Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

On the parametric behavior of $ A$-hypergeometric series


Authors: Christine Berkesch, Jens Forsgård and Laura Felicia Matusevich
Journal: Trans. Amer. Math. Soc. 370 (2018), 4089-4109
MSC (2010): Primary 33C70; Secondary 14M25, 32A10, 52B20
DOI: https://doi.org/10.1090/tran/7071
Published electronically: December 27, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We describe the parametric behavior of the series solutions of an $ A$-hypergeometric system. More precisely, we construct explicit stratifications of the parameter space such that, on each stratum, the series solutions of the system are holomorphic.


References [Enhancements On Off] (What's this?)

  • [Ado94] Alan Adolphson, Hypergeometric functions and rings generated by monomials, Duke Math. J. 73 (1994), no. 2, 269-290. MR 1262208, https://doi.org/10.1215/S0012-7094-94-07313-4
  • [AS14] Alan Adolphson and Steven Sperber, On logarithmic solutions of A-hypergeometric systems, arXiv:1402.5173, 2014.
  • [Ber11] Christine Berkesch, The rank of a hypergeometric system, Compos. Math. 147 (2011), no. 1, 284-318. MR 2771133, https://doi.org/10.1112/S0010437X10004811
  • [BFM14] Christine Berkesch, Jens Forsgård, and Laura Felicia Matusevich, Hypergeometric functions for projective toric curves, Adv. Math. 300 (2016), 835-867. MR 3534846, https://doi.org/10.1016/j.aim.2016.03.032
  • [BMW15] Christine Berkesch, Laura Felicia Matusevich, and Uli Walther, Singularities and holonomicity of binomial $ D$-modules, J. Algebra 439 (2015), 360-372. MR 3373376, https://doi.org/10.1016/j.jalgebra.2015.04.030
  • [DMM12] Alicia Dickenstein, Federico N. Martínez, and Laura Felicia Matusevich, Nilsson solutions for irregular $ A$-hypergeometric systems, Rev. Mat. Iberoam. 28 (2012), no. 3, 723-758. MR 2949617, https://doi.org/10.4171/RMI/689
  • [GG97] I. M. Gelfand and M. I. Graev, GG-functions and their connection with general hypergeometric functions, Uspekhi Mat. Nauk 52 (1997), no. 4(316), 3-48 (Russian); English transl., Russian Math. Surveys 52 (1997), no. 4, 639-684. MR 1480889, https://doi.org/10.1070/RM1997v052n04ABEH002055
  • [GG99] I. M. Gelfand and M. I. Graev, GG functions and their relations to general hypergeometric functions, Lett. Math. Phys. 50 (1999), no. 1, 1-27. MR 1751616, https://doi.org/10.1023/A:1007653012080
  • [GGZ87] I. M. Gelfand, M. I. Graev, and A. V. Zelevinskiĭ, Holonomic systems of equations and series of hypergeometric type, Dokl. Akad. Nauk SSSR 295 (1987), no. 1, 14-19 (Russian); English transl., Soviet Math. Dokl. 36 (1988), no. 1, 5-10. MR 902936
  • [GZK88] I. M. Gelfand, A. V. Zelevinskiĭ, and M. M. Kapranov, Equations of hypergeometric type and Newton polyhedra, Dokl. Akad. Nauk SSSR 300 (1988), no. 3, 529-534 (Russian); English transl., Soviet Math. Dokl. 37 (1988), no. 3, 678-682. MR 948812
  • [GKZ89] I. M. Gelfand, A. V. Zelevinskiĭ, and M. M. Kapranov, Hypergeometric functions and toric varieties, Funktsional. Anal. i Prilozhen. 23 (1989), no. 2, 12-26 (Russian); English transl., Funct. Anal. Appl. 23 (1989), no. 2, 94-106. MR 1011353, https://doi.org/10.1007/BF01078777
  • [Hot91] Ryoshi Hotta, Equivariant $ D$-modules, Proceedings of ICPAM Spring School in Wuhan (1991), available at arXiv:math/9805021.
  • [KW91] Heinz Kredel and Volker Weispfenning, Parametric Gröbner bases in rings of solvable type, Proc. IV International Conference on Computer Algebra in Physical Research, Joint Institute for Nuclear Research Dibna, USSR, May 1990, World Scientific, Singapore, 1991, pp. 236-244.
  • [MMW05] Laura Felicia Matusevich, Ezra Miller, and Uli Walther, Homological methods for hypergeometric families, J. Amer. Math. Soc. 18 (2005), no. 4, 919-941. MR 2163866, https://doi.org/10.1090/S0894-0347-05-00488-1
  • [NOT16] Katsusuke Nabeshima, Katsuyoshi Ohara, and Shinichi Tajima, Comprehensive Gröbner systems in rings of differential operators, holonomic $ D$-modules and b-functions, Proceedings of the 2016 ACM International Symposium on Symbolic and Algebraic Computation, ACM, New York, 2016, pp. 349-356. MR 3565734
  • [OT09] Katsuyoshi Ohara and Nobuki Takayama, Holonomic rank of $ \mathcal{A}$-hypergeometric differential-difference equations, J. Pure Appl. Algebra 213 (2009), no. 8, 1536-1544. MR 2517990, https://doi.org/10.1016/j.jpaa.2008.11.018
  • [Sai01] Mutsumi Saito, Isomorphism classes of $ A$-hypergeometric systems, Compositio Math. 128 (2001), no. 3, 323-338. MR 1858340, https://doi.org/10.1023/A:1011877515447
  • [Sai02] Mutsumi Saito, Logarithm-free $ A$-hypergeometric series, Duke Math. J. 115 (2002), no. 1, 53-73. MR 1932325, https://doi.org/10.1215/S0012-7094-02-11512-9
  • [SST00] Mutsumi Saito, Bernd Sturmfels, and Nobuki Takayama, Gröbner deformations of hypergeometric differential equations, Algorithms and Computation in Mathematics, vol. 6, Springer-Verlag, Berlin, 2000. MR 1734566
  • [SW08] Mathias Schulze and Uli Walther, Irregularity of hypergeometric systems via slopes along coordinate subspaces, Duke Math. J. 142 (2008), no. 3, 465-509. MR 2412045, https://doi.org/10.1215/00127094-2008-011
  • [W92] Volker Weispfenning, Comprehensive Gröbner bases, J. Symbolic Comput. 14 (1992), no. 1, 1-29. MR 1177987, https://doi.org/10.1016/0747-7171(92)90023-W

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 33C70, 14M25, 32A10, 52B20

Retrieve articles in all journals with MSC (2010): 33C70, 14M25, 32A10, 52B20


Additional Information

Christine Berkesch
Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
Email: cberkesc@math.umn.edu

Jens Forsgård
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
Email: jensf@math.tamu.edu

Laura Felicia Matusevich
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
Email: laura@math.tamu.edu

DOI: https://doi.org/10.1090/tran/7071
Received by editor(s): May 20, 2016
Received by editor(s) in revised form: September 13, 2016, and September 15, 2016
Published electronically: December 27, 2017
Additional Notes: The first author was partially supported by NSF Grant DMS 1440537
The second author was partially supported by the G. S. Magnusson Fund of the Royal Swedish Academy of Sciences
The third author was partially supported by NSF grants DMS 1001763 and DMS 1500832
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society