A symmetric approach to cyclization for systems over \(\mathbb{C} [y]\) (Q1368769)

Let \(R\) be a commutative ring and \((A,B)\) a pair of matrices over \(R\), where \(A\) is \(n \times n\) and \(B\) is \(n \times m\). The pair \((A,B)\) is said to satisfy the (feedback)-cyclization \((FC)\) property if there exist a \(m \times n\) matrix \(F\) over \(R\) and a vector \(u \in R^m\) such that the matrix \([Bu, (A + BF)Bu, \ldots , (A+BF)^{(n-1)}Bu]\) is invertible. A commutative ring which satisfies \(FC\) for all reachable pairs \((A,B)\) is said to be \(FC\) and one which satisfies \(FC\) for any reachable pair of a specific dimension \(n\) is called \(FC_n\). It was conjectured by \textit{R. Bumby} et al. [J. Pure Appl. Algebra 20, 113-127 (1981; Zbl 0455.15009)] that \({C}[y]\) is an \(FC\) ring and it has since been proved that \({C}[y]\) is \(FC_n\) for \(n \leq 3\). In this paper it is shown that \({C}[y]\) is \(FC_4\). The way this is done is to derive and solve a symmetrized and equivalent version of the \(FC\) problem. This version decreases the number of parameters needed to solve the \(FC\) problem as compared to the form stated above. The author notes in a concluding remark that he believes that, unlike solutions for \(FC_n\), \(n \leq 3\), the solution strategy followed here indicates the way to the general solution.