Singularities and holonomicity of binomial \(D\)-modules. (English) Zbl 1320.32014

Summary: We study binomial \(D\)-modules, which generalize \(A\)-hypergeometric systems. We determine explicitly their singular loci and provide three characterizations of their holonomicity. The first of these is an equivalence of holonomicity and \(L\)-holonomicity for these systems. The second refines the first by giving more detailed information about the \(L\)-characteristic variety of a non-holonomic binomial \(D\)-module. The final characterization states that a binomial \(D\)-module is holonomic if and only if its corresponding singular locus is proper.


32C38 Sheaves of differential operators and their modules, \(D\)-modules
14B05 Singularities in algebraic geometry
33C70 Other hypergeometric functions and integrals in several variables
14M25 Toric varieties, Newton polyhedra, Okounkov bodies


Full Text: DOI arXiv


[1] Adolphson, Alan, Hypergeometric functions and rings generated by monomials, Duke Math. J., 73, 269-290, (1994) · Zbl 0804.33013
[2] Berkesch Zamaere, Christine; Matusevich, Laura Felicia; Walther, Uli, Torus equivariant D-modules and hypergeometric systems, (2013), available at · Zbl 1320.32014
[3] Bernšteĭn, I. N., Analytic continuation of generalized functions with respect to a parameter, Funktsional. Anal. i Prilozhen., 6, 4, 26-40, (1972)
[4] Björk, Jan-Erik, Rings of differential operators, North-Holland Math. Library, vol. 21, (1979), North-Holland Publishing Co. Amsterdam-New York
[5] Fernández-Fernández, María-Cruz; Castro-Jiménez, Francisco-Jesús, On irregular binomial D-modules, Math. Z., 272, 3-4, 1321-1337, (2012) · Zbl 1262.32011
[6] Dickenstein, Alicia; Matusevich, Laura Felicia; Miller, Ezra, Combinatorics of binomial primary decomposition, Math. Z., 264, 4, 745-763, (2010) · Zbl 1190.13017
[7] Dickenstein, Alicia; Matusevich, Laura Felicia; Miller, Ezra, Binomial D-modules, Duke Math. J., 151, 3, 385-429, (2010) · Zbl 1205.13031
[8] Eisenbud, David; Sturmfels, Bernd, Binomial ideals, Duke Math. J., 84, 1, 1-45, (1996) · Zbl 0873.13021
[9] Gel’fand, I. M.; Gel’fand, S. I., Generalized hypergeometric equations, Dokl. Akad. Nauk SSSR, 288, 2, 279-283, (1986) · Zbl 0634.58030
[10] Gel’fand, I. M.; Graev, M. I.; Zelevinskiĭ, A. V., Holonomic systems of equations and series of hypergeometric type, Dokl. Akad. Nauk SSSR, 295, 1, 14-19, (1987)
[11] Gel’fand, I. M.; Zelevinskiĭ, A. V.; Kapranov, M. M., Equations of hypergeometric type and Newton polyhedra, Dokl. Akad. Nauk SSSR, Soviet Math. Dokl., 37, 3, 678-682, (1988), translation in · Zbl 0667.33010
[12] Gel’fand, I. M.; Zelevinskiĭ, A. V.; Kapranov, M. M., Hypergeometric functions and toric varieties, Funktsional. Anal. i Prilozhen., Funktsional. Anal. i Prilozhen., 27, 4, 91-26, (1993), correction in
[13] Gelfand, I. M.; Kapranov, M.; Zelevinsky, A. V., Generalized Euler integrals and A-hypergeometric functions, Adv. Math., 84, 2, 255-271, (1990) · Zbl 0741.33011
[14] Gelfand, I. M.; Kapranov, M.; Zelevinsky, A. V., Discriminants, resultants and multidimensional determinants, Math. Theory Appl., (1994), Birkhäuser Boston, Inc. Boston, MA · Zbl 0827.14036
[15] Kapranov, M. M., A characterization of A-discriminantal hypersurfaces in terms of the logarithmic Gauss map, Math. Ann., 290, 277-285, (1991) · Zbl 0714.14031
[16] Grayson, Daniel R.; Stillman, Michael E., Macaulay 2, a software system for research in algebraic geometry, available at · Zbl 0973.00017
[17] Matusevich, Laura Felicia; Miller, Ezra; Walther, Uli, Homological methods for hypergeometric families, J. Amer. Math. Soc., 18, 4, 919-941, (2005) · Zbl 1095.13033
[18] Passare, Mikael; Sadykov, Timur; Tsikh, August, Singularities of hypergeometric functions in several variables, Compos. Math., 141, 3, 787-810, (2005) · Zbl 1080.33012
[19] Saito, Mutsumi; Sturmfels, Bernd; Takayama, Nobuki, Gröbner deformations of hypergeometric differential equations, (2000), Springer-Verlag Berlin · Zbl 0946.13021
[20] Schulze, Mathias; Walther, Uli, Irregularity of hypergeometric systems via slopes along coordinate subspaces, Duke Math. J., 142, 3, 465-509, (2008) · Zbl 1144.13012
[21] Schulze, Mathias; Walther, Uli, Resonance equals reducibility for A-hypergeometric systems, Algebra Number Theory, 6, 3, 527-537, (2012) · Zbl 1251.13023
[22] Smith, Gregory G., Irreducible components of characteristic varieties, J. Pure Appl. Algebra, 165, 3, 291-306, (2001) · Zbl 1036.16025
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.