A note on ``Constructing matrices with prescribed off-diagonal submatrix and invariant polynomials'' (Q947631)

The paper deals with the following matrix completion problem. Given an arbitrary field \(F\), \(s\) monic nonconstant polynomials \(f_1,f_2 \ldots, f_s \in F[x]\) with \(f_s | \cdots | f_1\), two nonnegative integers \(p,q\) such that \(p+q=\sum_{i=1}^s \text{deg}(f_i)\) and a matrix \(B \in F^{q \times p}\), is there a matrix \(A=[A_{ij}]_{i,j \in \{1,2\}} \in F^{(p+q) \times (p+q)}\) with invariant polynomials \(f_s | \cdots | f_1\) and such that \(A_{21}=B\)? In the earlier paper mentioned in the title [\textit{A. Borobia} and \textit{R. Canogar}, ibid. 424, No. 2-3, 615--633 (2007; Zbl 1123.15010)], the authors obtained, for fields of characteristic different from 2, a finite step algorithm to construct matrix \(A\) when it exists. In this paper they extend the algorithm to any field.