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Singularities of the hypergeometric system associated with a monomial curve. (English) Zbl 1060.33023
The aim of this article is to obtain results for the solution of the hypergeometric system associated with a monomial curve computing the slopes of the corresponding ideal. This computation using the ACG algorithm [A. Assi, F. J. Castro-Jimenez and J. M. Granger Compositio Math. 104, No. 2, 107–123 (1996; Zbl 0862.32005)]. More precisely, let \(A_n=\mathbb{C}\langle x_1, \dots,x_n,\partial_1,\dots,\partial_n \rangle\) the Weyl algebra over \(\mathbb{C}\), \(\mathbb{C}[\partial]=\mathbb{C} [\partial_1,\dots,\partial_n]\) the subring of \(A_n\) of linear operators with constant coefficient, \(A=(a_{ij})\) an \((d\times n)\)-matrix of rank \(d\) with integer entires, \(I_A\) the toric ideal associated to \(A\) (i.e., the ideal generated by \(\{\partial^u-\partial^v|i,v\in\mathbb{N}^n\), \(Au^T=Ar^T\}\) where \(T\) means “transpare”). Let \(\theta=(\theta_1,\dots, \theta_k)^T\) with \(\theta_i=x_i\partial_i\). For \(\beta=(\beta_1, \beta_d)^T\) consider the column vector (in \(A^d_n)A\theta-\beta\), and denote by \(\langle A\theta-\beta\rangle\) the left ideal (of \(A_n)\) generated by the entries of \(A\theta-\beta\). Let \(H_A (\beta)\) the ideal of \(A_n)\) generated by \(I_A\cup\langle A\theta-\beta\rangle\). It is called the GKZ-hypergeometric system associated to \((A,\beta)\) (see I. M. Gelfand, A. V. Zelevinskii and M. M. Kapranov, Funct. Anal Appl. 29, No. 2, 94–106 (1989; Zbl 0787.33012)] \(A_n/H_A(\beta)= \chi_A(\beta)\) is a holonomic \(A_n\)-module. The \(AA\) treat the case \(d=1\), \(A=(a_1,\dots,a_n) \in\mathbb{Z}^n\), \(a_1=1\). If \(I\) is a left ideal in \(A_n\), \(r\in\mathbb{R}\) is a geometric slope (simples, called slopes in the sequel) of \(I\) (or of \(A_n/I)\) with respect to \(x_n=0\) if and only if \(\sqrt{\sigma^{(-r)F+V}(I)}\) is not bihomogeneous with respect to \(F= (0,\dots,1,\dots 1)\) and \(V=(0,\dots,0,-1,0, \dots,0,1)\).
They evaluate the geometric slopes of \({\mathcal H}_A(\beta)\) by succesive restriction of the number of variables. For this they translate a result of Y. Laurent and Z. Mebkhout [Ann. Sci. Éc. Norm. Super. 32, No. 1, 39–69 (1999; Zbl 0994.14007)] on restriction and slopes of \({\mathcal D}\)-modules into an algorithm using ACG algorithm and an algorithm devised by T. Oaku [Adv. Appl. Math. 19, 61–105 (1997; Zbl 0938.32005)]. The \(AA\) give a preprossesing method for the ACG algorithm to accelerate the original and apply a method for the ACG algorithm to accelerate the original and apply the general algorithm to the general algorithm to the system associated to \(A-(1,a_2,\dots,a_n)\). This system is so nice that the algorithm outputs the slopes without use of computers. It is the Laurent-Mekbhout theorem mentioned above that allows induction on the number of variables to calculate the slopes. Let \(A\) be \((1,a_2,\dots,a_n)\), with \(1<a_2<\cdots <a_k\), \(\beta\in\mathbb{C}\), \(H_A(\beta)\) the associated ideal. To apply the algorithm one has so find the non microcharacteristic varieties and compute the restrictiones of \({\mathcal H}_A (\beta)\) to these varieties. For \(f_1,\dots,f_m\) polynomials in \(\mathbb{C}[x_1, \dots,x_n,\xi_1, \dots,\xi_n]\) let \(\nu(f_1,\dots,f_m)\) be the affine subvariety of \(\mathbb{C}^{2n}\) defined by the \((f_i)\). Then the characteristic variety of \({\mathcal H}_A(\beta)\) in just \(\nu(\xi_1,\dots,\xi_{n-1},x_n\xi_n)\) and the \(AA\) show that for \({\mathcal H}_A(\beta)\) the variety \(y_i=0(1\leq i\leq n-2)\) is non microcharacteristic. Finally (Th. 4.5) the geometric slopes of \({\mathcal H}_A (\beta)\) along \(x_n=0\) at the origin and those of \({\mathcal H}_{(1,a_{n-1},a_n)} (\beta)\) along \(x_3=0\) at the origin coincide. The \(AA\) show also that any rational solution of the hypergeometric system \(H_{(1,a_2,\dots,a_n)}(\beta)\) is a polynomial, and that this system has a polynomial solution iff \(\beta\in\mathbb{N}= (0,1,\dots,\}\). This polynomial solution is the residue of \(\exp(\sum x_i-t^{d_i})t^{-\beta}\) at the \(t=0\). Finally, the \(AA\) show a system of differential equations \(RH_A(\beta)\) is reducible iff \(\beta\in\mathbb{Z}\). (Here \(R\) is the ring of differential operators in \(n\) variables with rational coefficients over \(\mathbb{C}\) and, in general, a left ideal \(J\) in \(R\) is irreducible of \(J\) is maximal in \(R)\).

33C99 Hypergeometric functions
32C38 Sheaves of differential operators and their modules, \(D\)-modules
Full Text: DOI
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