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Hypergeometric functions and rings generated by monomials. (English) Zbl 0804.33013
The author studies generalized hypergeometric functions associated with a set of integral vectors $$d^{(1)}, \dots, d^{(N)}\in \mathbb{Z}^ n$$ with span $$\mathbb{R}^ n$$. Such functions were studied by I. M. Gel’fand, A. V. Zelevinskij and M. M. Kapranov [Funct. Anal. Appl. 23, No. 2, 94-106 (1989); translation from Funkts. Anal. Prilozh. 23, No. 2, 12-26 (1989; Zbl 0787.33012); see also Sov. Math., Dokl. 37, No. 3, 678- 682 (1988); translation from Dokl. Akad. Nauk SSSR 300, No. 3, 529-534 (1988; Zbl 0667.33010)] under the assumption that the vectors $$d^{(j)}= (d_ i^{(j)})$$ satisfy the relation $$\sum_{i=1}^ n b_ i d_ i^{(j)} =1$$ for some integers $$b_ i$$. In the present paper, the last assumption was removed, and there is associated a holonomic system of partial differential equations in $$N$$ variables. This includes the generalization of confluent hypergeometric functions. The author studies the characteristic variety and $${\mathcal D}$$-module structure of this system and proves that its rank equals a simple multiple of the volume of convex hull of the $$d^{(j)}$$. This last statement is proved under the assumption either that certain rings are Cohen-Macaulay, or that the parameters are “non-resonant”.

##### MSC:
 33C70 Other hypergeometric functions and integrals in several variables 32S40 Monodromy; relations with differential equations and $$D$$-modules (complex-analytic aspects) 13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) 14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials 32C38 Sheaves of differential operators and their modules, $$D$$-modules 13C14 Cohen-Macaulay modules
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