Algebraic \(A\)-hypergeometric functions
The author considers \(A\)-hypergeometric functions, where as usual \(A\subset \mathbb{Z}^r\), \(r\in \mathbb{N}\), consists of \(N\) vectors \(\mathbf{a}_1, \ldots, \mathbf{a}_N\) whose \(\mathbb{Z}\)-span equals \(\mathbb{Z}^r\) and for which there exists a linear form \(h\) on \(\mathbb{R}^r\) such that \(h(\mathbf{a}_i) = 1\), for all \(i = 1, \ldots, N\), and \(A\)-hypergeometric equations of the from \[ \square_{\mathbf{l}} \Phi = \prod_{l_i > 0} \partial_i^{l_i}\Phi - \prod_{l_i=0}\partial_i^{|l_i|}\Phi = 0, \] and \[ Z_i \Phi = \left(\sum_{j=1}^N a_{ij} v_j \partial_j - \alpha_i\right)\Phi = 0, \quad i = 1, \ldots, r. \] Here, \(\alpha\) is a vector of parameters in \(\mathbb{Q}^r\), \(a_{ij}\) are the components of vector \(\mathbf{a}_i\), and \(\mathbf{l} = (l_1, \ldots, l_N)\in L\subset \mathbb{Z}^N\) , where \(L\) denotes the lattice of relations such that \(\sum_{i=1}^N l_i \mathbf{a}_i = 0\). The above system of equations is denoted by \(H_A(\alpha)\). The author requires the system \(H_A(\alpha)\) to satisfy one additional condition, namely that the \(\mathbb{R}_{\geq 0}\)-span of \(A\) intersected with \(\mathbb{Z}^r\) equals the \(\mathbb{Z}_{\geq 0}\)-span of \(A\). This condition ensures that the dimension of \(H_A(\alpha)\) equals the volume of the \(A\)-polytope \(Q(A)\). A combinatorial criterion is formulated and proved to decide whether an \(A\)-hyperbolic system \(H_A(\alpha)\) has a full set of algebraic solutions or not. In the case of the one-dimensional setting of hypergeometric functions, this criterion generalizes the so-called interlacing condition.
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