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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”.

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
Full Text: DOI
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