The coarsest lattice that determines a discrete multidimensional system

TL;DR

This paper identifies the minimal lattice that fully determines a discrete multidimensional system of linear difference equations, linking it to symmetry groups and applications in system analysis.

Contribution

It introduces the concept of a unique coarsest sublattice that defines the system and connects it to Galois symmetries, advancing understanding of system properties.

Findings

The coarsest sublattice uniquely determines the system.

Galois groups characterize the defining sublattice.

Applications include controllability, autonomy, and order reduction.

Abstract

A discrete multidimensional system is the set of solutions to a system of linear partial difference equations defined on the lattice $\Z^n$. This paper shows that it is determined by a unique coarsest sublattice, in the sense that the solutions of the system on this sublattice determine the solutions on $\Z^n$; it is therefore the correct domain of definition of the discrete system. In turn, the defining sublattice is determined by a Galois group of symmetries that leave invariant the equations defining the system. These results find application in understanding properties of the system such as controllability and autonomy, and in its order reduction.