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The $${\mathcal A}$$-hypergeometric system associated with a monomial curve. (English) Zbl 0952.33009
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 I. M. Gel’fand, A. V. Zelevinskij and 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 M. Saito, B. Sturmfels and N. Takayama [Gröbner deformations of hypergeometric differential equations. Springer-Verlag (2000; Zbl 0946.13021)].

##### MSC:
 33C70 Other hypergeometric functions and integrals in several variables 14D99 Families, fibrations in algebraic geometry 32G99 Deformations of analytic structures 33D15 Basic hypergeometric functions in one variable, $${}_r\phi_s$$
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