Finite Directional Coefficient Modules over Euler Fibers of $A$-Hypergeometric Systems

Abstract

Let A be a d-by-n integer matrix of rank d, let beta belong to the d-dimensional complex vector space, and consider the A-hypergeometric system defined by the ideal H_A(beta), with relation lattice L equal to the integer kernel of A. No homogeneity, pointedness, or positivity assumption is imposed on A. We study directed formal logarithmic solutions simultaneously over all exponent lattices in the Euler fiber over beta, which form an uncountable family when L has positive rank. For a fixed L-generic direction, negative-support strata are described by rational sign polyhedra, and their directional boundedness is characterized by recession cones. On each exponent lattice, normalization reduces the infinitely indexed coefficient equations to a finite system of Euler, transit, and directional vanishing relations, represented by a finitely presented directional coefficient module. We prove that only finitely many exponent lattices contribute nonzero directed solution spaces and that every contributing coefficient module has finite length. Macaulay inverse-system duality then identifies the dimension of the all-lattice directed formal logarithmic solution space with the length of the all-lattice coefficient module. We also give an explicit formal-series realization and a polynomial colon-ideal criterion for all realizable lowest exponents.