Limit key polynomials as $p$-polynomials

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 $p$.