, restriction and algebraic local cohomology groups of $D$-modules, Advances in Applied Mathematics 19 (1997), 61-105. MR 98d:14031
  • 16. Oaku, T., Takayama, N., and Walther, U., A localization algorithm for $D$-modules, Journal of Symbolic Computation 29 (2000), 721-728. MR 2001g:13056
  • 17. Saito, M., Sturmfels, B., and Takayama, N., Gröbner deformations of hypergeometric differential equations, Algorithms and Computation in Mathematics, 6. Springer-Verlag, Berlin, 2000. MR 2001i:13036
  • 18. Takayama, N., Kan: A system for computation in algebraic analysis, 1991 version 1, 1994 version 2, the latest version is 3.000726. Source code available for Unix computers. Download from http://www.openxm.org

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Additional Information

Francisco Jesús Castro-Jiménez
Affiliation: Universidad de Sevilla, Depto. de Álgebra, Apdo. 1160, E-41080 Sevilla, Spain
Email: castro@us.es

Nobuki Takayama
Affiliation: Department of Mathematics, Faculty of Science, Kobe University, 1-1, Rokkodai, Nada-ku, Kobe 657-8501, Japan
Email: takayama@math.kobe-u.ac.jp

DOI: https://doi.org/10.1090/S0002-9947-03-03300-2
Keywords: Algebraic geometry, $\mathcal{D}$-modules, toric varieties, hypergeometric systems
Received by editor(s): November 15, 2002
Published electronically: May 29, 2003
Additional Notes: The first author was partially supported by BFM-2001-3164, FQM-218 and FQM-813
Article copyright: © Copyright 2003 American Mathematical Society

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