Geodesic
Maximally reducible monodromy of bivariate hypergeometric systems
Izvestiya. Mathematics , Tome 80 (2016) no. 1, pp. 221-262.

Voir la notice de l'article provenant de la source Math-Net.Ru

We investigate the branching of solutions of holonomic bivariate Horn-type hypergeometric systems. Special attention is paid to invariant subspaces of Puiseux polynomial solutions. We mainly study Horn systems defined by simplicial configurations and Horn systems whose Ore–Sato polygons are either zonotopes or Minkowski sums of a triangle and segments proportional to its sides. We prove a necessary and sufficient condition for the monodromy representation to be maximally reducible, that is, for the space of holomorphic solutions to split into a direct sum of one-dimensional invariant subspaces.
Keywords: hypergeometric system of equations, monodromy representation, monodromy reducibility, intertwining operator. 
@article{IM2_2016_80_1_a7,
     author = {T. M. Sadykov and S. Tanab\'e},
     title = {Maximally reducible monodromy of bivariate hypergeometric systems},
     journal = {Izvestiya. Mathematics },
     pages = {221--262},
     publisher = {mathdoc},
     volume = {80},
     number = {1},
     year = {2016},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2016_80_1_a7/}
}
TY  - JOUR
AU  - T. M. Sadykov
AU  - S. Tanabé
TI  - Maximally reducible monodromy of bivariate hypergeometric systems
JO  - Izvestiya. Mathematics 
PY  - 2016
SP  - 221
EP  - 262
VL  - 80
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_2016_80_1_a7/
LA  - en
ID  - IM2_2016_80_1_a7
ER  - 
%0 Journal Article
%A T. M. Sadykov
%A S. Tanabé
%T Maximally reducible monodromy of bivariate hypergeometric systems
%J Izvestiya. Mathematics 
%D 2016
%P 221-262
%V 80
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_2016_80_1_a7/
%G en
%F IM2_2016_80_1_a7
T. M. Sadykov; S. Tanabé. Maximally reducible monodromy of bivariate hypergeometric systems. Izvestiya. Mathematics , Tome 80 (2016) no. 1, pp. 221-262. http://geodesic.mathdoc.fr/item/IM2_2016_80_1_a7/

[1] T. M. Sadykov, “Hypergeometric systems of equations with maximally reducible monodromy”, Dokl. Math., 78:3 (2008), 880–882 | DOI | MR | Zbl

[2] T. M. Sadykov, “Hypergeometric systems with polynomial bases”, Zhurn. Sib. fed. un-ta. Matem. i fiz., 1:1 (2008), 25–32; http://elib.sfu-kras.ru/handle/2311/615.

[3] F. Beukers, G. Heckman, “Monodromy for the hypergeometric function ${}_nF_{n-1}$”, Invent. Math., 95:2 (1989), 325–354 | DOI | MR | Zbl

[4] F. Beukers, “Algebraic $A$-hypergeometric functions”, Invent. Math., 180:3 (2010), 589–610 | DOI | MR | Zbl

[5] F. Beukers, Monodromy of A-hypergeometric functions, 2013, 29 pp., arXiv: 1101.0493v2

[6] E. Cattani, A. Dickenstein, B. Sturmfels, “Rational hypergeometric functions”, Compos. Math., 128:2 (2001), 217–240 | DOI | MR | Zbl

[7] I. M. Gel'fand, A. V. Zelevinskii, M. M. Kapranov, “Hypergeometric functions and toral manifolds”, Funct. Anal. Appl., 23:2 (1989), 94–106 | DOI | MR | Zbl

[8] A. Dickenstein, L. F. Matusevich, T. Sadykov, “Bivariate hypergeometric D-modules”, Adv. Math., 196:1 (2005), 78–123 | DOI | MR | Zbl

[9] M. Passare, T. Sadykov, A. Tsikh, “Singularities of hypergeometric functions in several variables”, Compos. Math., 141:3 (2005), 787–810 | DOI | MR | Zbl

[10] I. M. Gel'fand, M. I. Graev, V. S. Retakh, “General hypergeometric systems of equations and series of hypergeometric type”, Russian Math. Surveys, 47:4 (1992), 1–88 | DOI | MR | Zbl

[11] M. Sato, “Theory of prehomogeneous vector spaces (algebraic part)–the English translation of Sato's lecture from Shintani's note”, Notes by T. Shintani, transl. from the Japanese by M. Muro, Nagoya Math. J., 120 (1990), 1–34 | MR | Zbl

[12] T. M. Sadykov, “On a multidimensional system of hypergeometric differential equations”, Sib. Math. J., 39:5 (1998), 986–997 | DOI | MR | Zbl

[13] A. Dickenstein, L. F. Matusevich, E. Miller, “Binomial $D$-modules”, Duke Math. J., 151:3 (2010), 385–429 | DOI | MR | Zbl

[14] R. M. Hain, R. MacPherson, “Higher logarithms”, Illinois J. Math., 34:2 (1990), 392–475 | MR | Zbl

[15] T. M. Sadykov, “On the Horn system of partial differential equations and series of hypergeometric type”, Math. Scand., 91:1 (2002), 127–149 | MR | Zbl

[16] V. P. Palamodov, Linear differential operators with constant coefficients, Grundlehren Math. Wiss., 168, Springer-Verlag, New York–Berlin, 1970, viii+444 pp. | MR | MR | Zbl | Zbl

[17] T. Sadykov, “The Hadamard product of hypergeometric series”, Bull. Sci. Math., 126:1 (2002), 31–43 | DOI | MR | Zbl

[18] M. Passare, A. Tsikh, “Amoebas: their spines and their contours”, Idempotent mathematics and mathematical physics, Contemp. Math., 377, Amer. Math. Soc., Providence, RI, 2005, 275–288 | DOI | MR | Zbl

[19] S. Tanabé, “On Horn–Kapranov uniformisation of the discriminantal loci”, Singularities in geometry and topology 2004, Adv. Stud. Pure Math., 46, Math. Soc. Japan, Tokyo, 2007, 223–249 | MR | Zbl

[20] B. Ya. Kazarnovskii, “c-fans and Newton polyhedra of algebraic varieties”, Izv. Math., 67:3 (2003), 439–460 | DOI | DOI | MR | Zbl