On stable and fixed polynomials

Let $\nu$ be a rank one valuation on $K[x]$ and ${\mathrm{\Psi }}_n$ the set of key polynomials for $\nu$ of degree $n\in \mathbb{N}$. We discuss the concepts of being ${\mathrm{\Psi }}_n$-stable and $({\mathrm{\Psi }}_n,Q)$-fixed. We discuss when these two concepts coincide. We use this discussion to present a simple proof of Proposition 8.2 of [5] and Theorem 1.2 of [5].