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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 $$R)$$.

##### MSC:
 33C99 Hypergeometric functions 32C38 Sheaves of differential operators and their modules, $$D$$-modules
##### Software:
Macaulay2; OpenXM; D-modules
Full Text:
##### References:
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