Limit key polynomials as $p$-polynomials

Abstract

The main goal of this paper is to characterize limit key polynomials for a valuation $ν$ on $K[x]$. We consider the set $Ψ_α$ of key polynomials for $ν$ of degree $α$. We set $p$ be the exponent characteristic of $ν$. Our first main result (Theorem 1.1) is that if $Q_α$ is a limit key polynomial for $Ψ_α$, then the degree of $Q_α$ is $p^rα$ for some $r\in\mathbb N$. Moreover, in Theorem 1.2, we show that there exist $Q\inΨ_α$ and $Q_α$ a limit key polynomial for $Ψ_α$, such that the $Q$-expansion of $Q_α$ only has terms which are powers of $p$.