Limit key polynomials as $p$-polynomials

Abstract

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