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
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