×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] A. Adolphson and S. Sperber, Exponential sums and Newton polyhedra: cohomology and estimates , Ann. of Math. (2) 130 (1989), no. 2, 367-406. JSTOR: · Zbl 0723.14017 · doi:10.2307/1971424 · links.jstor.org
[2] J.-E. Björk, Rings of Differential Operators , North-Holland Math. Library, vol. 21, North-Holland, Amsterdam, 1979. · Zbl 0499.13009
[3] B. Dwork, On the zeta function of a hypersurface, II , Ann. of Math. (2) 80 (1964), 227-299. JSTOR: · Zbl 0173.48601 · doi:10.1007/BF02684275 · numdam:PMIHES_1962__12__5_0 · eudml:103828
[4] B. Dwork, Generalized Hypergeometric Functions , Oxford Mathematical Monographs, Clarendon, New York, 1990. · Zbl 0747.33001
[5] B. Dwork and F. Loeser, Hypergeometric series , Japan. J. Math. (N.S.) 19 (1993), no. 1, 81-129. · Zbl 0796.12005
[6] I. M. Gelfand, A. V. Zelevinskii, and M. M. Kapranov, Hypergeometric functions and toral manifolds , Functional Anal. Appl. 23 (1989), 94-106, English translation. · Zbl 0721.33006 · doi:10.1007/BF01078777
[7] I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinsky, Generalized Euler integrals and \(A\)-hypergeometric functions , Adv. Math. 84 (1990), no. 2, 255-271. · Zbl 0741.33011 · doi:10.1016/0001-8708(90)90048-R
[8] M. Hochster and J. A. Eagon, Cohen-Macaulay rings, invariant theory, and the generic perfection of determinantal loci , Amer. J. Math. 93 (1971), 1020-1058. JSTOR: · Zbl 0244.13012 · doi:10.2307/2373744 · links.jstor.org
[9] M. Hochster, Rings of invariants of tori, Cohen-Macaulay rings generated by monomials, and polytopes , Ann. of Math. (2) 96 (1972), 318-337. JSTOR: · Zbl 0237.14019 · doi:10.2307/1970791 · links.jstor.org
[10] M. Hochster, 22 May 1992, Personal communication.
[11] A. G. Kouchnirenko, Polyèdres de Newton et nombres de Milnor , Invent. Math. 32 (1976), no. 1, 1-31. · Zbl 0328.32007 · doi:10.1007/BF01389769 · eudml:142365
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.