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During recent years, linear system theory has intensively been applied in estimation and control. At the same time, image processing has attracted increasing interest and attempts have been made to extend the techniques of systems theory to multidimensional problems, among others, by Bose, Attasi, Givone and Roesser, and Mitra. Part I of our results is centered around polynomial descriptions of systems. The notion of minimality in connection with state space requires the concept of coprimeness of 2-D polynomial matrices. For this purpose, we have extended the existing 1-D results on greatest common right divisor (GCRD) extraction, Sylvester resultants, matrix fraction descriptions (MFD) to the 2-D case. In addition we have results that appear to be unique for multidimensional problems such as existence and uniqueness of so-called "primitive factorizations" and existence of general factorizations. Part II will appear in a companion paper presenting results on a comparison between the different state space models that have been proposed, using what we consider to be proper definitions of state, controllability and observability and their relation to minimality of 2-D systems. We also represent new implementations of 2-D transfer functions using a minimal number of dynamic elements.