On stable and fixed polynomials

Abstract

Let ν be a rank one valuation on K [ x ] and Ψ n the set of key polynomials for ν of degree n ∈ N . We discuss the concepts of being Ψ n -stable and ( Ψ n , Q ) -fixed. We discuss when these two concepts coincide. We use this discussion to present a simple proof of Proposition 8.2 of [3] and Theorem 1.2 of [5] .