Slopes of hypergeometric systems of codimension one
The author computes the slopes for an irregular hypergeometric system of codimension one. More precisely, if \(\mathcal{D} = \mathcal{D}_n = \mathbb{C} \{x_1, \ldots, x_n\}/\langle \partial_1, \ldots, \partial_n\rangle\) and \(I\) an ideal in \(\mathcal{D}\), the slopes of the \(\mathcal{D}\)-module \(\mathcal{D}/I\) were defined by \textit{Y. Laurent} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 32, No. 1, 39--69 (1999; Zbl 0944.14007)]. Let \(A\) be an \(n \times (n + 1)\) integer matrix, \(\beta \in \mathbb{C}^n\), such that all the \(n \times n\) minors of \(A\) are not zero. The hypergeometric system of \textit{I. M. Gelfand, N. N. Kapranov} and \textit{A. V. Zelevinsky} [Discriminants, resultants, and multidimensional determinants. Mathematics, and multidimensional determinants. Boston, Birkhäuser (2000; Zbl 0827.14036)] is the system defined by the ideal \(H_A (\beta) = \langle I_A, A \theta^t - \beta \rangle\); the \(A_n\)-module \(\mathcal{H}_A (\beta) = A_n / H_A (\beta)\) where \(\theta = (\theta_1, \ldots, \theta_n)\), with \(\theta_i = x_i \partial_i\) and \(A_n\) is the Weyl algebra. It is known that the \(A_n\)-module \(\mathcal{H}_A(\beta)\) is a holomonic module [Gelfand, Kapranov and Zelevinski (loc. cit.), \textit{A. Adolphson}, Duke Math. J. 73, 269--290 (1994; Zbl 0804.33013)]. The author calculates the slopes of the \(\mathcal{D}\)-module \(\mathcal{H}_A(\beta)\), assumed to be irregular. With a special ordering of the coordinates it is shown that there is just one slope of \(\mathcal{H}_A(\beta)\) with respect to \(x_j = 0\), which is calculated explicitly. The case when one of the \(m \times n\) minors of \(A\) is zero is also considered, and by renaming the variables, this case is reduced to the previous one.
scientific article; zbMATH DE number 2109605
scientific article