Notes on n-Dimensional System Theory

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
n = 1
and
2
are no longer applicable for
n \geqslant 3
. In fact, for
n \geqslant 3
, 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
n \geqslant 3
but a counterexample is not at hand. The matter is therefore unresolved.