\(\mathbb{C}[y]\) is a CA-ring and coefficient assignment is properly weaker than feedback cyclization over a PID (Q1346187)

The authors investigate the connection between the following properties of linear systems: cyclizability, coefficient assignability, pole assignability, and reachability. Over an arbitrary commutative ring each of the properties implies those following it. The main result of the paper is that for an algebraically closed field \(K\) the polynomial ring \(K[X]\) is a CA-ring: reachability implies coefficient assignability. (For example, this fails for \(\mathbb{R} [X]\).) An important step in the proof is a `canonical form' for linear systems over a principal ideal domain derived in a previous article of the authors [J. Pure Appl. Algebra 92, No. 3, 213-225 (1994; Zbl 0791.13008)]. It is known that for a field \(F\) of positive characteristic the polynomial ring \(F[X]\) is not an FC-ring: reachability does not imply cyclizability. This provides an example of a CA-ring that is not FC, answering a question of \textit{R. Bumby}, \textit{E. D. Sontag}, \textit{H. J. Sussman} and \textit{W. Vasconcelos} [J. Pure Appl. Algebra 20, 113-127 (1981; Zbl 0455.15009)].