On Arbitrary Assignability of Initial Conditions for a Discrete Autonomous $n$-D System

The issue of initial/boundary conditions for a general system of partial difference equations (PDEs)—called a discrete
$n$
-D system—is resolved through the notion of characteristic sets: the restriction of a solution trajectory to a minimal characteristic set can be considered to be the initial/boundary condition to the corresponding trajectory. In the recent paper (Mukherjee and Pal, 2021), it has been shown that every autonomous
$n$
-D system admits a minimal characteristic set that is the union of a sublattice (of rank equal to the Krull dimension of the system) and its finitely many parallel translates. Treating the restrictions of every solution trajectories to such a characteristic set as initial/boundary conditions, in this article, we provide a full parametrization of the set of all initial/boundary conditions of a discrete autonomous
$n$
-D system. The key to this parametrization is that the set of allowable initial/boundary conditions itself can be viewed as a discrete
$d$
-D system, where
$d$
is the Krull dimension of the system: we prove this result here. An upshot of this parametrization is the answer to free assignability of these initial/boundary conditions.