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Rational hypergeometric functions. (English) Zbl 0990.33013
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.

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