A remark on a polynomial matrix factorization theorem (Q2918683)

Let \(\Gamma\) be a closed and bounded Jordan curve in the complex plane such that the inner domain \(D_{+}\) contains \(0\) and the outer domain \(D_{-}\) contains \(\infty\). A factorization of an \(n\times n\) matrix function \(G\) defined on \(\Gamma\) is a representation of the form NEWLINE\[NEWLINEG(t)=G_+(t)\, \text{diag}\, [t^{k_1}, t^{k_2}, \dots , t^{k_n}]\,G_-(t) \qquad (t\in\Gamma),NEWLINE\]NEWLINE where the factors \(G_\pm (t)\) are boundary values of \(n\times n\) matrix functions \(G_\pm (z)\), which are analytic and invertible in \(D_\pm\). It is well known that, if \(G\) is a rational matrix function, then it admits a factorization of the above form and the factors \(G_\pm (t)\) are rational functions as well. Actually, it follows from the standard proof of this statement (see \textit{I. Gohberg, M. A. Kaashoek}, and \textit{I. M. Spitkovsky} [Oper. Theory, Adv. Appl. 141, 1--102 (2003; Zbl 1049.47001)]) that \(G_{+}\) is a polynomial function in this case. However, the authors claim that from the mentioned proof it is not clear that \(G_{-}\) is a (trigonometric) polynomial matrix as well (i.e., its entries might have poles only at the origin). Their aim is to prove this fact explicitly. They also estimate the partial indices of \(G\) and the orders of the matrix polynomials \(G_\pm (t)\). The proof is based on a procedure described in the paper mentioned above, which basically consists in removing the singularities of the determinant of \(G_{\pm}\) in \(D_\pm\). The same idea can be used to obtain a factorization theorem of polynomial matrices on the real line.