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

中文翻译：

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限关键字多项式为p -polynomials

本文的主要目的是为极限ν刻画极限键多项式的特征。$ķ[X]$。我们考虑一套${\mathrm{\Psi }}_{\alpha }$的关键多项式ν度的α。我们将p设为ν的指数特征。我们的第一个主要结果（定理1.1）是，如果F是${\mathrm{\Psi }}_{\alpha }$，则F的度为$p^{\mathrm{[R}}\alpha$ 对于一些 $\mathrm{[R}\in \mathbb{ñ}$。而且，在定理1.2中，我们证明存在$问\in {\mathrm{\Psi }}_{\alpha }$和F为的极限键多项式${\mathrm{\Psi }}_{\alpha }$，使得F的Q展开仅具有p的幂的项。