On characteristic cones of discrete nD autonomous systems: theory and an algorithm

In this paper, we provide a complete answer to the question of characteristic cones for discrete autonomous nD systems, with arbitrary n⩾2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\geqslant 2$$\end{document}, described by linear partial difference equations with real constant coefficients. A characteristic cone is a special subset (having the structure of a cone) of the domain (here Zn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}^n$$\end{document}) such that the knowledge of the trajectories on this set uniquely determines them over the whole domain. Despite its importance in numerous system-theoretic issues, the question of characteristic sets for multidimensional systems has not been answered in its full generality except for Valcher’s seminal work for the special case of 2D systems (Valcher in IEEE Trans Circuits and Syst Part I Fundam Theory Appl 47(3):290–302, 2000). This apparent lack of progress is perhaps due to inapplicability of a crucial intermediate result by Valcher to cases with n⩾3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\geqslant 3$$\end{document}. We illustrate this inapplicability of the above-mentioned result in Sect. 3 with the help of an example. We then provide an answer to this open problem of characterizing characteristic cones for discrete nD autonomous systems with general n; we prove an algebraic condition that is necessary and sufficient for a given cone to be a characteristic cone for a given system of linear partial difference equations with real constant coefficients. In the second part of the paper, we convert this necessary and sufficient condition to another equivalent algebraic condition, which is more suited from algorithmic perspective. Using this result, we provide an algorithm, based on Gröbner bases, that is implementable using standard computer algebra packages, for testing whether a given cone is a characteristic cone for a given discrete autonomous nD system.