Every discrete 2D autonomous system admits a finite union of parallel lines as a characteristic set

Abstract

In this paper, we show that every discrete 2D autonomous system, that is described by a set of linear partial difference equations with constant real coefficients, admits a finite union of parallel lines as a characteristic set. In order to prove our claim, we first look at a special class of scalar discrete 2D systems and provide such characteristic sets for systems in this class. This special class has the property that systems in this class have their quotient rings to be finitely generated modules over a one-variable Laurent polynomial subring of the original two-variable Laurent polynomial ring in the shift operators. We show that such systems always admit a finite collection of horizontal lines for a characteristic set. We then extend this result to non-scalar discrete 2D autonomous systems. We achieve this in two steps: first, we show that every scalar discrete 2D system can be converted into a system in the above-mentioned class by a coordinate transformation on the independent variables set, Z 2 . Using this we then show that characteristic sets for the original system can be found by applying the inverse coordinate transformation on characteristic sets of the transformed system. Since the transformed system, by virtue of being in the special class, admits a finite union of horizontal lines as a characteristic set, the original system is guaranteed to admit a characteristic set that is a coordinate transformation applied to a finite union of horizontal lines. The coordinate transformation maps this union of horizontal lines to a union of parallel, but possibly tilted, lines. In the next step, we generalize the scalar case to the general vector case: that is, systems with more than one dependent variables. The main motivation for studying characteristic sets that are unions of finitely many parallel lines is that, arguably, such sets can be called "thin" in Z 2 in comparison to the prevalent notions of convex cones and half-spaces as characteristic sets (see "Appendix 1").