Slopes of a Hypergeometric System Associated to a Monomial Curve

where () means “transpose”. We denote by θ the vector (θ1, . . . , θn) T with θi = xi∂i. For a given β = (β1, . . . , βd) T ∈ C we consider the column vector (in An) Aθ−β and we denote by 〈Aθ − β〉 the left ideal of An generated by the entries of Aθ − β. Following Gel’fand, Kapranov and Zelevinsky [6], we denote by HA(β) the left ideal of An generated by IA∪〈Aθ−β〉. It is called the GKZ-hypergeometric system associated to the pair (A, β). The quotient HA(β) = An/HA(β) is a holonomic An-module (see e.g. [17]). If the toric ideal IA is homogeneous, i.e., if the Q-row span of A contains (1, . . . , 1), it is known ([8], see also [17]) that HA(β) is regular holonomic and the book [17] is devoted to an algorithmic study of such systems. Especially, the book gives an algorithmic method to construct series solutions around singular points of the system. In this article, we would start a study of singularities of GKZ-hypergeometric systems for non homogeneous toric ideals IA by treating the “first” case when d = 1, A = (a1, a2, . . . , an) ∈ Z, a1 = 1. We evaluate the geometric slopes of HA(β) by successive restrictions of the number of variables. The slopes characterize Gevrey class solutions around singular locus. Our evaluation of the slopes is done as follows: (1) We translate Laurent and Mebkhout’s theorem [12] on restrictions and slopes of D-modules