zbMATH — the first resource for mathematics

Hypergeometric functions and toric varieties. (Hypergeometric functions and toral manifolds.) (English. Russian original) Zbl 0721.33006
Funct. Anal. Appl. 23, No. 2, 94-106 (1989); translation from Funkts. Anal. Prilozh. 23, No. 2, 12-26 (1989); correction Funct. Anal. Appl. 27, No. 4, 295 (1993); translation from Funkts. Anal. Prilozh. 27, No. 4, 91 (1993).
The paper studies the holonomy systems of linear differential equations connected with linear representations of complex tori. The characteristic manifold, the characteristic cycle of the system and, in particular, the number of independent solutions in a neighbourhood of a given point are expressed in terms of the volume of the corresponding Newton polyhedron. The basis of the space of solutions is expressed explicitly using the series of hypergeometric type. The paper contains also a number of examples which include many classical hypergeometric functions of one or several variables.
Reviewer: V.Müller

33C70 Other hypergeometric functions and integrals in several variables
34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
14Q99 Computational aspects in algebraic geometry
Full Text: DOI
[1] I. M. Gel’fand and S. I. Gel’fand, ”Generalized hypergeometric equations,” Dokl. Akad. Nauk SSSR,288, No. 2, 279-283 (1986).
[2] I. M. Gel’fand, M. I. Graev, and A. V. Zelevinskii, ”Holonomic systems of equations and series of hypergeometric type,” Dokl. Akad. Nauk SSSR,295, No. 1, 14-19 (1987).
[3] I. M. Gel’fand, A. V. Zelevinskii, and M. M. Kapranov, ”Equations of hypergeometric type and Newton polyhedra,” Dokl. Akad. Nauk SSSR,300, No. 3, 529-534 (1988).
[4] M. Kashiwara, System of Microdifferential Equations, Birkhäuser, Boston (1983). · Zbl 0521.58057
[5] V. A. Vasil’ev, I. M. Gel’fand, and A. V. Zelevinskii, ”General hypergeometric functions in the complex domain,” Funkts. Anal. Prilozhen.,21, No. 1, 23-38 (1987).
[6] V. I. Danilov, ”Geometry of toral manifolds,” Usp. Mat. Nauk,33, No. 2, 85-134 (1978).
[7] S. Kleiman, ”Real theory of singularities,” Usp. Mat. Nauk,35, No. 6, 69-148 (1980). · Zbl 0451.14020
[8] J.-L. Brylinsky, ”Transformations canoniques et transformation de Fourier,” Astérisque, No. 140 (1985).
[9] D. Mumford, Algebraic Geometry. Complex Projective Manifolds [Russian translation], Mir, Moscow (1979).
[10] D. N. Bernshtein, ”Number of roots of a system of equations,” Funkts. Anal. Prilozhen.,9, No. 3, 1-4 (1975). · Zbl 0395.60076 · doi:10.1007/BF01078167
[11] P. T. Bateman and A. Erdelyi, Higher Transcendental Functions [Russian translation], Vol. 1, Nauka, Moscow (1973).
[12] P. Gabriel and M. Zisman, Category of Fractions and Homotopy Theory [Russian translation], Mir, Moscow (1981).
[13] W. Miller, Symmetry and Separation of Variables [Russian translation], Mir, Moscow (1981).
[14] I. M. Gel’fand and V. V. Serganova, ”Combinatorial geometry and strata of a torus in compact homogeneous spaces,” Usp. Mat.,42, No. 2, 107-134 (1987). · Zbl 0629.14035
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.