×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Alan Adolphson, Hypergeometric functions and rings generated by monomials, Duke Math. J. 73 (1994), no. 2, 269 – 290. · Zbl 0804.33013
[2] A. Assi, F. J. Castro-Jiménez, and J. M. Granger, How to calculate the slopes of a \?-module, Compositio Math. 104 (1996), no. 2, 107 – 123. · Zbl 0862.32005
[3] A. Assi, F. J. Castro-Jiménez, and M. Granger, The Gröbner fan of an \?_{\?}-module, J. Pure Appl. Algebra 150 (2000), no. 1, 27 – 39. · Zbl 0967.32008
[4] Frits Beukers, W. Dale Brownawell, and Gert Heckman, Siegel normality, Ann. of Math. (2) 127 (1988), no. 2, 279 – 308. · Zbl 0652.10027
[5] Eduardo Cattani, Carlos D’Andrea, and Alicia Dickenstein, The \?-hypergeometric system associated with a monomial curve, Duke Math. J. 99 (1999), no. 2, 179 – 207. · Zbl 0952.33009
[6] I. M. Gel\(^{\prime}\)fand, A. V. Zelevinskiĭ, and M. M. Kapranov, Hypergeometric functions and toric varieties, Funktsional. Anal. i Prilozhen. 23 (1989), no. 2, 12 – 26 (Russian); English transl., Funct. Anal. Appl. 23 (1989), no. 2, 94 – 106. · Zbl 0721.33006
[7] Grayson, D. and Stillman, M., Macaulay2, a software system for research in algebraic geometry, available at http://www.math.uiuc.edu/Macaulay2.
[8] Hotta, R., Equivariant \(D\)-modules. Preprint math.RT/9805021.
[9] Yves Laurent, Théorie de la deuxième microlocalisation dans le domaine complexe, Progress in Mathematics, vol. 53, Birkhäuser Boston, Inc., Boston, MA, 1985 (French). · Zbl 0561.32013
[10] Yves Laurent, Polygône de Newton et \?-fonctions pour les modules microdifférentiels, Ann. Sci. École Norm. Sup. (4) 20 (1987), no. 3, 391 – 441 (French). · Zbl 0646.58021
[11] Yves Laurent and Zoghman Mebkhout, Pentes algébriques et pentes analytiques d’un \?-module, Ann. Sci. École Norm. Sup. (4) 32 (1999), no. 1, 39 – 69 (French, with English and French summaries). · Zbl 0944.14007
[12] Laurent, Y. and Mebkhout, Z., Image inverse d’un \({{\mathcal D} }\)-module et polygone de Newton, Compositio Math. 131 (2002), no. 1, 97-119. · Zbl 0993.35007
[13] Leykin, A. and Tsai, H., D-module package for Macaulay 2. http://www.math.cornell. edu/htsai
[14] Zoghman Mebkhout, Le théorème de positivité de l’irrégularité pour les \?_{\?}-modules, The Grothendieck Festschrift, Vol. III, Progr. Math., vol. 88, Birkhäuser Boston, Boston, MA, 1990, pp. 83 – 132 (French). · Zbl 0731.14007
[15] Toshinori Oaku, Algorithms for \?-functions, restrictions, and algebraic local cohomology groups of \?-modules, Adv. in Appl. Math. 19 (1997), no. 1, 61 – 105. · Zbl 0938.32005
[16] Toshinori Oaku, Nobuki Takayama, and Uli Walther, A localization algorithm for \?-modules, J. Symbolic Comput. 29 (2000), no. 4-5, 721 – 728. Symbolic computation in algebra, analysis, and geometry (Berkeley, CA, 1998). · Zbl 1012.13010
[17] Mutsumi Saito, Bernd Sturmfels, and Nobuki Takayama, Gröbner deformations of hypergeometric differential equations, Algorithms and Computation in Mathematics, vol. 6, Springer-Verlag, Berlin, 2000. · Zbl 0946.13021
[18] Takayama, N., Kan: A system for computation in algebraic analysis, 1991 version 1, 1994 version 2, the latest version is 3.000726. Source code available for Unix computers. Download from http://www.openxm.org
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.