Metrizability of spaces of valuation domains associated to pseudo-convergent sequences

The Zariski-Riemann space of valuation domains associated to pseudo-convergent sequences

Let $V$ be a valuation domain with quotient field $K$. Given a pseudo-convergent sequence $E$ in $K$, we study two constructions associating to $E$ a valuation domain of $K(X)$ lying over $V$, especially when $V$ has rank one. The first one has been introduced by Ostrowski, the second one more recently by Loper and Werner. We describe the main properties of these valuation domains, and we give a notion of equivalence on the set of pseudo-convergent sequences of $K$ characterizing when the associated valuation domains are equal. Then, we analyze the topological properties of the Zariski-Riemann…

Key polynomials and pseudo-convergent sequences

In this paper we introduce a new concept of key polynomials for a given valuation $ν$ on $K[x]$. We prove that such polynomials have many of the expected properties of key polynomials as those defined by MacLane and Vaquié, for instance, that they are irreducible and that the truncation of $ν$ associated to each key polynomial is a valuation. Moreover, we prove that every valuation $ν$ on $K[x]$ admits a sequence of key polynomials that completely determines $ν$ (in the sense which we make precise in the paper). We also establish the relation between these key polynomials and pseudo-convergent…

Augmented Valuation and Minimal Pair

Let $(K, ν)$ 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 $μ$ of $K [x]$ extending $ν$. 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 $(K, ν)$ be a henselian valued field and $\barν$ the unique extension of $ν$ to the algebraic closure $\bar K$ of $K$ and let $μ$ be a valuation of $ K [x]$ extending $ν$…

Extending valuations to the field of rational functions using pseudo-monotone sequences

Let $V$ be a valuation domain with quotient field $K$. We show how to describe all extensions of $V$ to $K(X)$ when the $V$-adic completion $\widehat{K}$ is algebraically closed, generalizing a similar result obtained by Ostrowski in the case of one-dimensional valuation domains. This is accomplished by realizing such extensions by means of pseudo-monotone sequences, a generalization of pseudo-convergent sequences introduced by Chabert. We also show that the valuation rings associated to pseudo-convergent and pseudo-divergent sequences (two classes of pseudo-monotone sequences) roughly corresp…

赋值与亨泽化

Valuations and henselization

本文研究以局部环为中心的赋值向其亨泽化的扩张，证明其可唯一确定亨泽化的极小素理想及同赋值群的扩张，还建立了亨泽化极小素理想与黎曼-扎里斯基空间连通分支的双射对应。

On the ranks and implicit constant fields of valuations induced by pseudo monotone sequences

Given a valued field $(K,v)$ and a pseudo monotone sequence $E$ in $(K,v)$, one has an induced valuation $v_E$ extending $v$ to $K(X)$. After fixing an extension of $v_E$ to a fixed algebraic closure $\overline{K(X)}$ of $K(X)$, we show that the implicit constant field of the extension $(K(X)|K,v_E)$ is simply the henselization of $(K,v)$. We consider the question: given a value transcendental extension $w$ of $v$ to $K(X)$ and a pseudo monotone sequence $E$ in $(K,v)$, under which precise conditions is $w$ induced by $E$? The dual nature of pseudo convergent sequences of algebraic type and ps…