Notes on n-Dimensional System Theory

Abstract

This paper makes three observations with regard to several issues of a fundamental nature that apparently must arise in any general theory of linear n-dimensional systems. It is shown, by means of three specific interrelated counterexamples, that certain decomposition techniques which have proven to be basic for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n = 1</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</tex> are no longer applicable for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n \geqslant 3</tex> . In fact, for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n \geqslant 3</tex> , at least three equally meaningful but inequivalent notions of polynomial coprimeness emerge, namely, zerocoprimeness (ZC), minor-coprimeness (MC), and factor-coprimeness (FC). Theorems I and 3 clarify the differences (and similarities) between these concepts, and Theorem 2 gives the ZC and MC properties a useful system formulation. (Unfortunately, FC, which in our opinion is destined to play a major role, has thus far eluded the same kind of characterization.) Theorem 4 reveals that the structure of 2-variable elementary polynomial matrices is completely captured by the ZC concept. However, there is reason to believe that ZC is insufficient for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n \geqslant 3</tex> but a counterexample is not at hand. The matter is therefore unresolved.