an:01425239
Zbl 0952.33009
Cattani, Eduardo; D'Andrea, Carlos; Dickenstein, Alicia
The \({\mathcal A}\)-hypergeometric system associated with a monomial curve
EN
Duke Math. J. 99, No. 2, 179-207 (1999).
00064801
1999
j
33C70 14D99 32G99 33D15
GKZ-systems; \({\mathcal A}\)-hypergeometric function; \({\mathcal A}\)-hypergeometric system
Let \({\mathcal A}\) be a spanning subset of \(\mathbb{Z}^{n+1}\) consisting of \(r\) elements, and let \(\alpha\in \mathbb{C}^{n+1}\). In the late eighties Gel'fand, Kapranov and Zelevinskij associated with \({\mathcal A}\) and \(\alpha\) a holonomic system of differential equations in \(\mathbb{C}^r\), called the \({\mathcal A}\)-hypergeometric system with exponent (or parameter) \(\alpha\). Its solutions are called the \({\mathcal A}\)-hypergeometric functions with parameter \(\alpha\) [see \textit{I. M. Gel'fand}, \textit{A. V. Zelevinskij} and \textit{M. M. Kapranov}, Funct. Anal. Appl. 23, No. 2, 94-106 (1989; Zbl 0721.33006); Adv. Math. 84, No. 2, 255-271 (1990; Zbl 0741.33011)]. In the literature \({\mathcal A}\)-hypergeometric systems are also called GKZ-systems. The paper under review studies the case of \({\mathcal A}\)-hypergeometric systems associated with monomial curves, which corresponds to the case \(n=1\). All rational \({\mathcal A}\)-hypergeometric functions with parameter \(\alpha\) are shown to be Laurent polynomials. This property is proven by counterexample not to be true in the general case \(n>1\). The rational \({\mathcal A}\)-hypergeometric functions with parameter \(\alpha\in \mathbb{Z}^2\) are shown to span a space of dimension at most 2. The value 2 is attained if and only if the monomial curve is not arithmetically Cohen-Macaulay. For all values of \(\alpha\), the holonomic rank \(r(\alpha)\) of the system is proven to satisfy the inequalities \(d\leq r(\alpha)\leq d+1\). Moreover \(r(\alpha)= d+1\) exactly for all \(\alpha\in \mathbb{Z}^2\) for which the space of rational solutions has dimension 2. The inequalities for the holonomic rank have also been obtained using different methods by \textit{M. Saito}, \textit{B. Sturmfels} and \textit{N. Takayama} [Gr??bner deformations of hypergeometric differential equations. Springer-Verlag (2000; Zbl 0946.13021)].
A.Pasquale (Clausthal-Zellerfeld)
Zbl 0946.13021; Zbl 0721.33006; Zbl 0741.33011