Dimensions of solution spaces of H-systems

Abstract

An H - system is a system of first-order linear homogeneous recurrence equations for a single unknown function T , with coefficients which are polynomials with complex coefficients. We consider solutions of H -systems which are of the form T : dom ( T ) → C where either dom ( T ) = Z d , or dom ( T ) = Z d ∖ S and S is the set of integer singularities of the system. It is shown that any natural number is the dimension of the solution space of some consistent H -system, and that in the case d ≥ 2 there are H -systems whose solution space is infinite dimensional. The relationship between dimensions of solution spaces in the two cases dom ( T ) = Z d and dom ( T ) = Z d ∖ S is investigated. We prove that every consistent H -system H has a non-zero solution T with dom ( T ) = Z d . Finally we give an appropriate corollary to the Ore–Sato theorem on possible forms of solutions of H -systems in this setting.