On arbitrary assignability of initial conditions for an overdetermined system of partial difference equations

In this paper, we provide an answer to the question of arbitrary assignability of trajectories on a characteristic set which serve as initial conditions for an overdetermined system of partial difference equations (pdes). A characteristic set is a subset of the domain ℤ
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with the defining property that trajectories restricted to this set allows to uniquely extend the trajectory over the whole domain. It is known in the literature that every overdetermined system of pdes admits a proper subset of the domain as a characteristic set. The objective of this paper is to address the question of freeness of a characteristic set when the characteristic set is given by a union of a sublattice and finitely many parallel translates of it. In other words, we examine if trajectories can be arbitrarily assigned on the union of a sublattice and finitely many parallel translates of it such that they form an initial condition for the given system of pdes. We show that this may not always be possible. We characterize the set of allowable initial conditions algebraically. With the help of this characterization, we further provide a necessary and sufficient condition for which the union of a sublattice and finitely many parallel translates of it is free.