Novel representation of discrete n-D autonomous systems

TL;DR

This paper introduces a new way to represent solutions of discrete n-dimensional autonomous systems using multidimensional flow operators, transforming complex systems into simpler special classes through coordinate changes.

Contribution

It provides a novel representation formula for autonomous systems and proves a discrete Noether's Normalization Lemma for Laurent polynomial rings.

Findings

Representation formula as multidimensional flow operators

Coordinate transformation reduces general systems to special class

Discrete Noether's Normalization Lemma for Laurent polynomial rings

Abstract

In this paper we address the problem of representing solutions of a system of scalar linear partial difference equations akin to state space equations of 1-D systems theory. We first obtain a representation formula for a special class of autonomous systems. Then we show every autonomous system can be converted into the special ones by a coordinate transformation on n-d integer grid. Using this conversion we provide representation formula for general autonomous systems. The representation formula we present can be viewed as multidimensional flow operators acting on initial conditions. These initial conditions are required to satisfy certain compatibility conditions. We give a full description of the set of allowable initial conditions. In our search for a general representation formula, one algebraic result plays a very crucial role. In this result we show that every quotient ring of the n-variable Laurent polynomial ring can be made a finitely generated faithful module over another Laurent polynomial ring of smaller dimension by doing a suitable change of coordinates. We call this result a discrete version of Noether's Normalization Lemma.