153 Coprime factorisation in the stable rational functions

Let $P(s),Q(s)\in 𝒮$ be non zero. We say that $P$ and $Q$ are coprime if there exist $X(s)$ and $Y(s)$ in $𝒮$ such that

Let $G(s)$ be a complex rational function. Then there exist $P(s)$ and $Q(s)$ in $𝒮$ such that

and $P(s)$ and $Q(s)$ are coprime in $𝒮$.