The 𝒜-hypergeometric system associated with a monomial curve
Eduardo H. Cattani, Carlos D’Andrea, Alicia Dickenstein · Duke Mathematical Journal · 1999
Introduction.In this paper we make a detailed analysis of the Ꮽ-hypergeometric system (or GKZ system) associated with a monomial curve and integral, hence resonant, exponents.We describe all rational solutions and show in Theorem 1.10 that they are, in fact, Laurent polynomials.We also show that for any exponent there are at most two linearly independent Laurent solutions and that the upper bound is reached if and only if the curve is not arithmetically Cohen-Macaulay.We then construct, for all integral parameters, a basis of local solutions in terms of the roots of the generic univariate polynomial (0.5) associated with Ꮽ.We also determine in Theorem 3.7 the holonomic rank r(α) for all α ∈ Z 2 and show that d ≤ r(α) ≤ d +1, where d is the degree of the curve.Moreover, the value d +1 is attained only for those exponents α for which there are two linearly independent rational solutions, and, therefore, r(α) = d for all α if and only if the curve is arithmetically Cohen-Macaulay.In order to place these results in their appropriate context, we recall the definition of the Ꮽ-hypergeometric systems.These were introduced in a series of papers in the mid-1980s by the Gel fand school, particularly Gel fand, Kapranov, and Zelevinsky (see [7] and [9] and the references therein).Let Ꮽ = {ν 1 , . . ., ν r } ⊂ Z n+1 be a finite subset that spans the lattice Z n+1 .Suppose, moreover, that there exists a vector λ = (λ 0 , . . ., λ n ) ∈ Q n+1 such that λ, ν j = 1 for all j = 1, . . ., r, that is, the set Ꮽ lies in a rational hyperplane.Let Ꮽ also denote the (n + 1) × r matrix whose columns are the vectors ν j .Let ᏸ ⊂ Z r be the sublattice of elements v ∈ Z r such that Ꮽ • v = 0. Given α ∈ C n+1 , the Ꮽ-hypergeometric system with exponent (or parameter) α iswhere Ꮽ = (a ij ) and Ᏸ v is the differential operator in C r :