On stable and fixed polynomials

TL;DR

This paper explores the properties of stable and fixed polynomials within valuation theory, providing insights into their equivalence and offering simplified proofs of existing theorems.

Contribution

It introduces the concepts of stability and fixedness for polynomials relative to key polynomial sets and clarifies their relationship, enhancing understanding in valuation theory.

Findings

Identifies conditions under which stability and fixedness coincide

Provides simplified proofs of key propositions and theorems

Clarifies the relationship between stability and fixedness in polynomial valuations

Abstract

Let $\nu$ be a rank one valuation on $K[x]$ and $\Psi_n$ the set of key polynomials for $\nu$ of degree $n\in\N$. We discuss the concepts of being $\Psi_n$-stable and $(\Psi_n,Q)$-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].