# zbMATH — the first resource for mathematics

Duality and monodromy reducibility of $$A$$-hypergeometric systems. (English) Zbl 1126.33006
The author studies two behaviors of a hypergeometric system. Let $$A$$ be an integer-entries $$d \times n$$ matrix, $$\beta \in \mathbb C^d$$ a parameter and $$H_A(\beta)$$ the $$A$$-hypergeometric system (or GKZ-system) [I. M. Gel’fand, I. M. Graev and A. V. Zelevinskii, Dokl. Akad. Nauk SSSR 295, 14–19 (1987; Zbl 0661.22005)]. First he studies whether $$H_A(\beta)$$ has irreducible monodromy, that is, $$H_A(\beta) \bigotimes_{\mathbb C[x]} \mathbb C(x)$$ is irreducible as a $$\mathbb C(x)$$-module. He proves that $$H_A(\beta)$$ has irreducible monodromy for almost all $$\beta \in \mathbb C^d$$, that is, except for a proper Zariski closed subset of $$\mathbb C^d$$. In the proof, the notion of rank-jumping plays a key role. It was introduced in [L. F. Matusevich, E. Miller and U. Walther, J. Am. Math. Soc. 18, No. 4, 919–941 (2005; Zbl 1095.13033)]. Next he studies the holonomic dual of $$H_A(\beta)$$. He proves that it is a GKZ-system for almost all $$\beta$$. He also studies the structure of the exceptional subset.

##### MSC:
 33C60 Hypergeometric integrals and functions defined by them ($$E$$, $$G$$, $$H$$ and $$I$$ functions) 33C80 Connections of hypergeometric functions with groups and algebras, and related topics
##### Keywords:
GKZ-system; holonomic D-module; rank-jumping
Full Text:
##### References:
 [1] Cattani E., D’Andrea C. and Dickenstein A. (1999). The A-hypergeometric system associated with a monomial curve. Duke Math. J. 99(2): 179–207 · Zbl 0952.33009 · doi:10.1215/S0012-7094-99-09908-8 [2] Dickenstein, A., Sadykov, T.: Bases in the solution space of the Mellin system. arXiv:math.AG/0609675, pp. 1–20 (2006) · Zbl 1159.33003 [3] Gel’fand I.M., Zelevinskiĭ A.V. and Kapranov M.M. (1989). Hypergeometric functions and toric varieties. Funktsional. Anal. i Prilozhen 23(2): 12–26 · Zbl 0737.35116 · doi:10.1007/BF01078569 [4] Gel’fand I.M., Zelevinskiĭ A.V. and Kapranov M.M. (1990). Generalized Euler integrals and A-hypergeometric functions. Adv. Math. 84: 255–271 · Zbl 0741.33011 · doi:10.1016/0001-8708(90)90048-R [5] Hotta, R.: Equivariant D-modules. arXiv:math.RT/9805021, pp. 1–36 (1998) [6] Matusevich L.F., Miller E. and Walther U. (2005). Homological methods for hypergeometric families. J. Amer. Math. Soc. 18: 919–941 · Zbl 1095.13033 · doi:10.1090/S0894-0347-05-00488-1 [7] Saito M. (2001). Isomorphism classes of A-hypergeometric systems. Compositio Math. 128(3): 323–338 · Zbl 1075.33009 · doi:10.1023/A:1011877515447 [8] Saito, M., Sturmfels, B., Takayama, N.: Gröbner deformations of hypergeometric differential equations, vol. 6 of Algorithms and Computation in Mathematics. Springer, Berlin Heidelberg New York (2000) · Zbl 0946.13021 [9] Schulze, M., Walther, U.: Regularity and slopes of A-hypergeometric systems. arXiv:math.AG/ 0608668, pp. 1–41 (2006) [10] Sturmfels, B.: Solving algebraic equations in terms of $$\mathcal A$$ -hypergeometric series. Discrete Math. 210(1–3), 171–181, 2000. Formal power series and algebraic combinatorics (Minneapolis) (1996) · Zbl 0963.33010 [11] Sturmfels, B., Takayama, N.: Gröbner bases and hypergeometric functions. In: Gröbner (ed.) Bases and Applications (Linz, 1998), vol. 251 of London Math. Soc. Lecture Note Ser., pp 246–258. Cambridge University Press, Cambridge (1998) · Zbl 0918.33004
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.