Augmented Valuation and Minimal Pair

Abstract

Let $(K, \nu)$ be a valued field, the notions of \emph{augmented valuation}, of \emph{limit augmented valuation} and of \emph{admissible family} of valuations enable to give a description of any valuation $\mu$ of $K [x]$ extending $\nu$. In the case where the field $K$ is algebraically closed, this description is particularly simple and we can reduce it to the notions of \emph{minimal pair} and \emph{pseudo-convergent family}. Let $(K, \nu )$ be a henselian valued field and $\bar\nu$ the unique extension of $\nu$ to the algebraic closure $\bar K$ of $K$ and let $\mu$ be a valuation of $ K [x]$ extending $\nu$, we study the extensions $\bar\mu$ from $\mu$ to $\bar K [x]$ and we give a description of the valuations $\bar\mu_i$ of $\bar K [x]$ which are the extensions of the valuations $\mu_i$ belonging to the admissible family associated with $\mu$.