Logarithm-free \(A\)-hypergeometric series. (English) Zbl 1031.33011

The author studies the solutions of an \(A\)-hypergeometric system cf. I. M. Gel’fand, A. V. Zelevinskij and M. M. Kapranov [Sov. Math. Dokl. 37, No. 3, 678-682 (1988; Zbl 0667.33010)]. He obtains an explicit formula for the vector space of logarithm-free \(A\)-hypergeometric series in functions of the volumes of some facets. It is proved that there exists a fundamental system of solutions in the case where \(\text{conv}(A)\) is a simplex and, in this case, there is given a rank formula for the system. The author also obtains a classification of \(A\)-hypergeometric systems as analytic \({\mathcal D}\)-modules.


33C70 Other hypergeometric functions and integrals in several variables
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
13N10 Commutative rings of differential operators and their modules
16S32 Rings of differential operators (associative algebraic aspects)


Zbl 0667.33010
Full Text: DOI arXiv


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