The 𝒜-hypergeometric system associated with a monomial curve

Abstract

Introduction. Inthis paper we make a detailed analysis of the -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 . 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. Inorder to place these results intheir appropriate con text, we recall the defin ition