Rational hypergeometric functions. (English) Zbl 0990.33013

Cattani, Eduardo; Dickenstein, Alicia; Sturmfels, Bernd

Compos. Math. 128, No. 2, 217-240 (2001).

Let \(A = (a_{ij})\) be an integer \(d \times s\)-matrix and \(\beta\) a vector in \(\mathbb{C}^d\). Define \(H_A(\beta)\) to be the left ideal in the Weyl algebra \(\mathbb{C}\langle x_1,\ldots,x_s,\partial_1,\ldots,\partial_s\rangle\) that is generated by the toric operators \(\partial^u-\partial^v\), \(u,v \in \mathbb{N}^s\) with \(A\cdot u = A\cdot v\), and the Euler operators \(\sum_{j=1}^s a_{ij}x_j\partial_j - \beta_j\). Following Gel’fand, Kapranov, and Zelevinsky, a holomorphic function on an open subset of \(\mathbb{C}^s\) is \(A\)-hypergeometric of degree \(\beta\) if it is annihilated by \(H_A(\beta)\). The authors conjecture that the denominator of any rational \(A\)-hypergeometric function is a product of resultants, that is, a product of special discriminants arising from Cayley configurations. They prove this conjecture for toric hypersurfaces and for toric varieties of dimension at most three. They apply toric residues to show that every toric resultant appears in the denominator of some rational hypergeometric function.

Reviewer: David M.Bressoud (Saint Paul)

MSC:

33C70	Other hypergeometric functions and integrals in several variables
14M25	Toric varieties, Newton polyhedra, Okounkov bodies

Keywords:

hypergeometric functions; toric varieties