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

TL;DR

This paper generalizes Ostrowski's result on valuation extensions to the field of rational functions by using pseudo-monotone sequences, linking valuation rings to topological structures in algebraic extensions.

Contribution

It introduces a framework using pseudo-monotone sequences to describe all valuation extensions to rational function fields, extending previous one-dimensional results.

Findings

Valuation rings correspond to closed and open balls in the topology induced by $V$.

Extensions are realized via pseudo-monotone sequences, generalizing pseudo-convergent sequences.

The approach applies when the $V$-adic completion of $K$ is algebraically closed.

Abstract

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 correspond, respectively, to the closed and the open balls of $K$ in the topology induced by $V$.

Equations

$$V_E=\{\varphi \in K(X)\mid \varphi (s_{\nu })\in V,\text{ for all sufficiently large }\nu \in \Lambda \}.$$

$$v_{\alpha ,\delta }(f)=\inf \{v(a_i)+i\delta \mid i=0,\dots ,n\},$$

$$I^v=\{\begin{array}{ll}cV,& \text{ifleavevmodenobreak }I=cM\text{leavevmodenobreak for someleavevmodenobreak }c\in K\\ I,& \text{otherwise}.\end{array}$$

$$\mathrm{Br}(E)=$$

$$v(\beta -s_{\nu })=v(\alpha -s_{\nu }+x)=v(\alpha -s_{\nu })={\delta }_{\nu }$$

$$v(\varphi (s_{\nu }))=\lambda {\delta }_{\nu }+\gamma .$$

$$v(\varphi (s_{\nu }))=\lambda v(s_{\nu }-\beta )+\gamma =\lambda {\delta }_{\nu }+\gamma ,$$

$$v(\varphi (s_{\nu }))$$

$$V_E=\{\varphi \in K(X)\mid \varphi (s_{\nu })\in V,\text{ for all sufficiently large }\nu \in \Lambda \}.$$

$$M_E=\{\varphi \in K(X)\mid \varphi (s_{\nu })\in M,\text{ for all sufficiently large }\nu \in \Lambda \}.$$

$$v_E(r_i{p}^i)=v_E(r_i)+i{\Delta }_E\ne v_E(r_j)+j{\Delta }_E=v_E(r_j{p}^j)$$

$$\varphi (X)=\frac{p(X)}{q(X)}=\frac{a(X)q_1(X)+b(X)}{q(X)}=\frac{a(X)}{q_2(X)}+\frac{b(X)}{q(X)},$$

$${\mathrm{degdom}}_E({\varphi }_2)={\mathrm{degdom}}_E(b)-{\mathrm{degdom}}_E(q_1)\le \mathrm{deg}b-\mathrm{deg}{q}_1<0,$$

$$\psi (X)={\varphi }_1(X)-{\varphi }_1(\alpha )=\frac{a(X)q_2(\alpha )-a(\alpha )q_2(X)}{q_2(\alpha )q_2(X)}.$$

$$v_E(\varphi (X)-\varphi (s_{\nu }))=v_E(({\varphi }_1(X)-{\varphi }_1(s_{\nu }))+({\varphi }_2(X)-{\varphi }_2(s_{\nu }))).$$

$$v_E(\varphi (X)-\varphi (s_{\nu }))=v_E({\varphi }_1(X)-{\varphi }_1(s_{\nu }))=v_E({\varphi }_1(s_{\nu })),$$

$${\mathcal{L}}_1(W)=$$

$${\mathcal{L}}_2(W)=$$

$$\mathcal{L}(W)=$$

$$w(X-\beta )=w(X-\alpha +\alpha -\beta )=\{\begin{array}{ll}w(X-\alpha ),& \text{ifleavevmodenobreak }v(\alpha -\beta )\ge w(X-\alpha )\\ v(\alpha -\beta ),& \text{ifleavevmodenobreak }v(\alpha -\beta )<w(X-\alpha )\end{array}$$

$$V_{\infty }=\{\varphi \in K(X)\mid \varphi (\infty )\in V\},$$

$$w(e(X-\alpha ))\ge 0⟺w(X-\alpha )\ge v(e^{-1})⟺w(X-\alpha +e^{-1})=v(e^{-1})$$

$$w\left(\frac{e}{X-\alpha }\right)\ge 0⟺w(X-\alpha )\le v(e)⟺w(X-\alpha +e)=w(X-\alpha ),$$

$$\rho (U_E)=$$

$$\{\rho \circ \varphi \in \overline{K}(X)\mid \sigma \circ \rho (\varphi (s_{\nu }))\in U\text{leavevmodenobreak eventually}\}.$$

$$\rho (U_E)=$$

$$\{\psi \in \overline{K}(X)\mid \sigma (\psi (s_{\nu }))\in U\text{leavevmodenobreak eventually}\}.$$

$$v(\beta -t_{\mu })=v(\beta -s_{\nu }+s_{\nu }-t_{\kappa }+t_{\kappa }-t_{\mu })={\delta }_{\mu }^{\prime }$$

$$v(s_{\nu }-t_{\mu })=v(s_{\nu }-\alpha +\alpha -t_{\mu })\ge \min \{{\delta }_{\nu },{\delta }_{\mu }^{\prime }\}.$$

$$u(s_{\mu }-s_{\nu })=u(c_{\mu }+\beta -c_{\nu }-\beta )=u(c_{\mu }-c_{\nu })={\delta }_{\nu }$$

Full text

Keywords: pseudo-convergent sequence, pseudo-limit, pseudo-monotone sequence, monomial valuation, extension of valuations.

MSC Primary 12J20, 13A18, 13F30.

1 Introduction

Throughout the paper, $V$ will denote a valuation domain with quotient field $K$ and maximal ideal $M$, $v$ will denote its valuation and $\Gamma_{v}$ its value group. We also fix an algebraic closure $\overline{K}$ of $K$. The study of extensions of $V$ is one of the central parts of valuation theory, which naturally splits into the study of algebraic and purely transcendental extensions. The former can be considered a generalization of the fundamental problems of algebraic number theory, and is well-studied through the concepts of inertia, decomposition and ramification (in what is known as ramification theory). The latter – which is essentially the study of extensions of $V$ to function fields – is less well understood, but plays a main role in several facets and applications of the theory (see [11] and the references therein). The first step of this problem is to classify all the extensions of $V$ to the rational function field $K(X)$.

In case $V$ has rank one, there are two classical approaches to this problem: the most famous one, due to MacLane, uses key polynomials and augmented valuations and works for arbitrary fields $K$, but requires the valuation ring to be discrete [14]; it has been recently generalized by Vaquiè in [23] for general valuation domains. The second approach, due to Ostrowski, “makes no discreteness assumptions” but “requires an elaborate construction to obtain values of $\overline{K}$ from those of $K$”, as MacLane acknowledged in his paper [14, p. 380]. More precisely, Ostrowski showed that, for a given extension $W$ of $V$ to $K(X)$, there exists a pseudo-convergent sequence $E=\{s_{n}\}_{n\in\mathbb{N}}\subset\overline{K}$ with respect to a extension $u$ of $v$ to $\overline{K}$ (we refer to §2.3 for the definition) such that the valuation $w$ associated to $W$ is given by the real limit $w(\phi)=\lim_{n\to\infty}u(\phi(s_{n}))$, for all $\phi\in K(X)$; for its importance, Ostrowski called this result Fundamentalsatz [16, §11, IX, p. 378]. To our knowledge, Ostrowski’s Fundamentalsatz seems to have been mostly forgotten (except in the survey [21]), even if pseudo-convergent sequences have enjoyed some success: for example, Kaplansky used them to characterize immediate extensions of a valued field and maximal fields in [10], and they are linked to the recently introduced notion of approximation type (see [12]).

In generalizing Ostrowski’s Fundamentalsatz, we realized that when dealing with the general case (i.e., when the rank of $V$ or of the extension of $V$ to $K(X)$ is not one), pseudo-convergent sequences are not enough to construct all extensions of $V$ to $K(X)$ (see Example 4.4): for this reason, we use the more general notion of pseudo-monotone sequences, used in [17] to encompass Ostrowski’s notion of pseudo-convergent sequence and the two other kinds of sequences introduced by Chabert in 2010 (namely pseudo-divergent and pseudo-stationary sequences) in order to characterize the so-called polynomial closure in the context of rings of integer-valued polynomials. We recall that, given a subset $S$ of $V$, the ring of integer-valued polynomials over $S$ is classically defined as $\textrm{Int}(S,V)=\{f\in K[X]\mid f(S)\subseteq V\}$, and the polynomial closure of $S$ is the largest subset $\overline{S}\subseteq V$ such that $\textrm{Int}(S,V)=\textrm{Int}(\overline{S},V)$. One of the main results of Chabert was to prove that, when $V$ has rank one, the polynomial closure is the closure operator associated to a topology on $K$ (extending the case when $V$ is discrete, originally proved by McQuillan in [15, Lemma 2]). Chabert obtained his result by describing $\overline{S}$ through the set of pseudo-limits of the pseudo-monotone sequences contained in $S$.

In this paper, continuing our earlier work in [19], we describe the extensions of $V$ to $K(X)$ by means of pseudo-monotone sequences of $K$, generalizing a natural construction of Loper and Werner, who were interested in studying when the ring of integer-valued polynomials over a pseudo-convergent sequence is a Prüfer domain [13]. More precisely, we associate to every pseudo-monotone sequence $E=\{s_{\nu}\}_{\nu\in\Lambda}\subset K$ (see §2.3 for the definition) the valuation domain

[TABLE]

We first study the properties of $V_{E}$ in relation with the properties of $E$; subsequently, we analyze when and how it is possible to associate to an arbitrary extension a pseudo-monotone sequence. Our main result (Theorem 6.2) proves that every extension of $V$ to $K(X)$ can be realized in this way if and only if the $v$-adic completion $\widehat{K}$ of $K$ is algebraically closed. In particular, the statement holds if $K$ is algebraically closed, giving a generalization of Ostrowski’s result. We also show that, under the same condition, every extension of $V$ to $K(X)$ which is not immediate is a monomial valuation, a natural way of constructing extensions to the field of rational functions (see §2.1).

The structure of the paper is as follows. In Section 2, after settling the notation used throughout the paper, and the notions of monomial valuation and divisorial ideal, we give the definition of pseudo-monotone sequence in a general valued field $(K,v)$; we note that Chabert’s original definitions of pseudo-divergent and pseudo-stationary sequences were given only for a rank one valuation, but they easily extend to the general case. We then introduce the notions of pseudo-limit, breadth ideal and gauge separately for the three different types of pseudo-monotone sequences: pseudo-convergent sequences (§2.3.1), pseudo-divergent sequences (§2.3.2) and pseudo-stationary sequences (§2.3.3). In the last part of that section, we characterize pseudo-limits and breadth ideals of pseudo-monotone sequences according to their type (Lemmas 2.5 and 2.6).

In Section 3 we show that the sequence of values of the images under a rational function of a pseudo-monotone sequence is eventually monotone (Proposition 3.2); the result is accomplished by introducing the notion of dominating degree of a rational function $\phi\in K(X)$ with respect to a pseudo-monotone sequence $E\subset K$ (Definition 3.1), which roughly speaking counts the number of roots of $\phi$ in $\overline{K}$ which are pseudo-limits of $E$. Through this result, we show that, for each pseudo-monotone sequence $E$, the ring $V_{E}$ is a valuation domain of $K(X)$ extending $V$ (Theorem 3.4). We then describe the main properties of $V_{E}$ (residue field, value group and associated valuation) in Proposition 3.7, and show that the image of a pseudo-convergent or a pseudo-divergent sequence under a rational function is eventually either pseudo-convergent or pseudo-divergent (Proposition 3.8), improving the analogous result of Ostrowski [16, III, §64, p. 371] on images of pseudo-convergent sequences under polynomial mappings.

In Section 4, we associate to each extension $W$ a subset $\mathcal{L}(W)$ of $K$ (which corresponds to the notion of pseudo-limit of a pseudo-monotone sequence) and show that if $K$ is algebraically closed, then $\mathcal{L}(W)$ (if nonempty) uniquely determines $W$ (Proposition 4.5). In Section 5, we use the results of the previous section to completely describe (for any field $K$) when two different pseudo-monotone sequences of $K$ give rise to the same associated extension of $V$ to $K(X)$. Subsequently, in Section 6 we give the proof of the aforementioned main Theorem 6.2.

In the final Section 7, we illustrate the different containments which may occur among the valuation domains $V_{E}$ of $K(X)$. We conclude with a modern proof of Ostrowski’s Fundamentalsatz (Theorem 7.4).

2 Background and notation

For an extension $U$ or $W$ of $V$ to a field $F$ containing $K$, we denote the associated valuation with the corresponding lower case letter (i.e., $u$ or $w$, respectively). We recall that an extension $V\subset U$ is immediate if $U$ and $V$ have the same value group and same residue field. We denote by $\widehat{K}$ and $\widehat{V}$, respectively, the completion of $K$ and $V$ with respect to the topology induced by the valuation $v$. The elements of $\widehat{K}$ can be constructed as limits of Cauchy sequences $\{a_{\nu}\}_{\nu\in\Lambda}$, where $\Lambda$ is a well-ordered set; $\Lambda$ is not necessarily countable, but can be considered of cardinality equal to the cofinality of the ordered set $\Gamma_{v}$. See for example [8, Section 2.4] for the details of the construction. For a sequence $\{s_{\nu}\}_{\nu\in\Lambda}$ of elements in $K$, the set of indices $\Lambda$ will always be a well-ordered set without a maximum.

2.1 Monomial valuations

We recall the definition of monomial valuations, a standard way of extending a valuation $v$ of $K$ to $K(X)$.

Definition 2.1**.**

Let $\Gamma$ be a totally ordered group containing $\Gamma_{v}$, and let $\alpha\in K$ and $\delta\in\Gamma$. For every polynomial $f(X)=a_{0}+a_{1}(X-\alpha)+\ldots+a_{n}(X-\alpha)^{n}\in K[X]$, define

[TABLE]

and, for a rational function $\phi=f/g$ (with $f,g$ polynomials), define $v_{\alpha,\delta}(\phi)=v_{\alpha,\delta}(f)-v_{\alpha,\delta}(g)$. Then, $v_{\alpha,\delta}$ is a valuation on $K(X)$, and it is called monomial valuation [4, Chapt. VI, §. 10, Lemma 1]. We denote by $V_{\alpha,\delta}$ the associated valuation domain of $K(X)$.

For example, the Gaussian extension $v_{G}=v_{0,0}$ of $v$, defined as $v_{G}(\sum_{i\geq 0}a_{i}X^{i})=\inf_{i}\{v(a_{i})\}$, is a monomial valuation. In general, $v_{\alpha,\delta}$ is residually transcendental over $v$ (i.e., the residue field of $V_{\alpha,\delta}$ is transcendental over the residue field of $V$) if and only if $\delta$ is torsion over $\Gamma_{v}$ [17, Lemma 3.5]. Furthermore, every residually transcendental extension of $V$ can be written as $W^{\prime}\cap K(X)$, where $W^{\prime}$ is a monomial valuation domain of $\overline{K}(X)$ with respect to an extension $w$ of $v$ to $\overline{K}$ ([1, 3]).

2.2 Divisorial ideals

Let $V$ be a valuation domain with maximal ideal $M$, and let $\mathcal{F}(V)$ be the set of fractional ideals of $V$. The $v$-operation (or divisorial closure) on $V$ is the map sending each $I\in\mathcal{F}(V)$ to the ideal $I^{v}$ equal to the intersection of all principal fractional ideals containing it; equivalently, $I^{v}=(V:(V:I))$, where, for a fractional ideal $I$ of $V$, we set $(V:I)=\{x\in K\mid xI\subseteq V\}$ [9, Theorem 34.1]. If $I=I^{v}$, we say that $I$ is a divisorial ideal.

If the maximal ideal $M$ of $V$ is principal, then each fractional ideal $I$ of $V$ is divisorial; on the other hand, if $M$ is not principal, then (see for example [9, §34, Exercise 12, p. 431])

[TABLE]

We say that $I$ is strictly divisorial if $I$ is equal to the intersection of all principal fractional ideals properly containing it; in particular, each strictly divisorial ideal is divisorial. We now characterize these ideals.

Lemma 2.2.

$I$* is not strictly divisorial if and only if $I=cM$ for some $c\in K$.*

Proof.

Suppose first that $I$ is not principal. Then, $I$ is strictly divisorial if and only if it is divisorial; furthermore, $I$ is not divisorial if and only if $I=cM$ for some $c\in K$ and $M$ is not principal, by the above remark. Hence, the claim holds in this case.

Suppose that $I=c^{\prime}V$ is principal: if also $I=cM$ for some $c$, then $cV$ is the minimal principal ideal properly containing $I$, and $I$ is not strictly divisorial. Conversely, if $I$ is not strictly divisorial, then there is a minimal principal ideal $cV$ properly containing $I$; this implies that $c^{\prime}/c$ is the generator of the maximal ideal of $V$, and so $I=cM$. ∎

2.3 Pseudo-monotone sequences

The central concept of the paper is the following, which along with Ostrowski’s notion of pseudo-convergent sequence includes also other two related notions introduced by Chabert in [5].

Definition 2.3**.**

Let $E=\{s_{\nu}\}_{\nu\in\Lambda}\subset K$ be a sequence. We say that the sequence $E$ is:

pseudo-convergent if $v(s_{\rho}-s_{\nu})<v(s_{\sigma}-s_{\rho})$ for all $\nu<\rho<\sigma\in\Lambda$;

• -

pseudo-divergent if $v(s_{\rho}-s_{\nu})>v(s_{\sigma}-s_{\rho})$ for all $\nu<\rho<\sigma\in\Lambda$;

• -

pseudo-stationary if $v(s_{\nu}-s_{\mu})=v(s_{\nu^{\prime}}-s_{\mu^{\prime}})$ for all $\nu\neq\mu\in\Lambda$, $\nu^{\prime}\neq\mu^{\prime}\in\Lambda$.

If $E$ satisfies any of these definitions, we say that $E$ is a pseudo-monotone sequence ([17]). We say that $E$ is strictly pseudo-monotone if $E$ is either pseudo-convergent or pseudo-divergent. If $E$ and $F$ are two pseudo-monotone sequences that are either both pseudo-convergent, both pseudo-divergent or both pseudo-stationary we say that $E$ and $F$ are of the same kind.

We note that Ostrowski’s and Chabert’s original definitions required the above condition to be valid only for all $\nu$ large enough. Instead, we adopt Kaplansky’s convention that the condition is valid for all $\nu$, both since it is not restrictive for our purposes (see Definition 3.3) and in view of the following remark. If $E=\{s_{\nu}\}_{\nu\in\Lambda}$ is a sequence in $K$ and $E^{\prime}=\{s_{\nu}\}_{\nu\geq N}$ is pseudo-monotone for some $N\in\Lambda$, we say that $E$ is eventually pseudo-monotone (and analogously for eventually pseudo-convergent, pseudo-divergent and pseudo-stationary).

Remark 2.4**.**

Strictly pseudo-monotone sequences are “rigid”, in the sense that, given a set $E$, there is at most one way to index $E$ to make it pseudo-monotone. Indeed, if the indexing $\{s_{\nu}\}_{\nu\in\Lambda}$ makes $E$ pseudo-convergent, then the equality $v(s_{\nu}-s_{\mu})=v(s_{\nu}-s_{\mu^{\prime}})$ (for $\mu\neq\mu^{\prime}$) implies that both $\mu$ and $\mu^{\prime}$ are greater than $\nu$; thus, the elements of $E$ that appear before $s_{\nu}$ are exactly the $t$ such that $v(s_{\nu}-t)\neq v(s_{\nu}-t^{\prime})$ for all $t\neq t^{\prime}$, and this condition depends only on the set $E$. In the same way, if $E$ is pseudo-divergent, then the elements of $E$ appearing after $s_{\nu}$ are the $t$ such that $v(s_{\nu}-t)\neq v(s_{\nu}-t^{\prime})$ for all $t\neq t^{\prime}$. In particular, if $E=\{s_{\nu}\}_{\nu\in\Lambda}$ and $F=\{t_{\nu}\}_{\nu\in\Lambda}$ are two strictly pseudo-monotone sequences that are equal as sets, then $s_{\nu}=t_{\nu}$ for every $\nu\in\Lambda$.

On the other hand, pseudo-stationary sequences are “flexible”: any permutation of $E=\{s_{\nu}\}_{\nu\in\Lambda}$ is again pseudo-stationary. For this reason, it may be more apt to call them “pseudo-stationary sets”, but we will continue to treat them as sequences for analogy with the strictly pseudo-monotone case.

In this paper, we shall treat pseudo-monotone sequences in a general framework in order to build extensions of the valuation domain $V$ to the field of rational functions $K(X)$, and to give theorems valid for all kind of such sequences. However, there are slight differences in how the main concepts concerning pseudo-monotone sequences (for example the breadth ideal, the pseudo-limit and the gauge) are defined in each of the three cases; hence, we shall describe them separately.

2.3.1 Pseudo-convergent sequences

Let $E=\{s_{\nu}\}_{\nu\in\Lambda}$ be a pseudo-convergent sequence in $K$. Then, if $\nu$ is fixed, the value $v(s_{\rho}-s_{\nu})$, for $\rho>\nu$, does not depend on $\rho$. We denote by $\delta_{\nu}\in\Gamma_{v}$ this value; the sequence $\{\delta_{\nu}\}_{\nu\in\Lambda}$ (which, by definition, is a strictly increasing sequence in $\Gamma_{v}$) is called the gauge of $E$.

The breadth ideal $\mathrm{Br}(E)$ of $E$ is the set

[TABLE]

this set is always a fractional ideal of $K$. If $c_{\nu}=s_{\rho}-s_{\nu}$, for some $\rho>\nu$, then $\mathrm{Br}(E)=\bigcap_{\nu\in\Lambda}c_{\nu}V$. If $\mathrm{Br}(E)$ is a principal ideal, say generated by an element $c\in K$, then $\delta_{\nu}$ converges to an element $\delta\in\Gamma_{v}$ (and, clearly, $v(c)=\delta$). When this happens, we call $\delta$ the breadth of $E$. Note, however, that the breadth of a pseudo-convergent sequence may not always be defined; if $V$ has rank $1$ (that is, if $\Gamma_{v}$ can be embedded as a totally ordered group into $\mathbb{R}$), then $\delta_{\nu}$ always converges to an element $\delta\in\mathbb{R}$, which may not belong to $\Gamma_{v}$. See [19] and [17, Lemma 2.3] for this case.

An element $\alpha\in K$ is a pseudo-limit of $E$ if $v(\alpha-s_{\nu})=\delta_{\nu}$ for all $\nu\in\Lambda$ or, equivalently, if $v(\alpha-s_{\nu})<v(\alpha-s_{\rho})$ for all $\nu<\rho\in\Lambda$. It also suffices that these conditions hold only for $\nu\geq N$, for some $N\in\Lambda$. If the gauge $\{\delta_{\nu}\}_{\nu\in\Lambda}$ is cofinal in $\Gamma_{v}$ (or, equivalently, if $E$ is a Cauchy sequence), then it is well-known that $E$ converges to a unique pseudo-limit $\alpha$ in the completion $\widehat{K}$, which in this case is called simply limit.

Following Kaplansky [10], we say that $E$ is of transcendental type if $v(f(s_{\nu}))$ eventually stabilizes for every $f\in K[X]$; on the other hand, if $v(f(s_{\nu}))$ is eventually increasing for some $f\in K[X]$, we say that $E$ is of algebraic type. As we have already remarked in [19], it follows from the work of Kaplansky in [10] that a pseudo-convergent sequence $E\subset K$ is of algebraic type if and only if $E$ admits pseudo-limits in $\overline{K}$, with respect to some extension $u$ of $v$. Note that any pseudo-convergent sequence satisfies either one of these two conditions, because the image of a pseudo-convergent sequence by a polynomial is an eventually pseudo-convergent sequence (see [16, III, §64, p. 371] or Proposition 3.8 below).

2.3.2 Pseudo-divergent sequences

Let $E=\{s_{\nu}\}_{\nu\in\Lambda}$ be a pseudo-divergent sequence in $K$. Symmetrically to the case of pseudo-convergent sequences, for a fixed $\nu$, we have that $v(s_{\rho}-s_{\nu})$ is constant for all $\rho<\nu$; if $\nu$ is not the minimum of $\Lambda$, we denote by $\delta_{\nu}\in\Gamma_{v}$ this value. The sequence $\{\delta_{\nu}\}_{\nu\in\Lambda}$ is a strictly decreasing sequence in $\Gamma_{v}$, called the gauge of $E$.

The breadth ideal $\mathrm{Br}(E)$ of $E$ is the set

[TABLE]

this set is a fractional ideal of $K$ if and only if the gauge of $E$ is bounded from below, while otherwise $\mathrm{Br}(E)=K$. In particular, unlike in the pseudo-convergent case, $\mathrm{Br}(E)$ may not be a fractional ideal. If for each non-minimal $\nu\in\Lambda$ we set $c_{\nu}=s_{\rho}-s_{\nu}$, for some $\rho<\nu$, then $\mathrm{Br}(E)=\bigcup_{\nu\in\Lambda}c_{\nu}V$. Contrary to the case of a pseudo-convergent sequence, it is easily seen that the breadth ideal of a pseudo-divergent sequence is never a principal ideal. However, if $\delta_{\nu}\searrow\delta$, for some $\delta\in\Gamma_{v}$, then $\mathrm{Br}(E)=\{x\in K\mid v(x)>c\}=cM$, where $c\in K$ has value $\delta$. As in the case of a pseudo-convergent sequence, when this condition holds we call $\delta$ the breadth of $F$.

An element $\alpha\in K$ is a pseudo-limit of $E$ if $v(\alpha-s_{\nu})=\delta_{\nu}$ for all (sufficiently large) $\nu\in\Lambda$ or, equivalently, if $v(\alpha-s_{\nu})>v(\alpha-s_{\rho})$ for all (sufficiently large) $\nu<\rho\in\Lambda$. Every element of $E$ is a pseudo-limit of $E$: see [17, §2.1.3] and Lemma 2.5 below.

2.3.3 Pseudo-stationary sequences

Let $E=\{s_{\nu}\}_{\nu\in\Lambda}$ be a pseudo-stationary sequence in $K$. Note that the residue field of $V$ is necessarily infinite (see [17, §2.1.2]). The element $\delta=v(s_{\nu}-s_{\mu})\in\Gamma_{v}$, for $\nu\neq\mu$, is called the breadth of $E$. In analogy with pseudo-convergent and pseudo-divergent sequences, we define the gauge of $E$ to be the constant sequence $\{\delta_{\nu}=\delta\}_{\nu\in\Lambda}$.

The breadth ideal $\mathrm{Br}(E)$ of $E$ is the set

[TABLE]

this set is always a principal fractional ideal of $K$, generated by any $c\in K$ whose value is $\delta$. In particular, we can take $c=s_{\nu^{\prime}}-s_{\nu}$ for any $\nu^{\prime}\neq\nu$.

An element $\alpha\in K$ is a pseudo-limit of $E$ if $v(\alpha-s_{\nu})=\delta$ for all sufficiently large $\nu\in\Lambda$ or, equivalently, if $v(\alpha-s_{\nu})=\delta$ for all but at most one $\nu\in\Lambda$. As in the pseudo-divergent case, every element of $E$ is a pseudo-limit of $E$: see [17, §2.1.2] and Lemma 2.5 below.

2.4 Pseudo-limits and the breadth ideal

In general, if $E\subset K$ is a pseudo-monotone sequence, we denote the set of pseudo-limits of $E$ in $K$ by $\mathcal{L}_{E}$ and the breadth ideal by $\mathrm{Br}(E)$ (or $\mathcal{L}_{E}^{v}$ and $\mathrm{Br}_{v}(E)$, respectively, if we need to underline the valuation). We will constantly use the following trivial remark: if $u$ is an extension of $v$ to an overfield $F$ of $K$, then $E$ is a pseudo-monotone sequence in the valued field $(F,u)$; in particular, $\mathcal{L}_{F}^{u}$ will denote the set of pseudo-limits of $E$ in the valued field $(F,u)$. We use the notation $\mathcal{L}_{E}$ and $\mathrm{Br}(E)$ also in the case $E$ is only eventually pseudo-monotone.

The first part of the next result generalizes the classical result of Kaplansky for pseudo-convergent sequences ([10, Lemma 3]) to pseudo-monotone sequences. The proof is actually the same, but for the sake of the reader we give it here.

Lemma 2.5.

Let $E=\{s_{\nu}\}_{\nu\in\Lambda}\subset K$ be a pseudo-monotone sequence and let $\alpha\in K$ be a pseudo-limit of $E$. Then the set of pseudo-limits $\mathcal{L}_{E}$ of $E$ is equal to $\alpha+\mathrm{Br}(E)$. Moreover, $E\cap\mathcal{L}_{E}=\emptyset$ if $E$ is a pseudo-convergent sequence and $E\subset\mathcal{L}_{E}$ if $E$ is either pseudo-divergent or pseudo-stationary.

Proof.

Let $\beta=\alpha+x$, for some $x\in\mathrm{Br}(E)$. If $E$ is either pseudo-convergent or pseudo-divergent, then it is easy to see that for any $\nu\in\Lambda$ we have

[TABLE]

so that $\beta$ is a pseudo-limit of $E$. If $E$ is pseudo-stationary, then we have $v(\beta-s_{\nu})\geq\delta=v(s_{\nu}-s_{\mu})=v(s_{\nu}-\beta+\beta-s_{\mu})$ and therefore for at most one $\nu\in\Lambda$ we may have the strict inequality $v(\beta-s_{\nu})>\delta$. So, also in this case $\beta$ is a pseudo-limit of $E$.

Conversely, if $\beta$ is a pseudo-limit of $E$, then $v(\alpha-\beta)=v(\alpha-s_{\nu}+s_{\nu}-\beta)\geq\delta_{\nu}$, so that $\alpha-\beta\in\mathrm{Br}(E)$, as we wanted to show.

We prove the last claim. If $E$ is a pseudo-convergent sequence, then it is clear (both if $E$ is of algebraic type or of transcendental type). If the sequence $E$ is either pseudo-divergent or pseudo-stationary, the claim is proved in [17, §2.1.2 & §2.1.3]. ∎

In particular, since pseudo-divergent and pseudo-stationary sequences always admit a pseudo-limit in $K$, in these cases there is no analogue of the notion of pseudo-convergent sequences of transcendental type.

The following result characterizes which fractional ideals of $V$ are breadth ideals for some pseudo-monotone sequence $E$ of $K$, and which cosets are the set of pseudo-limits for some pseudo-monotone sequence.

Lemma 2.6.

Let $I$ be a fractional ideal of $V$ and let $\alpha\in K$; let $\mathcal{L}=\alpha+I$.

1. (a)

$\mathcal{L}=\mathcal{L}_{E}$* for some pseudo-convergent sequence $E$ if and only if $I$ is strictly divisorial; in particular, if the maximal ideal of $V$ is not principal this happens if and only if $I$ is divisorial.* 2. (b)

$\mathcal{L}=\mathcal{L}_{E}$* for some pseudo-divergent sequence if and only if $I$ is not principal.* 3. (c)

If $V/M$ is infinite, $\mathcal{L}=\mathcal{L}_{E}$ for some pseudo-stationary sequence if and only if $I$ is principal.

Proof.

It is easily seen that, if $\mathcal{L}_{E}\neq\emptyset$ for some pseudo-monotone sequence $E=\{s_{\nu}\}_{\nu\in\Lambda}$, for every $\beta\in K$ the set $\beta+\mathcal{L}_{E}$ is the set of pseudo-limits of $\beta+E=\{\beta+s_{\nu}\}_{\nu\in\Lambda}$; hence, it is enough to prove the claims for $\alpha=0$. Furthermore, by Lemma 2.5, under this hypothesis we have $\mathcal{L}_{E}=\mathrm{Br}(E)$, and thus we only need to find which ideals are breadth ideals.

If $I=\mathrm{Br}(E)$ for some pseudo-convergent $E=\{s_{\nu}\}_{\nu\in\Lambda}$, for each $\nu$ let $c_{\nu}=s_{\rho}-s_{\nu}$, for some $\rho>\nu$; then $I=\bigcap_{\nu}c_{\nu}V$, and each $c_{\nu}V$ properly contains $I$. Therefore $I$ is a strictly divisorial ideal. Conversely, if $I=\bigcap_{a\in A}aV$, where for each $a\in A$ we have $I\subsetneq aV$, we can take a well-ordered subset $\{a_{\nu}\}_{\nu\in\Lambda}$ such that $I=\bigcap_{\nu}a_{\nu}V$ and $a_{\rho}V\subsetneq a_{\nu}V$ for all $\rho>\nu$; then, $\{a_{\nu}\}_{\nu\in\Lambda}$ is a pseudo-convergent sequence having 0 as a pseudo-limit and breadth ideal $I$. The last remark follows from Lemma 2.2.

Likewise, if $I=\mathrm{Br}(E)$ for some pseudo-divergent $E=\{s_{\nu}\}_{\nu\in\Lambda}$, for each $\nu$ let $c_{\nu}=s_{\rho}-s_{\nu}$, for some $\rho<\nu$; then $I=\bigcup_{\nu}c_{\nu}V$, while if $I$ is not principal we can find a well-ordered sequence $E=\{a_{\nu}\}_{\nu\in\Lambda}\subset V$ which generates $I$ and such that $a_{\nu}V\subset a_{\rho}V$ for every $\nu<\rho$, so that $E$ is a pseudo-divergent sequence and $I$ is its breadth ideal.

If $I=\mathrm{Br}(E)$ for some pseudo-stationary sequence $E=\{s_{\nu}\}_{\nu\in\Lambda}$, then $I=(s_{\nu}-s_{\mu})V$, for any $\nu\not=\mu$; conversely, if $I=cV$, then we can find a well-ordered set $E=\{s_{\nu}\}_{\nu\in\Lambda}$ of distinct elements of valuation $v(c)$ whose cosets modulo $cM$ are different (because the residue field of $V$ is infinite); then, $E$ is pseudo-stationary with breadth ideal $E$. ∎

3 Valuation domains associated to pseudo-monotone sequences

Let $\phi\in K(X)$ be a rational function: if $\alpha\in\overline{K}$ is a zero or a pole of $\phi$, we say that $\alpha$ is a critical point of $\phi$. We denote by $\Omega_{\phi}$ the multiset of critical points of $\phi$. Let $S=\{\alpha_{1},\ldots,\alpha_{k}\}$ be a submultiset of $\Omega_{\phi}$. The weighted sum of $S$ is the sum $\sum_{\alpha_{i}\in S}\epsilon_{i}$, where $\epsilon_{i}=1$ if $\alpha_{i}$ is a zero of $\phi$ and $\epsilon_{i}=-1$ if $\alpha_{i}$ is a pole of $\phi$. The $S$-part of $\phi$ is the rational function $\phi_{S}(X)=\prod_{\alpha_{i}\in S}(X-\alpha_{i})^{\epsilon_{i}}$, where $\epsilon_{i}$ is as above.

The following definition generalizes [19, Definition 3.5] to pseudo-monotone sequences.

Definition 3.1**.**

Let $E=\{s_{\nu}\}_{\nu\in\Lambda}$ be a pseudo-monotone sequence in $K$, let $u$ be an extension of $v$ to $\overline{K}$ and let $\phi\in K(X)$. The dominating degree $\mathrm{degdom}_{E,u}(\phi)$ of $\phi$ with respect to $E$ and $u$ is the weighted sum of the elements of $\Omega_{\phi}$ which are pseudo-limits of $E$ with respect to $u$.

The next proposition is a generalization to pseudo-monotone sequences of [19, Theorem 3.3]; in particular, it shows that the dominating degree does not depend on the chosen extension of $v$ to $\overline{K}$.

Proposition 3.2.

Let $E=\{s_{\nu}\}_{\nu\in\Lambda}\subset K$ be a pseudo-monotone sequence of gauge $\{\delta_{\nu}\}_{\nu\in\Lambda}$, and let $\phi\in K(X)$. Let $u$ be an extension of $v$ to $\overline{K}$ and let $\lambda=\mathrm{degdom}_{E,u}(\phi)$. Then there exist $\gamma\in\Gamma_{v}$ and $\nu_{0}\in\Lambda$ such that for each $\nu\geq\nu_{0}$ we have

[TABLE]

Furthermore, if $\beta\in\overline{K}$ is a pseudo-limit of $E$ with respect to $u$, then $\gamma=u\left(\frac{\phi}{\phi_{S}}(\beta)\right)$, where $S$ is the set of critical points of $\phi$ which are pseudo-limits of $E$ with respect to $u$.

Moreover, the dominating degree of $\phi$ does not depend on $u$; that is, if $u^{\prime}$ is another extension of $v$ to $\overline{K}$, then $\mathrm{degdom}_{E,u}(\phi)=\mathrm{degdom}_{E,u^{\prime}}(\phi)$.

Proof.

If $E$ is a pseudo-convergent sequence, then the statement is the same as in [19, Proposition 3.6].

If the sequence $E$ is pseudo-divergent, then the proof is essentially the same as when $E$ is pseudo-convergent: let $\beta\in K$ be a pseudo-limit of $E$ and let $\Delta=\Delta_{E}$ be the least final segment of $\mathbb{Q}\Gamma_{v}$ containing the gauge of $E$ (if $\mathrm{Br}(E)=K$, just take $\Delta=\Gamma_{v}$). Take $\tau\in\Gamma_{v}\cap\Delta$ such that $C=\{s\in\overline{K}\mid u(s-\beta)\in\Delta\cap(-\infty,\tau)\}$ contains no critical points of $\phi$. Then, $s_{\nu}\in C$ for all large $\nu$ and, by construction, the weighted sum of the subset $S$ of $\Omega_{\phi}$ of those elements $\alpha$ such that $u(\alpha-\beta)>\Delta\cap(-\infty,\tau)$ is exactly $\lambda=\mathrm{degdom}_{E,u}(\phi)$. Therefore, we can apply [19, Theorem 3.3] to the convex set $\Delta\cap(-\infty,\tau)$, and thus there is a $\nu_{0}\in\Lambda$ such that for each $\nu\geq\nu_{0}$ we have

[TABLE]

where $\gamma=u\left(\frac{\phi}{\phi_{S}}(\beta)\right)$, as in the statement of the proposition, again by [19, Theorem 3.3]. Since $v(\phi(s_{\nu}))\in\Gamma_{v}$ and does not depend on $\beta$, the same happens for $\gamma$. For the final claim the proof is analogous to [19, Proposition 3.6(c)].

If $E$ is pseudo-stationary, we cannot apply directly [19, Theorem 3.3], but the same general method works: let $\phi\in K(X)$ and write $\phi(X)=c\prod_{i=1}^{n}(X-\alpha_{i})^{\epsilon_{i}}$, where $c\in K$, $\alpha_{i}\in\overline{K}$ and $\epsilon_{i}\in\{1,-1\}$. Let $u$ be a fixed extension of $v$ to $\overline{K}$, let $\beta\in K$ be a pseudo-limit of $E$ and let $S$ be the multiset of critical points of $\phi$ which are pseudo-limits of $E$ with respect to $u$. If $\alpha\in\Omega_{\phi}\setminus S$, then $u(s_{\nu}-\alpha)=u(\beta-\alpha)<\delta$ for all sufficiently large $\nu\in\Lambda$; on the other hand, if $\alpha\in S$, then there is at most one $\nu$ (say $\nu_{0}$) such that $u(s_{\nu_{0}}-\alpha)>\delta$, while $u(s_{\nu}-\alpha)=\delta$ for all $\nu\neq\nu_{0}$. Hence, for all large $\nu$ we have $u(s_{\nu}-\alpha)=\delta$. Note that, if $\alpha\notin S$, then $u(\beta-\alpha)$ does not depend on the chosen pseudo-limit $\beta$ of $E$. In particular, $u(s_{\nu}-\alpha)\leq\delta$ and equality holds if and only if $\alpha$ is a pseudo-limit of $E$, in complete analogy with [19, Remark 4.7(a)]. Now, let $\lambda$ be the weighted sum of $S$ (which is equal to $\mathrm{degdom}_{E,u}(\phi)$) and $\gamma=u\left(\frac{\phi}{\phi_{S}}(\beta)\right)$: then, for all large $\nu$, $s_{\nu}$ is not a critical point of $\phi$ and we have

[TABLE]

It is clear as before that $\gamma\in\Gamma_{v}$ and does not depend on the chosen pseudo-limit $\beta$ of $E$, by the above remark. To conclude, we only need to prove that the dominating degree of $\phi$ with respect to a pseudo-stationary sequence $E$ does not depend on the extension of $v$ to $\overline{K}$. Let $u,u^{\prime}$ be two extensions of $v$ to $\overline{K}$. By Lemma 2.5, it follows that $\mathcal{L}_{E}=s+cV$, where a pseudo-limit $s$ of $E$ can be chosen in $K$ and $c\in K$ has value $\delta_{E}$. Now, by the same Lemma we also have that $\mathcal{L}_{E}^{u}=s+cU$ and $\mathcal{L}_{E}^{u^{\prime}}=s+cU^{\prime}$; in particular, $\mathcal{L}_{E}^{u}$ and $\mathcal{L}_{E}^{u^{\prime}}$ are conjugate under the action of the Galois group of $\overline{K}$ over $K$. It is then clear that $\Omega_{\phi}\cap\mathcal{L}_{E}^{u}$ and $\Omega_{\phi}\cap\mathcal{L}_{E}^{u^{\prime}}$ are conjugate too, so $\mathrm{degdom}_{E,u}(\phi)=\mathrm{degdom}_{E,u^{\prime}}(\phi)$, as desired ∎

Note that by Proposition 3.2 we may drop the suffix $u$ in the dominating degree of a rational function. However, note that given a pseudo-monotone sequence $E\subset K$ without pseudo-limits in $K$, different extensions of $v$ to $\overline{K}$ give rise to different set of pseudo-limits, which are conjugate under the action of the Galois group of $\overline{K}$ over $K$.

Moreover, if $E=\{s_{\nu}\}_{\nu\in\Lambda}$ is a pseudo-stationary sequence and $\phi\in K(X)$, the values of $\phi$ on $E$ are eventually constant, namely $v(\phi(s_{\nu}))=\lambda\delta+\gamma$, where $\lambda=\mathrm{degdom}_{E}(\phi)$, $\delta=\delta_{E}$ and $\gamma\in\Gamma_{v}$, for all sufficiently large $\nu$.

Definition 3.3**.**

Let $E=\{s_{\nu}\}_{\nu\in\Lambda}\subset K$ be a pseudo-monotone sequence. We define

[TABLE]

Theorem 3.4.

Let $E=\{s_{\nu}\}_{\nu\in\Lambda}\subset K$ be a pseudo-monotone sequence. Then $V_{E}$ is a valuation domain with maximal ideal

[TABLE]

Proof.

The proof is exactly as the one of [19, Theorem 3.8], but we repeat it here for completeness.

The set $V_{E}$ is a ring since if $\phi(s_{\nu}),\psi(s_{\nu})\in V$ for all sufficiently large $\nu$, then also $(\phi+\psi)(s_{\nu})$ and $(\phi\psi)(s_{\nu})$ are eventually in $V$.

Let $\phi\in K(X)$. By Proposition 3.2, we have $v(\phi(s_{\nu}))=\lambda\delta_{\nu}+\gamma$, for all $\nu\in\Lambda$ sufficiently large, for some $\lambda\in\mathbb{Z}$ and $\gamma\in\Gamma_{v}$. In particular, the values of $\phi$ over $E$ are either eventually positive, eventually negative or eventually constant, so either $\phi(s_{\nu})\in V$ or $\phi(s_{\nu})^{-1}=\phi^{-1}(s_{\nu})\in V$ (in both cases for all $\nu\in\Lambda$ sufficiently large), which shows that $V_{E}$ is a valuation domain.

The claim about the maximal ideal of $V_{E}$ follows immediately. ∎

We call $V_{E}$ the extension of $V$ associated to the pseudo-monotone sequence $E$. Note that, if $E$ is a pseudo-convergent sequence and its gauge is cofinal in $\Gamma_{v}$ (or, equivalently, $E$ is a Cauchy sequence), then $V_{E}=V_{\alpha}=\{\phi\in K(X)\mid\phi(\alpha)\in\widehat{V}\}$, where $\alpha$ is the (unique) limit of $E$ in the completion $\widehat{K}$. See [18] for a study of these valuation domains.

The main properties of the valuation domain $V_{E}$ and its associated valuation $v_{E}$ are summarized in Proposition 3.7 below, which is a generalization of [19, Proposition 3.11]. We need to introduce another definition.

Definition 3.5**.**

Let $E\subset K$ be a pseudo-monotone sequence. We denote by $\mathcal{P}_{E}$ the set of the irreducible monic polynomials $p\in K[X]$ which have at least one root in $\overline{K}$ which is a pseudo-limit of $E$ (with respect to some extension of $v$ to $\overline{K}$), or, equivalently, such that $\mathrm{degdom}_{E}(p)>0$.

We note that $\mathcal{P}_{E}$ is nonempty if and only if $E$ has a pseudo-limit in $\overline{K}$; that is, $\mathcal{P}_{E}$ is empty if and only if $E$ is a pseudo-convergent sequence of transcendental type. If $E$ is a pseudo-convergent sequence of algebraic type which is also a Cauchy sequence, then $\mathcal{P}_{E}$ contains a unique element, namely the minimal polynomial of the (unique) limit of $E$ in $\widehat{K}$ (and by Lemma 2.5 this is the only case in which $\mathcal{P}_{E}$ has only one element).

Lemma 3.6.

Let $E$ be a strictly pseudo-monotone sequence having a pseudo-limit in $\overline{K}$, and let $p\in K[X]$. Then:

1. (a)

$v_{E}(p)\notin\Gamma_{v}$* if and only if some irreducible factor of $p$ is in $\mathcal{P}_{E}$;* 2. (b)

if $v_{E}(p)\notin\Gamma_{v}$, then $v_{E}(p)$ is not torsion over $\Gamma_{v}$; 3. (c)

if $p_{1},p_{2}\in\mathcal{P}_{E}$ are of minimal degree, then $v_{E}(p_{1})=v_{E}(p_{2})$.

Proof.

Let $p\in K[X]$. Then, $v_{E}(p)=v(t)$ for some $t\in K$ if and only if $v(t)=v(p(s_{\nu}))=\mathrm{degdom}_{E}(p)\delta_{\nu}+\gamma$ for all $\nu$ sufficiently large (Proposition 3.2); since $E$ is strictly pseudo-monotone, it follows that $v_{E}(p)\in\Gamma_{v}$ if and only if $\mathrm{degdom}_{E}(p)=0$. Since $\mathrm{degdom}_{E}(q_{1}\cdots q_{n})=\sum_{i}\mathrm{degdom}_{E}(q_{i})$, a follows.

b is a consequence of the previous point applied to the powers $p^{n}$ of $p$.

Finally, if $p_{1},p_{2}\in\mathcal{P}_{E}$ are polynomials of minimal degree, then $p_{1}-p_{2}=r$ for some $r\in K[X]$ of lower degree, because $p_{1},p_{2}$ are monic; by minimality, no factor of $r$ belongs to $\mathcal{P}_{E}$, and so $v_{E}(r)\in\Gamma_{v}$. Hence, it must be $v_{E}(p_{1})=v_{E}(p_{2})$ (otherwise $v_{E}(p_{1}-p_{2})=\min\{v_{E}(p_{1}),v_{E}(p_{2})\}$ which is not in $\Gamma_{v}$), and c holds. ∎

Proposition 3.7.

Let $E=\{s_{\nu}\}_{\nu\in\Lambda}\subset K$ be a pseudo-monotone sequence. If $\mathcal{P}_{E}$ is nonempty, we let $\Delta_{E}=v_{E}(p)$ for some $p\in\mathcal{P}_{E}$ of minimal degree.

1. (a)

If $E$ is either pseudo-convergent of algebraic type or pseudo-divergent, then $\Gamma_{v_{E}}=\Delta_{E}\mathbb{Z}\oplus\Gamma_{v}$ (as groups) and $V_{E}/M_{E}\cong V/M$. 2. (b)

If $E$ is pseudo-convergent of transcendental type, then $V\subset V_{E}$ is immediate. 3. (c)

If $E$ is pseudo-stationary, then $\Gamma_{v_{E}}=\Gamma_{v}$ and $V_{E}/M_{E}$ is a purely transcendental extension of $V/M$: more precisely, $V_{E}/M_{E}=V/M(t)$, where $t$ is the residue of $\frac{X-\alpha}{c}$ modulo $M_{E}$, where $c\in K$ satisfies $v(c)=\delta_{E}$ and $\alpha\in\mathcal{L}_{E}$. 4. (d)

If $E$ is not pseudo-convergent of transcendental type, then $\Gamma_{v_{E}}=\langle\Gamma_{v},\Delta_{E}\rangle$. Furthermore, $\Delta_{E}$ does not depend on $p$ and, if $E$ is a pseudo-stationary sequence, $\Delta_{E}=\delta_{E}$. 5. (e)

If $E$ has a pseudo-limit $\beta\in K$, then $v_{E}=v_{\beta,\Delta_{E}}$.

Proof.

a In both cases, $E$ has a pseudo-limit in $\overline{K}$ with respect to some extension of $v$, and so $\mathcal{P}_{E}\neq\emptyset$. Fix a polynomial $p\in\mathcal{P}_{E}$ of minimal degree, and let $\Delta_{E}=v_{E}(p)$, which does not depend on $p$ and is not torsion over $\Gamma_{v}$ by Lemma 3.6. For every $q\in K[X]$, we can write $q=r_{0}+r_{1}p+r_{2}p^{2}+\cdots+r_{n}p^{n}$, for some (uniquely determined) $r_{0},\ldots,r_{n}\in K[X]$ such that $\deg r_{i}<\deg p$. Since $\Delta_{E}$ is not torsion over $\Gamma_{v}$ and $v_{E}(r_{i})\in\Gamma_{v}$ for each $i$ by minimality of the degree of $p$, we have

[TABLE]

for every $i\neq j$; therefore, $v_{E}(q)=\min\{v_{E}(r_{0}),v_{E}(r_{1}p),\ldots,v_{E}(r_{n}p^{n})\}$, and in particular $v_{E}(q)\in\Gamma_{v}\oplus\Delta_{E}\mathbb{N}$. Hence, $\Gamma_{v_{E}}=\Gamma_{v}\oplus\Delta_{E}\mathbb{Z}$.

We now show that $V_{E}/M_{E}=V/M$. If $E$ has a pseudo-limit $\alpha$ in $K$, then as in [19, Proposition 3.11] by Lemma 3.6 we have $v_{E}=v_{\alpha,\Delta_{E}}$ and by [4, Chap. VI, §10, 1., Proposition 1] $V_{E}$ and $V$ have the same residue field. Suppose instead $\mathcal{L}_{E}=\emptyset$ (in particular, $E$ must be a pseudo-convergent sequence, by Lemma 2.5), and let $\phi$ be a unit of $V_{E}$. Let $u$ be an extension of $v$ to $\overline{K}$ and let $\alpha\in\mathcal{L}_{E}^{u}$. Then, the residue field of $U_{E}$ is equal to the residue field of $U$ (by the previous case); hence, there is a unit $\beta$ of $U$ such that $\phi-\beta\in M_{U_{E}}$. Thus, $\phi(s_{\nu})\in\beta+M_{U}$ for all $\nu$ bigger or equal than some $N\in\Lambda$.

Since $\phi$ is a unit of $V_{E}$, $\phi(s_{\nu})$ is a unit of $V$ for all large $\nu$; without loss of generality, for $\nu\geq N$. Let $a$ be such that $\phi(s_{N})\in a+M$: then, for every $\nu>N$ we have $\phi(s_{\nu})-\phi(s_{N})\in M_{U}\cap V=M$, and thus also $\phi(s_{\nu})\in a+M$. Hence, the image of $\phi$ is in $V/M$, and so $V/M=V_{E}/M_{E}$. The claim is proved.

b This follows from Kaplansky’s results in [10].

c Suppose that $E$ is a pseudo-stationary sequence. It is clear that, without loss of generality, we may suppose that $K$ is algebraically closed. In order to prove the claim, by [16, §11, IV, p. 366] it is sufficient to show that $v_{E}(X-\alpha-\beta)=\min\{v_{E}(X-\alpha),v(\beta)\}$ for each $\beta\in K$. By Proposition 3.2, we have $v_{E}(X-\alpha)=\delta$. If $\delta\not=v(\beta)$ we are done. If $\delta=v(\beta)$, then by Lemma 2.5, $\alpha+\beta$ is a pseudo-limit of $E$, so again by Proposition 3.2 we have $v_{E}(X-\alpha-\beta)=\delta$.

d For pseudo-convergent sequences of algebraic type or pseudo-divergent sequences the claim follows from the proof of part a. For a pseudo-stationary sequence $E$, $\Delta_{E}=v_{E}(X-\alpha)=\delta_{E}$ for all pseudo-limits $\alpha\in\mathcal{L}_{E}$, and we are done. e follows in the same way. ∎

The next proposition constitutes an important generalization of [10, Lemma 5] and [16, III, §64, p. 371], which says that the image under a polynomial of a pseudo-convergent sequence is an eventually pseudo-convergent sequence.

Proposition 3.8.

Let $E=\{s_{\nu}\}_{\nu\in\Lambda}\subset K$ be a strictly pseudo-monotone sequence and let $\phi\in K(X)$ be non-constant. Then $\phi(E)=\{\phi(s_{\nu})\}_{\nu\in\Lambda}$ is an eventually strictly pseudo-monotone sequence, which is of the same kind of $E$ if $\mathrm{degdom}_{E}(\phi)>0$, and not of the same kind if $\mathrm{degdom}_{E}(\phi)<0$; if $\mathrm{degdom}_{E}(\phi)\not=0$, then $\mathcal{L}_{\phi(E)}=\mathrm{Br}(\phi(E))$. Furthermore, if $\phi(E)$ is eventually pseudo-convergent, then $\phi(X)$ is a pseudo-limit of $\phi(E)$ with respect to $v_{E}$.

Proof.

Let $\lambda=\mathrm{degdom}_{E}(\phi)$. Suppose first that $\lambda>0$ and $E$ is a pseudo-convergent sequence. By Proposition 3.2 we have $v(\phi(s_{\nu}))=\lambda\delta_{\nu}+\gamma<v(\phi(s_{\mu}))=\lambda\delta_{\mu}+\gamma$ for all $\nu<\mu$ sufficiently large (say greater than some $\nu_{0}\in\Lambda$), which shows that $\phi(E)$ is an eventually pseudo-convergent sequence with gauge $\{\lambda\delta_{\nu}+\gamma\}_{\nu\in\Lambda}$. Since $v(\phi(s_{\nu}))$ increases, [math] is a pseudo-limit of $\phi(E)$, and thus by Lemma 2.5 we have the equality $\mathcal{L}_{\phi(E)}=\mathrm{Br}(\phi(E))$. Since $v(\phi(s_{\rho}))>v(\phi(s_{\nu}))$ if $\rho>\nu$ (sufficiently large), we have $v_{E}\left(\frac{\phi(X)}{\phi(s_{\nu})}\right)>0$ for all $\nu$ sufficiently large; hence, eventually, $v_{E}(\phi(X)-\phi(s_{\nu}))=v_{E}(\phi(s_{\nu}))$, and in particular $\{v_{E}(\phi(X)-\phi(s_{\nu}))\}_{\nu\in\Lambda}$ is strictly increasing. Hence, $\phi(X)$ is a pseudo-limit of $\phi(E)$.

If $\lambda>0$ and $E$ is a pseudo-divergent sequence, then as above $\phi(E)$ is eventually pseudo-divergent. If $\lambda<0$, then in the same way we can prove that $\phi(E)$ is strictly pseudo-monotone, not of the same kind of $E$, and $\phi(X)$ is a pseudo-limit of $\phi(E)$ with respect to $v_{E}$.

Suppose now that $\lambda=0$ and $E$ is a pseudo-convergent sequence. Without loss of generality, we may also suppose that $K=\overline{K}$. Let $\phi(X)=p(X)/q(X)$, where $p,q\in K[X]$. Since $K$ is algebraically closed, we can write $q(X)=q_{1}(X)q_{2}(X)$ in such a way that all zeros of $q_{1}$ are pseudo-limits of $E$ while no zero of $q_{2}$ is a pseudo-limit of $E$ (if $E$ has no pseudo-limits, then $q(X)=q_{2}(X)$ and $q_{1}(X)=1$). In particular, $\deg q_{1}=\mathrm{degdom}_{E}(q_{1})$. Dividing $p$ by $q_{1}$, we have

[TABLE]

where $a,b\in K[X]$ and $\deg b<\deg q_{1}$. The rational function $\phi_{2}(X)=\frac{b(X)}{q(X)}$ has dominating degree

[TABLE]

and thus, by the previous part of the proof, $\{\phi_{2}(s_{\nu})\}_{\nu\in\Lambda}$ is an eventually pseudo-divergent sequence.

Consider now $\phi_{1}(X)=\frac{a(X)}{q_{2}(X)}$. If $E$ has a pseudo-limit in $K=\overline{K}$, let $\alpha\in\mathcal{L}_{E}$. If not, then $E$ is a pseudo-convergent sequence of transcendental type, and we can extend $v$ to a transcendental extension $K(z)$ of $K$ such that $z$ is a pseudo-limit of $E$ ([10, Theorem 2]), and we set $\alpha=z$; with a slight abuse of notation, we still denote by $v$ this extension to $K(z)$. Note that in any case $q_{2}(\alpha)\neq 0$ since $\mathrm{degdom}_{E}(q_{2})=0$. Consider the following rational function over $K(\alpha)$:

[TABLE]

Since $\psi(\alpha)=0$, the dominating degree of the numerator of $\psi$ is positive; on the other hand, $\mathrm{degdom}_{E}(q_{2}(\alpha)q_{2})=\mathrm{degdom}_{E}(q_{2})=0$. Hence, $\mathrm{degdom}_{E}(\psi)>0$, and by the previous part of the proof $\{\psi(s_{\nu})\}_{\nu\in\Lambda}$ is an eventually pseudo-convergent sequence in $K(\alpha)$. Thus, also $\{\phi_{1}(s_{\nu})\}_{\nu\in\Lambda}=\{\psi(s_{\nu})+\phi_{1}(\alpha)\}_{\nu\in\Lambda}$ is eventually pseudo-convergent in $K(\alpha)$; however, $\phi_{1}(s_{\nu})\in K$ for every $\nu$, and thus $\{\phi_{1}(s_{\nu})\}_{\nu\in\Lambda}$ is a eventually pseudo-convergent sequence in $K$.

By definition, $\phi(s_{\nu})=\phi_{1}(s_{\nu})+\phi_{2}(s_{\nu})$ and, by the previous points, the sequences $\{\phi_{1}(s_{\nu})\}_{\nu\in\Lambda}$ and $\{\phi_{2}(s_{\nu})\}_{\nu\in\Lambda}$ are eventually pseudo-convergent and eventually pseudo-divergent, respectively. In particular, for large $\nu$, $v(\phi_{1}(s_{\rho})-\phi_{1}(s_{\nu}))$, $\rho>\nu$, is increasing and $v(\phi_{2}(s_{\rho})-\phi_{2}(s_{\nu}))$, $\rho>\nu$, is decreasing; it follows that $v(\phi(s_{\rho})-\phi(s_{\nu}))$, $\rho>\nu$, is eventually equal to one of the two. Hence, $\phi(s_{\nu})$ is eventually strictly pseudo-monotone, as claimed.

Suppose in particular that $\phi(E)$ is eventually pseudo-convergent: then,

[TABLE]

By the case $\lambda>0$, we have $v_{E}((\phi_{1}(X)-\phi_{1}(s_{\nu}))=v_{E}(\phi_{1}(s_{\nu}))$ for all large $\nu$. On the other hand, since $\phi(E)$ is pseudo-convergent we have $v_{E}(\phi_{1}(s_{\nu}))<v_{E}(\phi_{2}(s_{\rho}))$ for all large $\nu<\rho$; in particular, we also have $v_{E}(\phi_{2}(X))\geq v_{E}(\phi_{1}(X))$ and so $v_{E}(\phi_{2}(X)-\phi_{2}(s_{\nu}))$ is bigger than both $v_{E}(\phi_{1}(X))$ and $v_{E}(\phi_{1}(s_{n}))$. Hence,

[TABLE]

which is eventually strictly increasing. Hence, $\phi(X)$ is a pseudo-limit of $\phi(E)$ with respect to $v_{E}$, as claimed.

If $E$ is pseudo-divergent, the same reasoning applies (with the only difference that $\phi_{1}(E)$ will be pseudo-divergent and $\phi_{2}(E)$ pseudo-convergent). ∎

4 Extensions

We now start the proof of our generalization of Ostrowski’s Fundamentalsatz (Theorem 6.2): we want to show that, under some hypothesis, we can obtain every extension $W$ of $V$ to $K(X)$ as a valuation domain $V_{E}$ associated to a pseudo-monotone sequence $E$ contained in $K$. In order to accomplish this objective, we want to associate to each such extension $W$ a subset of $K$ which is the analogue of the set of pseudo-limits of a pseudo-monotone sequence.

Definition 4.1**.**

Let $W$ be an extension of $V$ to $K(X)$. We define the following subsets of $K$:

[TABLE]

Equivalently, $\alpha\in\mathcal{L}_{2}(W)$ if $w(X-\alpha)=v(c)$ for some $c\in K$, and the image of $\frac{X-\alpha}{c}$ in the residue field of $W$ does not belong to the residue field of $V$.

Proposition 4.2.

Let $W$ be an extension of $V$ to $K(X)$.

1. (a)

Suppose $K$ is algebraically closed. Then $V\subset W$ is immediate if and only if $\mathcal{L}(W)=\emptyset$. 2. (b)

If $\alpha\in\mathcal{L}(W)$, then $w(X-\alpha)\geq w(X-\beta)$ for each $\beta\in K$, and equality occurs if and only if $\beta\in\mathcal{L}(W)$. 3. (c)

If $\mathcal{L}(W)\neq\emptyset$, then exactly one of $\mathcal{L}_{1}(W)$ and $\mathcal{L}_{2}(W)$ is nonempty. 4. (d)

If $\mathcal{L}_{1}(W)\neq\emptyset$ is nonempty, then it is equal to $K$ or to $\alpha+I$ for some $\alpha\in K$ and some (fractional) ideal $I$. 5. (e)

If $\mathcal{L}_{2}(W)\neq\emptyset$ is nonempty, then it is equal to $\alpha+cV$ for some $\alpha,c\in K$ with $v(c)=w(X-\alpha)$.

Note that (b) above is a generalization of [19, Proposition 3.11, (a)].

Proof.

a Suppose $K$ is algebraically closed. If $V\subset W$ is immediate, then $\Gamma_{w}=\Gamma_{v}$ (so $\mathcal{L}_{1}(W)=\emptyset$); furthermore, since $W/M_{W}=V/M$, also $\mathcal{L}_{2}(W)=\emptyset$. Conversely, suppose that $V\subset W$ is not immediate. If $\Gamma_{v}\neq\Gamma_{w}$, then $w(p)\notin\Gamma_{v}$ for some $p\in K[X]$, and thus $w(p^{\prime})\notin\Gamma_{v}$ for some irreducible factor $p^{\prime}$ of $p$; since $K$ is algebraically closed, $p^{\prime}(X)=X-\alpha$ and $\alpha\in\mathcal{L}_{1}(W)$. If $\Gamma_{v}=\Gamma_{w}$, then $V/M\subsetneq W/M_{W}$ and this extension must be transcendental (since $K$ is algebraically closed, so is $V/M$). By the proof of [1, Proposition 2], we can find $\alpha,c\in K$ such that $w(X-\alpha)=v(c)$ and the image of $\frac{X-\alpha}{c}$ is transcendental over $V/M$; it follows that $\alpha\in\mathcal{L}_{2}(W)$, which in particular is nonempty.

b-e If $\mathcal{L}(W)\neq\emptyset$ and $\mathcal{L}(W)\neq K$, let $\alpha\in\mathcal{L}(W)$. Then, if $\beta\in K$ we have:

[TABLE]

Suppose $\alpha\in\mathcal{L}_{1}(W)$. Since $w(X-\beta)$ is equal either to $w(X-\alpha)\notin\Gamma_{v}$ or to $v(\alpha-\beta)$, in the former case $\beta\in\mathcal{L}_{1}(W)$, while in the latter $\beta\notin\mathcal{L}_{1}(W)$ and $w(X-\beta)<w(X-\alpha)$. Moreover, $\mathcal{L}_{1}(W)=\alpha+\{x\in K\mid v(x)>w(X-\alpha)\}$, and the latter set is an ideal.

If $\alpha\in\mathcal{L}_{2}(W)$ and $v(\alpha-\beta)\geq w(X-\alpha)$, then $w(X-\beta)=w(X-\alpha)=v(c)\in\Gamma_{v}$, for some $c\in K$, so $\beta\in\mathcal{L}_{2}(W)$ because $(X-\beta)/c=(X-\alpha)/c+(\beta-\alpha)/c$: over the residue field of $W$ $(X-\alpha)/c$ is not in $V/M$ so it follows that the same holds for $(X-\beta)/c$. Similarly, if $\beta\in\mathcal{L}_{2}(W)$ it can be proved that $v(\alpha-\beta)\geq w(X-\alpha)$ and so $w(X-\beta)=w(X-\alpha)$. If $v(\alpha-\beta)<w(X-\alpha)$, then as before $w(X-\beta)<w(X-\alpha)$. In particular, $\mathcal{L}_{2}(W)=\alpha+\{x\in K\mid v(x)\geq w(X-\alpha)\}=\alpha+cV$, and $\mathcal{L}_{1}(W)=\emptyset$ (because $\alpha\in\mathcal{L}_{2}(W)$ and (1)). Note that this argument shows that at most one of the sets $\mathcal{L}_{i}(W)$, $i=1,2$, can be non-empty.

In all cases, $w(X-\alpha)\geq w(X-\beta)$ for all $\alpha\in\mathcal{L}(W)$ and $\beta\in K$, and equality occurs if and only if $\beta\in\mathcal{L}(W)$. ∎

Proposition 4.3.

Let $E\subset K$ be a pseudo-monotone sequence.

1. (a)

If $E$ is a strictly pseudo-monotone sequence, then $\mathcal{L}_{1}(V_{E})=\mathcal{L}_{E}$. 2. (b)

If $E$ is pseudo-stationary, then $\mathcal{L}_{2}(V_{E})=\mathcal{L}_{E}$.

In both cases, $\mathcal{L}(V_{E})$ is the set of pseudo-limits of $E$ in $K$.

Proof.

Suppose first that $E=\{s_{\nu}\}_{\nu\in\Lambda}$ is a strictly pseudo-monotone sequence. Let $\alpha\in K$, and suppose $w(X-\alpha)=w(c)$ for some $c\in K$. Then, $(X-\alpha)/c$ is a unit of $W$, and in particular for large $\nu$ both $(s_{\nu}-\alpha)/c$ and $c/(s_{\nu}-\alpha)$ belong to $V$. Therefore, $v(s_{\nu}-\alpha)=v(c)$ for large $\nu$; hence, $w(X-\alpha)\in\Gamma_{v}$ if and only if $\alpha\notin\mathcal{L}_{E}$. Thus, $\mathcal{L}_{1}(V_{E})=\mathcal{L}_{E}$; furthermore, by Proposition 3.7a, $V_{E}/M_{E}=V/M$, and so $\mathcal{L}_{2}(V_{E})=\emptyset$. Hence, $\mathcal{L}(V_{E})=\mathcal{L}_{E}$.

Suppose now that $E$ is pseudo-stationary: then, by Proposition 3.7e, $v_{E}=v_{\alpha,\delta_{E}}$. By Proposition 4.2e, $\mathcal{L}(V_{E})=\mathcal{L}_{2}(V_{E})=\alpha+cV$, where $c\in K$ has value $v_{E}(X-\alpha)=\delta_{E}$. By Lemma 2.5 this is precisely $\mathcal{L}_{E}$. ∎

Example 4.4**.**

Proposition 4.3 allows to show that there are extensions of $V$ to $K(X)$ which cannot be realized as $V_{E}$, for any pseudo-convergent sequence $E\subset K$. For example, consider the following valuation domain of $K(X)$ introduced in [18]:

[TABLE]

where $\phi(\infty)$ is defined as $\psi(0)$, where $\psi(X)=\phi(1/X)$. Then, $V_{\infty}$ is the image of $V_{0}=\{\phi\in K(X)\mid\phi(0)\in V\}$ under the $K$-automorphism $\Phi$ of $K(X)$ sending $X$ to $1/X$. The valuation domain $V_{0}$ is equal to $V_{F}$, where $F=\{t_{\nu}\}_{\nu\in\Lambda}$ is a Cauchy sequence with limit [math]. Consider $E=\{s_{\nu}=t_{\nu}^{-1}\}_{\nu\in\Lambda}$: by Proposition 3.8, $E$ is pseudo-divergent with $\mathrm{Br}(E)=K$ (since, as $v(t_{\nu})$ is cofinal in $\Gamma_{v}$, $v(s_{\nu})$ is coinitial). Thus, $V_{\infty}=V_{E}$ has $\mathcal{L}(V_{\infty})=\mathcal{L}_{E}=K$, which is different from $\mathcal{L}(V_{G})=\mathcal{L}_{G}$ for every pseudo-convergent sequence $G$ (by Lemma 2.6). In particular, $V_{\infty}\neq V_{G}$. Note also that $V_{\infty}$ is contained in the DVR $K[1/X]_{(1/X)}$ ([18, Proposition 2.2]).

Proposition 4.2a is false without the assumption on $K$: in fact, if $E\subset K$ is a pseudo-convergent sequence of algebraic type without pseudo-limits in $K$, then, for some extension $u$ of $v$ to $K$, by Proposition 4.3 we have $\mathcal{L}(U_{E})=\mathcal{L}^{u}_{E}\not=\emptyset$, so by contracting down to $K$ we have $\mathcal{L}(V_{E})=\mathcal{L}_{E}=\emptyset$ while $V\subset V_{E}$ is not immediate by Proposition 3.7.

Proposition 4.5.

Suppose $K$ is algebraically closed, and let $W_{1},W_{2}$ be two extensions of $V$ to $K(X)$. If either $\mathcal{L}_{1}(W_{1})=\mathcal{L}_{1}(W_{2})\neq\emptyset$ or $\mathcal{L}_{2}(W_{1})=\mathcal{L}_{2}(W_{2})\neq\emptyset$, then $W_{1}=W_{2}$.

Proof.

Let $\mathcal{L}=\mathcal{L}(W_{1})=\mathcal{L}(W_{2})$; we shall use $w$ to indicate either $w_{1}$ or $w_{2}$. Fix also $\alpha\in\mathcal{L}$.

Let $\phi\in K(X)$, and write it as $\phi(X)=c\prod_{\gamma\in\Omega}(X-\gamma)^{\epsilon_{\gamma}}$, where $\Omega$ is the multiset of critical points of $\phi$, $c\in K$ and $\epsilon_{\gamma}\in\{-1,+1\}$.

For every $\gamma\notin\mathcal{L}$, by Proposition 4.2b $w(X-\gamma)<w(X-\alpha)$, so $w(X-\gamma)=v(\alpha-\gamma)$; furthermore, if $\gamma_{1},\gamma_{2}\in\mathcal{L}$, then $w(X-\gamma_{1})=w(X-\gamma_{2})$. Hence, $w(\phi)=w(\psi)$, where $\psi(X)=d(X-\alpha)^{t}$ for some $d\in K$, $t\in\mathbb{Z}$ (more precisely, $d=c\prod_{\gamma\in\Omega\setminus\mathcal{L}}(\alpha-\gamma)^{\epsilon_{\gamma}}$ and $t=\sum_{\gamma\in\Omega\cap\mathcal{L}}\epsilon_{\gamma}$.) Note that, in particular, we have both $w_{1}(\phi)=w_{1}(\psi)$ and $w_{2}(\phi)=w_{2}(\psi)$.

If $t=0$, then $w(\phi)=v(d)$ and so its sign does not depend on whether $w=w_{1}$ or $w=w_{2}$; i.e., $\phi\in W_{1}$ if and only if $\phi\in W_{2}$. If $t\neq 0$, then $\psi=(e(X-\alpha)^{\epsilon})^{|t|}$, where $e\in K$ is such that $e^{|t|}=d$ and $\epsilon=t/|t|$; thus, $\psi\in W_{i}$ if and only if $e(X-\alpha)^{\epsilon}\in W_{i}$, for $i=1,2$, since a valuation domain is integrally closed.

Suppose now that $\alpha\in\mathcal{L}_{1}(W)$ and $t>0$. Then,

[TABLE]

(since $w(X-\alpha)\notin\Gamma_{v}$), i.e., if and only if $\alpha-e^{-1}\notin\mathcal{L}_{1}(W)$. Since $\mathcal{L}_{1}(W_{1})=\mathcal{L}_{1}(W_{2})$, it follows that $w_{1}(e(X-\alpha))\geq 0$ if and only if $w_{2}(e(X-\alpha))\geq 0$, i.e., $\phi\in W_{1}$ if and only if $\phi\in W_{2}$, as claimed. Analogously, if $t<0$, then

[TABLE]

that is, if and only if $\alpha-e\in\mathcal{L}_{1}(W)$. As before, this implies that $\phi\in W_{1}$ if and only if $\phi\in W_{2}$; hence, $W_{1}=W_{2}$.

Suppose now that $\alpha\in\mathcal{L}_{2}(W)$. If $t>0$, then $w(e(X-\alpha))\geq 0$ if and only if $w(X-\alpha)>w(f)$ for all $f\in K$ such that $v(e^{-1})>v(f)$; that is, if and only if $w(X-\alpha+f)=v(f)$ for all such $f$. This happens if and only if $\alpha-f\notin\mathcal{L}$ for all these $f$; since $v(e^{-1})>v(f)$ depends only on $V$, it follows as before that $w_{1}(e(X-\alpha))\geq 0$ if and only if $w_{2}(e(X-\alpha))\geq 0$, i.e., $\phi\in W_{1}$ if and only if $\phi\in W_{2}$, as claimed. If $t<0$, then, in the same way, $w(e/(X-\alpha))\geq 0$ if and only if $w(X-\alpha)<v(f)$ for all $f$ such that $v(f)<v(e)$; as above, this implies that $\phi\in W_{1}$ if and only if $\phi\in W_{2}$. Hence, $W_{1}=W_{2}$. ∎

Example 4.6**.**

In Proposition 4.5 we can’t drop the hypothesis that $K$ is algebraically closed: for example, take $\alpha\in K$ and let $\delta\in\mathbb{Q}\Gamma_{v}\setminus\Gamma_{v}$. Let $E\subset K$ be a pseudo-convergent sequence having a pseudo-limit $\alpha$ and such that $\mathrm{Br}(E)=I=\{x\in K\mid v(x)>\delta\}$; by Proposition 4.3a, $\mathcal{L}_{1}(V_{E})=\alpha+I\neq\emptyset$. Take now the monomial valuation $w=v_{\alpha,\delta}$: then, $\mathcal{L}_{1}(W)=\alpha+I=\mathcal{L}_{1}(V_{E})$, but $W\neq V_{E}$ since the value group of $w$ is contained in the divisible hull of the value group of $v$, while $\Gamma_{v_{E}}=\Gamma_{v}\oplus\Delta_{E}\mathbb{Z}$ is not (by Proposition 3.7 and Lemma 3.6).

Joining the previous propositions, we can prove that if $K$ is algebraically closed, then any extension of $V$ to $K(X)$ is in the form $V_{E}$ for some pseudo-monotone sequence $E$; however, we postpone this result to Theorem 6.2 in order to cover a more general case.

Proposition 4.7.

Let $E\subset K$ be a pseudo-monotone sequence, and let $U$ be an extension of $V$ to $\overline{K}$. Then $U_{E}$ is the unique common extension of $U$ and $V_{E}$ to $\overline{K}(X)$. Moreover, if $F\subset K$ is another pseudo-monotone sequence such that $E$ and $F$ are either both pseudo-stationary or both strictly pseudo-monotone, then $V_{E}=V_{F}$ if and only if $U_{E}=U_{F}$.

Proof.

The first claim can be proved in the same way as [19, Theorem 5.7], but we repeat the proof for clarity. Clearly, $U_{E}$ extends both $U$ and $V_{E}$. Suppose there is another extension $W$ of $U$ and $V_{E}$ to $\overline{K}(X)$: then, by [4, Chapt. VI, §8, 6., Corollary 1], there is a $K(X)$-automorphism $\sigma$ of $\overline{K}(X)$ such that $U_{E}=\sigma(W)$. Let $\rho=\sigma^{-1}$: then,

[TABLE]

Since $s_{\nu}\in K$ and $\rho|_{K}$ is the identity, $\rho(\phi(s_{\nu}))=(\rho\circ\phi)(s_{\nu})$; hence,

[TABLE]

In particular, note that $\rho(U_{E})=\rho(U)_{E}$.

Since both $U_{E}$ and $W$ are extensions of $U$, for any $t\in\overline{K}$ we have that $t\in U$ if and only if $\sigma(t)\in U$; in particular, this happens for $t=\psi(s_{\nu})$. It follows that $\rho(U_{E})=W=U_{E}$, as claimed.

We prove now the last claim. One direction is clear, since $V_{E}=U_{E}\cap K$ and $V_{F}=U_{F}\cap K$. The other implication follows from the previous claim, since $U_{E}$ is the unique common extension of $V_{E}$ and $U$ and $U_{F}$ is the unique common extension of $V_{F}$ and $U$. ∎

5 Equivalence of pseudo-monotone sequences

Using the results of the previous sections, we can now tackle the problem of when two pseudo-monotone sequences have the same associated extension of $V$ to $K(X)$.

Proposition 5.1.

Let $E,F\subset K$ be two pseudo-monotone sequences that are either both pseudo-stationary or both strictly pseudo-monotone. Let $u$ be an extension of $v$ to $\overline{K}$. If $\mathcal{L}_{E}^{u}\neq\emptyset$, then $V_{E}=V_{F}$ if and only if $\mathcal{L}_{E}^{u}=\mathcal{L}_{F}^{u}$. Furthermore, if $\mathcal{L}_{E}\neq\emptyset$, then the previous condition is also equivalent to the corresponding one over $K$.

Proof.

By Proposition 4.7, it is enough to show that $U_{E}=U_{F}$ if and only if $\mathcal{L}_{E}^{u}=\mathcal{L}_{F}^{u}$.

Suppose $\mathcal{L}_{E}^{u}\neq\emptyset$. Then $U\subset U_{E}$ is not immediate by Proposition 3.7, and by Proposition 4.3 $\mathcal{L}_{E}^{u}=\mathcal{L}_{2}(U_{E})$ if $E$ is pseudo-stationary and $\mathcal{L}_{E}^{u}=\mathcal{L}_{1}(U_{E})$ if $E$ is strictly pseudo-monotone. Hence, if $\mathcal{L}_{E}^{u}=\mathcal{L}_{F}^{u}$, then also $\mathcal{L}^{u}_{F}\neq\emptyset$; if $E$ and $F$ are both pseudo-stationary, then $\mathcal{L}_{2}(U_{F})=\mathcal{L}_{2}(U_{E})\neq\emptyset$ and so $U_{E}=U_{F}$ by Proposition 4.5, while if $E$ and $F$ are strictly pseudo-monotone the same conclusion holds by the same proposition. Conversely, if $U_{E}=U_{F}$, then $\mathcal{L}^{u}_{E}=\mathcal{L}(U_{E})=\mathcal{L}(U_{F})=\mathcal{L}^{u}_{F}$ and so $E$ and $F$ have the same pseudo-limits (in $\overline{K}$).

Suppose now $\mathcal{L}_{E}\neq\emptyset$. If $\mathcal{L}_{E}^{u}=\mathcal{L}_{F}^{u}$ then $\mathcal{L}_{E}=\mathcal{L}_{F}$. Conversely, if $\mathcal{L}_{E}=\mathcal{L}_{F}$, then by Lemma 2.5 $\mathrm{Br}(E)=\mathrm{Br}(F)$. In particular, $\mathrm{Br}_{u}(E)=\mathrm{Br}_{u}(F)$ so by the same Lemma $\mathcal{L}_{E}^{u}=\mathcal{L}_{F}^{u}$. ∎

Remark 5.2**.**

Note that, under the same assumptions of Proposition 5.1, by Lemma 2.5 $E$ and $F$ have the same set of pseudo-limits (either over $K$ or over $\overline{K}$) if and only if they have the same breadth ideal and they have at least one pseudo-limit in common. 2. 2.

It is possible to have $V_{E}=V_{F}$ even if $E$ is pseudo-convergent and $F$ is pseudo-divergent: for example, if $I$ is not finitely generated and it is not equal to $cM$ for any $c\in K$, we can find both a pseudo-convergent sequence $E$ and a pseudo-divergent sequence $F$ such that $I=0+I$ is the set of pseudo-limits of $E$ and $F$ (Lemmas 2.5 and 2.6). By Proposition 5.1, $V_{E}=V_{F}$. 3. 3.

If $E,F$ are pseudo-divergent sequences with $\mathrm{Br}(E)=K=\mathrm{Br}(F)$ (that is, if the gauges of $E,F$ are not bounded from below, see §2.3.2), then $\mathcal{L}_{E}=K=\mathcal{L}_{F}$, and so $V_{E}=V_{F}$. This extension is exactly the valuation domain $V_{\infty}$ considered in Example 4.4.

Let $E,F$ be two Cauchy sequences with limits $x_{E},x_{F}\in K$, respectively. By Proposition 5.1, $V_{E}=V_{F}$ if and only if $x_{E}=x_{F}$; by extending $v$ to the completion $\widehat{K}$, we see that this can happen even if the limits are not in $K$. Thus, the condition $V_{E}=V_{F}$ generalizes the notion of equivalence between Cauchy sequences: for this reason, we say that two pseudo-monotone sequences are equivalent if $V_{E}=V_{F}$. We now want to characterize this notion in a more intrinsic way, but we need to distinguish between the different types. The first result, involving pseudo-convergent sequences, is a generalization of [19, Theorem 5.4].

Proposition 5.3.

Let $E=\{s_{\nu}\}_{\nu\in\Lambda},F=\{t_{\mu}\}_{\mu\in\Lambda}\subset K$ be pseudo-convergent sequences. Then $E$ and $F$ are equivalent if and only if $\mathrm{Br}(E)=\mathrm{Br}(F)$ and, for every $\kappa\in\Lambda$, there are $\nu_{0},\mu_{0}\in\Lambda$ such that, whenever $\nu\geq\nu_{0}$, $\mu\geq\mu_{0}$, we have $v(s_{\nu}-t_{\mu})>v(t_{\rho}-t_{\kappa})$, for any $\rho>\kappa$.

Note that the condition of the proposition is not symmetrical in $E$ and $F$, despite the fact that the definition of the equivalence relation is symmetric.

Proof.

By Proposition 5.1, without loss of generality we can suppose that $K$ is algebraically closed. Let $\{\delta_{\nu}\}_{\nu\in\Lambda},\{\delta^{\prime}_{\nu}\}_{\nu\in\Lambda}$ be the gauges of $E$ and $F$, respectively. We will use the following remark: $\mathrm{Br}(E)\subseteq\mathrm{Br}(F)$ if and only if for each $\mu\in\Lambda$ there exists $\nu\in\Lambda$ such that $\delta^{\prime}_{\mu}\leq\delta_{\nu}$.

We assume first that the conditions of the statement hold. Suppose that $E$ is of algebraic type: then, $E$ has a pseudo-limit $\beta\in K$. Fix $\mu\in\Lambda$. By the above remark, there exists $\nu_{0}\in\Lambda$ such that for all $\nu\geq\nu_{0}$, $\delta_{\nu}>\delta^{\prime}_{\mu}$. There also exist $\iota_{0},\mu_{0}\in\Lambda$ such that for all $\nu\geq\iota_{0}$, $\kappa\geq\mu_{0}$, we have $v(s_{\nu}-t_{\kappa})>\delta^{\prime}_{\mu}$. Then, for $\nu\geq\max\{\iota_{0},\nu_{0}\}$ and $\kappa>\max\{\mu,\mu_{0}\}$ we have

[TABLE]

so that $\beta$ is a pseudo-limit of $F$. Therefore, $F$ is of algebraic type and $\mathcal{L}_{E}\subseteq\mathcal{L}_{F}$. The reverse inclusion is proved symmetrically, and $V_{E}=V_{F}$ follows from Proposition 5.1.

Suppose now that $E$ is of transcendental type: by the previous part of the proof, also $F$ must be of transcendental type. We can repeat the previous reasoning by using $X$ instead of $\beta$ (since $X$ is a pseudo-limit of $E$ with respect to $v_{E}$: see [19, Theorem 3.8] or Theorem 3.4); this proves that $X$ is a pseudo-limit of $F$ with respect to $v_{F}$. The fact that $V_{E}=V_{F}$ now follows from [10, Theorem 2].

Assume now that $V_{E}=V_{F}$. Suppose first that $E$ is of algebraic type: then, $\mathcal{L}_{E}\neq\emptyset$, and by Proposition 5.1 we must have $\mathcal{L}_{F}=\mathcal{L}_{E}$, and thus $F$ is also of algebraic type. In particular, $\mathrm{Br}(E)=\mathrm{Br}(F)$. Let $\alpha\in\mathcal{L}_{E}=\mathcal{L}_{F}$. Then,

[TABLE]

By the remark, for every $\kappa$ there is an $\iota_{0}$ such that $\delta_{\iota_{0}}>\delta^{\prime}_{\kappa}$; choosing $\mu_{0}>\kappa$ we have that $E$ and $F$ satisfy the conditions of the statement.

Suppose now that $E$ is of transcendental type; as before, this implies that also $F$ is of transcendental type. Without loss of generality we may suppose that $\mathrm{Br}(F)\subseteq\mathrm{Br}(E)$. If this containment is strict, then there exists a $c\in\mathrm{Br}(E)\setminus\mathrm{Br}(F)$. Then, $\frac{c}{X-\alpha}$ is in $V_{E}$ for each $\alpha\in K$ (because $X$ is a pseudo-limit of $E$ with respect to $v_{E}$ and $E$ has no pseudo-limits in $K$). On the other hand, for every $\nu$ we have $\frac{c}{X-t_{\nu}}\notin V_{F}$, a contradiction. Therefore $\mathrm{Br}(E)=\mathrm{Br}(F)$. We know that $X$ is a pseudo-limit of $F$ with respect to $v_{F}$, so that $\{v_{F}(X-t_{\mu})\}_{\mu\in\Lambda}$ is a (eventually) strictly increasing sequence. In particular, since $V_{E}=V_{F}$ implies that $\lambda\circ v_{E}=v_{F}$ for some isomorphism of totally ordered groups $\lambda:\Gamma_{v_{E}}\to\Gamma_{v_{F}}$, it follows that $\{v_{E}(X-t_{\mu})\}_{\mu\in\Lambda}$ is a (eventually) strictly increasing sequence, so that $X$ is a pseudo-limit of $F$ with respect to $v_{E}$. Thus $v_{E}(X-t_{\mu})=\delta^{\prime}_{\mu}$, for each $\mu\in\Lambda$ (sufficiently large). The proof now proceeds as above, replacing a pseudo-limit $\alpha$ of $E$ and $F$ by $X$ (which is a pseudo-limit of $E$ and $F$ with respect to $v_{E}$). Hence, the conditions of the statement holds. ∎

The cases of pseudo-divergent and pseudo-stationary sequences are very similar, with the further simplification that in these cases we do not need to consider sequences of transcendental type (which do not exist).

Proposition 5.4.

Let $E=\{s_{\nu}\}_{\nu\in\Lambda},F=\{t_{\mu}\}_{\mu\in\Lambda}\subset K$ be pseudo-divergent sequences. Then $E$ and $F$ are equivalent if and only if $\mathrm{Br}(E)=\mathrm{Br}(F)$ and there exist $\nu_{0},\mu_{0}\in\Lambda$ such that for all $\nu\geq\nu_{0},\mu\geq\mu_{0}$ there exists $\kappa\in\Lambda$ such that $v(s_{\nu}-t_{\mu})\geq v(t_{\rho}-t_{\kappa})$, for any $\rho<\kappa$.

Note that the above condition amounts to saying that $s_{\nu}-t_{\mu}$ is eventually in the breadth ideal $\mathrm{Br}(E)=\mathrm{Br}(F)$.

The following is the analogous result for pseudo-stationary sequences.

Proposition 5.5.

Let $E=\{s_{\nu}\}_{\nu\in\Lambda},F=\{t_{\mu}\}_{\mu\in\Lambda}\subset K$ be pseudo-stationary sequences with breadth $\delta_{E}$ and $\delta_{F}$, respectively. Then $E$ and $F$ are equivalent if and only if $\delta_{E}=\delta_{F}=\delta$ and $v(s_{\nu}-t_{\mu})\geq\delta$ for all $\nu,\mu\in\Lambda$.

Proof.

The conditions of the statement say (using Lemma 2.5) that $E\subset\mathcal{L}_{F}$ and $F\subset\mathcal{L}_{E}$. By the same Lemma, this is equivalent to $\mathcal{L}_{E}=\mathcal{L}_{F}$, which is equivalent to $V_{E}=V_{F}$ by Proposition 5.1. ∎

6 A generalized Fundamentalsatz

In general, not all the extensions of $V$ to $K(X)$ can be realized via a pseudo-monotone sequence contained in $K$. For example, let $V$ be the ring of $p$-adic integers $\mathbb{Z}_{p}$, for some prime $p\in\mathbb{Z}$. It is not difficult to see that for $\alpha\in\overline{\mathbb{Q}_{p}}\setminus\mathbb{Q}_{p}$, the valuation domain $V_{p,\alpha}=\{\phi\in\mathbb{Q}_{p}(X)\mid\phi(\alpha)\in\overline{\mathbb{Z}_{p}}\}$ of $\mathbb{Q}_{p}(X)$, where $\overline{\mathbb{Z}_{p}}$ is the unique valuation domain of $\overline{\mathbb{Q}_{p}}$, is not of the form $V_{E}$, for any pseudo-monotone sequence $E\subset\mathbb{Q}_{p}$ (for example, by Proposition 3.7 and [18, Proposition 2.2 & Theorem 3.2], see also the proof of Theorem 6.2).

In this section, we show when all extensions of $V$ to $K(X)$ are induced by pseudo-monotone sequences in $K$. We start with a lemma which allows us to reduce to the algebraically closed case.

Lemma 6.1.

Let $L$ be an extension of $K$ and $U$ a valuation domain of $L$ lying over $V$ such that $\Gamma_{u}=\Gamma_{v}$. Let $F\subset L$ be a pseudo-monotone sequence with respect to $u$ having a pseudo-limit $\beta\in K$. Then:

1. (a)

if $F$ is strictly pseudo-monotone, there is a sequence $E\subset K$ of the same kind as $F$ that is equivalent to $F$ (with respect to $u$); 2. (b)

if $F$ is pseudo-stationary and the residue field of $V$ is infinite, there is a pseudo-stationary sequence $E\subset K$ that is equivalent to $F$ (with respect to $u$).

Proof.

Let $F=\{t_{\nu}\}_{\nu\in\Lambda}$.

a For every $\nu$, there is a $c_{\nu}\in K$ such that $u(t_{\nu}-\beta)=u(c_{\nu})=\delta_{\nu}$; let $s_{\nu}=c_{\nu}+\beta$ and let $E=\{s_{\nu}\}_{\nu\in\Lambda}$. Then, $E\subset K$ (since $\beta\in K$) and

[TABLE]

for every $\mu>\nu$, so $E$ is pseudo-monotone of the same kind as $F$ and the gauges of $E$ and $F$ coincide; in particular, $\mathrm{Br}_{u}(E)=\mathrm{Br}_{u}(F)$. By Proposition 5.1, $E$ and $F$ are equivalent.

b Since $u(t_{\nu}-\beta)=\delta\in\Gamma_{v}$ and the residue field of $V$ is infinite, we can find an infinite set $\{c_{\nu}\}_{\nu\in\Lambda}\subset V$ such that $u(c_{\nu}-\beta)=\delta$ and such that $u(c_{\nu}-c_{\mu})=\delta$ for every $\nu\neq\mu$. Setting $s_{\nu}=c_{\nu}+\beta$, as in the previous case we can take $E=\{s_{\nu}\}_{\nu\in\Lambda}$, and $E$ and $F$ are equivalent by Proposition 5.1. ∎

Theorem 6.2.

Let $V$ be a valuation domain with quotient field $K$. Then, every extension $W$ of $V$ to $K(X)$ is of the form $W=V_{E}$ for some pseudo-monotone sequence $E\subset K$ if and only if $\widehat{K}$ is algebraically closed. In this case, we have the following.

1. (a)

If $V\subset W$ is immediate, then $E$ is necessarily a pseudo-convergent sequence of transcendental type. 2. (b)

If $V\subset W$ is not immediate, then:

1. (b1)

if $\mathcal{L}(W)=\emptyset$, then $E$ is a pseudo-convergent Cauchy sequence of algebraic type whose limit is in $\widehat{K}\setminus K$; 2. (b2)

if $\mathcal{L}_{1}(W)=\alpha+I\neq\emptyset$ and $I$ is a divisorial fractional ideal, then $E$ can be taken to be pseudo-convergent of algebraic type; 3. (b3)

if $\mathcal{L}_{1}(W)=\alpha+I\neq\emptyset$ and $I$ is not a principal ideal, then $E$ can be taken to be pseudo-divergent; 4. (b4)

if $\mathcal{L}_{2}(W)=\alpha+I\neq\emptyset$, then $E$ is necessarily a pseudo-stationary sequence.

Note that, since every nondivisorial ideal is nonprincipal, cases b(b2) and b(b3) cover all possibilities. Furthermore, these two cases are not mutually exclusive: see Remark 5.2.

Proof.

Throughout the proof we will use the fact that $\widehat{K}$ is algebraically closed if and only if $\overline{K}$ embeds in $\widehat{K}$ (which in turn follows from the fact that the completion of an algebraically closed field is algebraically closed [22, §15.3, Theorem 1]). Loosely speaking, this condition holds if and only if $K$ is dense in its algebraic closure $\overline{K}$.

Suppose that $\widehat{K}$ is not algebraically closed. Then by above there exists $\alpha\in\overline{K}$ such that $K(\alpha)$ cannot be embedded into $\widehat{K}$, that is, $\alpha$ is not the limit of any Cauchy sequence in $K$. Let $U$ be an extension of $V$ to $\overline{K}$ and let $F\subset\overline{K}$ be a pseudo-convergent Cauchy sequence with limit $\alpha$. Let $W=U_{F}\cap K(X)$: we claim that $W\neq V_{E}$ for any pseudo-monotone sequence $E$. Indeed, if $W=V_{E}$ for some pseudo-monotone sequence $E\subset K$, by Proposition 4.7 $U_{E}$ is the only common extension of $U$ and $V_{E}$ to $\overline{K}(X)$, so that $U_{E}=U_{F}$. By Proposition 5.1, we must have $\mathcal{L}_{E}^{u}=\mathcal{L}_{F}^{u}=\{\alpha\}$ and $\mathrm{Br}_{u}(E)=\mathrm{Br}_{u}(F)=(0)$ and thus $\mathcal{L}_{E}=\mathcal{L}_{E}^{u}\cap K=\emptyset$; hence, $E\subset K$ should be a pseudo-convergent Cauchy sequence with limit $\alpha$ (Lemma 2.5). However, this is impossible by the choice of $\alpha$, and so $W\neq V_{E}$ for any pseudo-monotone sequence $E$.

Suppose now that $\widehat{K}$ is algebraically closed, and let $\mathcal{W}$ be a common extension of $\widehat{V}$ and $W$ to $\widehat{K}(X)$.

If $\widehat{V}\subset\mathcal{W}$ is immediate, then also $V\subset W$ is immediate (since $V\subset\widehat{V}$ is); by Kaplansky’s Theorem [10, Theorem 2], there is a pseudo-convergent sequence $E\subset K$ such that $W=V_{E}$.

Suppose $\widehat{V}\subset\mathcal{W}$ is not immediate. By Proposition 4.2a, $\mathcal{L}(\mathcal{W})\subseteq\widehat{K}$ is nonempty, say equal to $\alpha+J$ for some $\alpha\in\widehat{K}$ and some $J$ that is either a fractional ideal of $\widehat{V}$ or the whole $\widehat{K}$.

If $J=(0)$ let $E\subset K$ be a pseudo-convergent Cauchy sequence having limit $\alpha$: then, $\mathcal{L}(\widehat{V}_{E})=\mathcal{L}_{1}(\widehat{V}_{E})=\{\alpha\}=\mathcal{L}_{1}(\mathcal{W})=\mathcal{L}(\mathcal{W})$, and by Proposition 4.5 it follows that $\mathcal{W}=\widehat{V}_{E}$. Hence, $W=\mathcal{W}\cap K=\widehat{V}_{E}\cap K=V_{E}$. In particular, if $\alpha\in K$ then $\mathcal{L}(W)=\{\alpha\}$, while if $\alpha\in\widehat{K}\setminus K$ then $\mathcal{L}(W)=\emptyset$; furthermore, by Proposition 3.7 if $V\subset W$ is not immediate, then $E$ must be a sequence of algebraic type.

Suppose now that $J\neq(0)$. Then, the open set $\alpha+J$ must contain an element $\beta$ of $K$, and in particular $\alpha+J=\beta+J$. Using Lemma 2.6, we construct a pseudo-monotone sequence $F\subset\widehat{K}$ with breadth ideal $J$ and with $\beta$ as pseudo-limit, with the following properties:

• •

if $\mathcal{L}_{1}(\mathcal{W})\neq\emptyset$ and $J$ is a strictly divisorial fractional ideal, we take $F$ to be a pseudo-convergent sequence;

• •

if $\mathcal{L}_{1}(\mathcal{W})\neq\emptyset$ and $J$ is a nondivisorial fractional ideal, we take $F$ to be a pseudo-divergent sequence (note that, in this case, $J=c\widehat{M}$ is not principal);

• •

if $\mathcal{L}_{1}(\mathcal{W})=\widehat{K}$, we take $F$ to be a pseudo-divergent sequence whose gauge is coinitial in $\Gamma_{v}$;

• •

if $\mathcal{L}_{2}(\mathcal{W})\neq\emptyset$, we take $F$ to be a pseudo-stationary sequence.

Note that the first case falls in b(b2), the second and the third ones in b(b3) and the fourth one in b(b4).

In all cases, $\mathcal{W}=\widehat{V}_{F}$ by Proposition 4.5 (in the first three cases using $\mathcal{L}_{1}$ and in the last one using $\mathcal{L}_{2}$). Since $V\subset\widehat{V}$ is immediate, we can apply Lemma 6.1 to find a pseudo-monotone sequence $E\subset K$ that is equivalent to $F$; hence, $V_{E}=\widehat{V}_{F}\cap K=\mathcal{W}\cap K=W$. The theorem is now proved. ∎

Remark 6.3**.**

By Proposition 3.7 and the main Theorem 6.2, if $\widehat{K}$ is algebraically closed, then every extension of $V$ to $K(X)$ which is not immediate is a monomial valuation. This result was already known to hold but only with the stronger assumption that $K$ is algebraically closed, see [2, pp. 286-289].

We remark that a more direct approach to the proof of Theorem 6.2 can be given by considering the set $w(X,K)=\{w(X-a)\mid a\in K\}$, which is a subset of $\Gamma_{w}$. If $w(X,K)$ has no maximum, then, exactly as in the original proof of Ostrowski, we can extract from $w(X,K)$ a cofinal sequence which determines a pseudo-convergent sequence $E$ in $K$ of transcendental type such that $W=V_{E}$. If instead $w(X,K)$ has a maximum $\Delta_{w}=w(X-a_{0})$, then, following again Ostrowski’s proof, one can show that $W$ is a monomial valuation of the form $V_{a_{0},\Delta_{w}}$: according to whether $\Delta_{w}$ is in $\Gamma_{v}$ or not (and, in the latter case, depending on the properties of the cut induced by $\Delta_{w}$ on $\Gamma_{v}$), we can find a pseudo-monotone sequence $E\subset K$ with $a_{0}$ as pseudo-limit and such that $W=V_{E}$. This approach can be connected to the one given above by noting that $\Gamma_{v}\setminus w(X,K)=v(J)$ (where $J$ is the ideal defined in the proof of Theorem 6.2), and that if $\Delta_{w}$ exists then we have $\{a\in K\mid w(X-a)=\Delta_{w}\}=\mathcal{L}(W)$.

When $w(X,K)$ has a maximum and $V$ has rank 1, Ostrowski proved in his Fundamentalsatz [16, p. 379] that the rank one valuation associated to $W$ can be realized through a pseudo-convergent sequence $F=\{s_{\nu}\}_{\nu\in\Lambda}\subset K$ by means of the map defined as $w_{F}(\phi)=\lim_{\nu\to\infty}v(\phi(s_{\nu}))$, for each $\phi\in K(X)$ (where the limit is taken in $\mathbb{R}$). If $W=V_{E}$, where $E\subset L$ is a pseudo-stationary sequence, as in Theorem 6.2b(b4), then $E$ and $F$ have the same set of pseudo-limits, and in particular they have the same breadth. Furthermore, by Proposition 7.1 below, in this case we have $V_{F}\subset W=V_{E}$. See also [19] for other results regarding the valuation $w_{F}$ introduced by Ostrowski.

An immediate corollary of Theorem 6.2 is that, for any field $K$, if $U$ is an extension of $V$ to $\overline{K}$, then every extension $W$ of $V$ to $K(X)$ can be written as the contraction of $U_{E}$ to $K(X)$, namely $U\cap K(X)$, where $E\subset\overline{K}$ is a pseudo-monotone sequence with respect to $U$; furthermore, in view of the examples above, we cannot always choose $E$ to be contained in $K$.

Remark 6.4**.**

The hypothesis that $\widehat{K}$ is algebraically closed is weaker than the hypothesis that $K$ is algebraically closed; we give a few examples.

Let $K=\overline{\mathbb{Q}}\cap\mathbb{R}$, where $\overline{\mathbb{Q}}$ is the algebraic closure of $\mathbb{Q}$, and let $V$ be an extension to $K$ of $\mathbb{Z}_{(5\mathbb{Z})}$. Then, $i$ belongs to the completion $\widehat{K}$, since the polynomial $X^{2}+1$ has a root in $\mathbb{Z}/5\mathbb{Z}$; therefore, $K(i)=\overline{\mathbb{Q}}$ can be embedded into $\widehat{K}$, so $\widehat{K}$ is algebraically closed while $K$ is not. 2. 2.

If $K$ is separably closed, then $\widehat{K}$ is algebraically closed (it is enough to adapt the proof of [20, Chapter 2, (N)] to the general case). 3. 3.

Suppose that $V$ has rank $1$ and that the residue field $k$ has characteristic [math]. Then, $\widehat{K}$ is algebraically closed if and only if $k$ is algebraically closed and the value group $\Gamma_{v}$ is divisible. Indeed, these two conditions are necessary, since completion preserves value group and residue field. Conversely, suppose that the two conditions hold. When $V$ has rank 1 then $\widehat{K}$ is henselian, i.e. $\widehat{v}$ has a unique extension to the algebraic closure of $\widehat{K}$. Since $k$ has characteristic [math], all finite extensions of $\overline{K}$ are defectless [7, Corollary 20.23], and thus the fundamental inequality is an equality and thus the degree $[\overline{\widehat{K}}:\widehat{K}]=1$, i.e., $\widehat{K}$ is algebraically closed.

7 Geometrical interpretation

Throughout this section, we suppose that the maximal ideal $M$ of $V$ is not finitely generated, and that its residue field $k$ is infinite. We also fix $\alpha\in K$. For any $\delta\in\Gamma_{v}$, we denote $B(\alpha,\delta)=\{x\in K\mid v(\alpha-x)\geq\delta\}$ and $\mathring{B}(\alpha,\delta)=\{x\in K\mid v(\alpha-x)>\delta\}$ the closed and open ball (respectively) of center $\alpha$ and radius $\delta$.

By Lemma 2.6, we can find both a pseudo-convergent sequence $E$ and a pseudo-stationary sequence $F$ such that $\mathcal{L}_{E}=\mathcal{L}_{F}=\alpha+cV=B(\alpha,\delta)$, where $\delta=v(c)$, for some $c\in K$; furthermore, again by Lemma 2.6, for every $z\in\alpha+cV$ we can find a pseudo-divergent sequence $D_{z}$ such that $\mathcal{L}_{D_{z}}=z+cM=\mathring{B}(z,\delta)$. Note that by Lemma 2.5 $D_{z}\subset\mathring{B}(z,\delta)$. In geometrical terms, $E$ and $F$ are associated to the closed ball $B(\alpha,\delta)$, while each $D_{z}$ is associated to the open ball $\mathring{B}(z,\delta)$, which is contained in $B(\alpha,\delta)$ and has the same radius. In the next proposition we show the containments among the valuation domains associated to these sequences.

Proposition 7.1.

Preserve the notation above, and let $z,w\in B(\alpha,\delta)$. Then, $V_{F}$ properly contains both $V_{E}$ and $V_{D_{z}}$, $V_{E}\neq V_{D_{z}}$, and $V_{D_{z}}=V_{D_{w}}$ if and only if $z-w\in cM$.

Proof.

Let $U$ be an extension of $V$ to $\overline{K}$ and let $z\in B(\alpha,\delta)$: then

[TABLE]

Let $\phi\in K(X)$ and, for any sequence $G$, let $\lambda_{G}$ be the dominating degree of $\phi$ with respect to $G$.

Since $\mathcal{L}^{u}_{E}=\mathcal{L}_{F}^{u}$, we have $\lambda_{E}=\lambda_{F}$; if $E=\{s_{\nu}\}_{\nu\in\Lambda}$ and $\{\delta_{\nu}\}_{\nu\in\Lambda}$ is the gauge of $E$, by Proposition 3.2 for large $\nu$ we have $v(\phi(s_{\nu}))=\lambda_{E}\delta_{\nu}+\gamma$, where $\gamma=u\left(\frac{\phi}{\phi_{S}}(\alpha)\right)$. If $\phi\in V_{E}$, then $v(\phi(s_{\nu}))\geq 0$ for all $\nu$ sufficiently large; since $\delta_{\nu}\nearrow\delta$, it follows that $\lambda_{E}\delta+\gamma\geq 0$. However, if $F=\{t_{\nu}\}_{\nu\in\Lambda}$, then applying again Proposition 3.2 we have $v(\phi(t_{\nu}))=\lambda_{F}\delta+\gamma=\lambda_{E}\delta+\gamma$, where $\gamma$ is the same as the previous case; it follows that $v(\phi(t_{\nu}))\geq 0$ for large $\nu$, i.e., $\phi\in V_{F}$.

Fix now $z\in B(\alpha,\delta)$ and let $D_{z}=\{r_{\nu}\}_{\nu\in\Lambda}$. Let $\{\delta_{\nu}^{\prime}\}_{\nu\in\Lambda}$ be the gauge of $D_{z}$; by mimicking the proof of Proposition 3.2, we have

[TABLE]

for some $d\in K$, where $S$ is the multiset of critical points of $\phi$. Hence, for large $\nu$, $v(\phi(r_{\nu}))=\lambda_{D_{z}}\delta_{\nu}^{\prime}+(\lambda_{F}-\lambda_{D_{z}})\delta+\gamma$. As in the previous case, if $\phi\in V_{D_{z}}$ then $v(\phi(r_{\nu}))\geq 0$ for large $\nu$, and so $0\leq\lambda_{F}\delta+\gamma=v(\phi(t_{\nu}))$, i.e., $\phi\in V_{F}$.

Thus $V_{E}$ and the $V_{D_{z}}$ are contained in $V_{F}$; the containment is strict by Proposition 3.7, since $V_{F}$ is residually transcendental over $V$ while the others are not. The last two claims follow from Lemma 2.5 and Proposition 5.1 by comparing the set of the pseudo-limits of the sequences involved. ∎

Consider now the quotient map $\pi:V_{F}\longrightarrow V_{F}/M_{F}$. By Proposition 3.7c, $V_{F}/M_{F}\simeq k(t)$, where $t$ is the image of $\frac{X-\alpha}{c}$. Let $W$ be either $V_{E}$ or $V_{D_{z}}$ for some $z\in B(\alpha,\delta)$: then, $M_{F}\subset W$, and thus we can consider the quotient $\pi(W)=W/M_{F}$, obtaining the following commutative diagram:

[TABLE]

In particular, $W/M_{F}$ is a (proper) valuation domain of $k(t)$ containing $k$: hence, by [6, Chapter 1, §3], $W/M_{F}$ must be equal either to $k[t]_{(f(t))}$, for some irreducible polynomial $f\in k[t]$, or to $k[1/t]_{(1/t)}$. In particular, $\pi$ induces a one-to-one correspondence between the valuation domains of $k(t)$ containing $k$ and the valuation domains of $K(X)$ contained in $V_{F}$. The strictly pseudo-monotone sequences we considered above are exactly the linear case, as we show next.

Proposition 7.2.

Preserve the notation above. Then:

1. (a)

$\pi(V_{E})=k[1/t]_{(1/t)}$; 2. (b)

$\pi(V_{D_{z}})=k[t]_{(t-\theta(z))}$, where $\theta(z)=\pi\left(\frac{z-\alpha}{c}\right)$; 3. (c)

$\pi^{-1}(k[t]_{(t-x)})=V_{D_{\alpha+yc}}$, where $y$ is an element of $V$ satisfying $\pi(y)=x$.

Proof.

Let $\phi(X)=\frac{X-\alpha}{c}$: then, as in the previous discussion, $t=\pi(\phi)$. The ring $k[1/t]_{(1/t)}$ is the only valuation domain of $k(t)$ containing $k$ such that $1/t$ belongs to the maximal ideal: hence, in order to show that $\pi(V_{E})=k[1/t]_{(1/t)}$ we only need to show that $1/\phi\in M_{E}$. This follows immediately from the fact that $v(\phi(s_{\nu}))=\delta_{\nu}-\delta<0$, where $E=\{s_{\nu}\}_{\nu\in\Lambda}$ and $\{\delta_{\nu}\}_{\nu\in\Lambda}$ is the gauge of $E$.

Analogously, in order to show that $\pi(V_{D_{z}})=k[t]_{(t-\theta(z))}$, we need to show that $t-\theta(z)$ is in the maximal ideal of $\pi(V_{D_{z}})$ or, equivalently, that

[TABLE]

This is an immediate consequence of the definition of $z$ and $c$, and the claim is proved.

The last point follows by the fact that $\theta(\alpha+yc)=\pi\left(\frac{\alpha+yc-\alpha}{c}\right)=\pi(y)=x$. ∎

If $k$ is algebraically closed (in particular, if $K$ is algebraically closed), then all irreducible polynomials of $k[t]$ are linear; thus, Proposition 7.2 describes all the subextensions of $V_{F}$. When $k$ is not algebraically closed, on the other hand, it follows that some of the valuation rings of $k(t)$ containing $k$ cannot be obtained by pseudo-divergent sequences contained in $K$ in the same way as in Proposition 7.2; however, we can construct them by using pseudo-divergent sequences in $\overline{K}$ with respect to a fixed extension of $V$.

Given an extension $u$ of $v$ to $\overline{K}$ we denote by $\mathcal{D}(U)$ the decomposition group of $U$ in $\mathrm{Gal}(\overline{K}/K)$, that is, $\mathcal{D}(U)=\{\sigma\in\mathrm{Gal}(\overline{K}/K)\mid\sigma(U)=U\}$.

Proposition 7.3.

Let $W$ be an extension of $V$ to $K(X)$ which is properly contained in $V_{F}$, and suppose that $\pi(W)=k[t]_{(f(t))}$ for some nonlinear irreducible $f\in k[t]$. Let $u$ be an extension of $v$ to $\overline{K}$.

1. (a)

There exists $z\in\mathcal{L}_{F}^{u}$ such that $W=U_{D_{z}}\cap K(X)$, where $D_{z}\subset\mathring{B}_{u}(z,\delta)\subset\overline{K}$ is pseudo-divergent.

Let $\overline{\pi}:U_{F}\to\overline{k}(t)$ be the canonical residue map.

1. (b)

$\overline{\theta}(z)=\overline{\pi}\left(\frac{z-\alpha}{c}\right)$* is a zero of $f(t)$.* 2. (c)

Let $z,w\in\mathcal{L}_{F}^{u}$. Then the following are equivalent:

1. (i)

$U_{D_{z}}\cap K(X)=U_{D_{w}}\cap K(X)$; 2. (ii)

$\overline{\theta}(z)$* and $\overline{\theta}(w)$ are conjugate over $k$;* 3. (iii)

$\rho(z)-w\in cM_{U}$* for some $\rho\in\mathcal{D}(U)$.*

In particular, the number of extensions of $W$ to $\overline{K}(X)$ is equal to the number of distinct roots of $f$ in $\overline{k}$.

Proof.

Let $\mathcal{W}$ be an extension of $W$ to $\overline{K}(X)$ and let $U=\mathcal{W}\cap\overline{K}$; then, $U$ is an extension of $V$. The diagram (2) lifts to

[TABLE]

By Proposition 7.2, $\mathcal{W}$ is equal to $U_{D_{z}}$, for some $z\in\mathcal{L}^{u}_{F}=\alpha+cU$, and thus $W=U_{D_{z}}\cap K(X)$, as desired.

b If $U_{D_{z}}$ is an extension of $W$ to $\overline{K}(X)$, then $\overline{\pi}(U_{D_{z}})=\overline{k}[t]_{(t-\overline{\theta}(z))}$ is an extension of $\pi(W)=k[t]_{(f(t))}$ to $\overline{k}(t)$. It is straightforward to see that this implies that $t-\overline{\theta}(z)$ is a factor of $f(t)$ in $\overline{k}[t]$, i.e., that $\overline{\theta}(z)$ is a zero of $f(t)$.

c The equivalence of ci and cii follows from the previous point.

cii $\iff$ ciii There is a surjective map from the decomposition group $\mathcal{D}(U)$ of $U$ to the Galois group $\mathrm{Gal}(\overline{k}/k)$, where $\rho\in\mathcal{D}(U)$ goes to the map $\overline{\rho}$ sending $x\in k$ to $\pi(\rho(y))$, where $y$ satisfies $\pi(y)=x$ [4, Chapt. V, §2.2, Proposition 6(ii)]. Hence, if $\overline{\theta}(z)$ and $\overline{\theta}(w)$ are conjugates there is a $\overline{\rho}\in\mathrm{Gal}(\overline{k}/k)$ such that $\overline{\rho}(\overline{\theta}(z))=\overline{\theta}(w)$, and we have

[TABLE]

Since $\alpha,c\in K$, the last condition holds if and only if $\rho(z)-w\in cM_{U}$. Conversely, if $\rho(z)-w\in cM_{U}$, then we can follow the same reasoning in the opposite order, and so $\overline{\theta}(z)$ and $\overline{\theta}(w)$ are conjugate over $k$. ∎

We conclude by reproving Ostrowski’s Fundamentalsatz. Recall that, if $V$ has rank $1$, we can always consider $v$ as a (not necessarily surjective) map from $K\setminus\{0\}$ to $\mathbb{R}$.

Theorem 7.4.

Suppose that $v$ is a valuation of rank $1$ and $K$ is algebraically closed. Let $w$ be an extension of $v$ to $K(X)$ of rank $1$. Then the following hold:

1. (a)

there is a pseudo-convergent sequence $E=\{s_{\nu}\}_{\nu\in\Lambda}\subset K$ such that

[TABLE]

for every nonzero $\phi\in K(X)$; 2. (b)

if $V\subset W$ is not immediate, there is also a pseudo-divergent sequence $F=\{t_{\nu}\}_{\nu\in\Lambda}$ such that

[TABLE]

for every nonzero $\phi\in K(X)$.

Proof.

If $V\subset W$ is immediate, then by [10, Theorems 1 and 3] $W=V_{E}$ for some pseudo-convergent sequence $E$ of transcendental type and $w(\phi)=v_{E}(\phi)=v(\phi(s_{\nu}))$ for $\nu\geq N(\phi)$.

Suppose now that $V\subset W$ is not immediate. By Theorem 6.2, there is a pseudo-monotone sequence $G\subset K$ such that $W=V_{G}$ with $\mathcal{L}_{G}\neq\emptyset$. We distinguish two cases.

Suppose first that $G$ is pseudo-stationary. Then, $\mathrm{Br}(G)=cV$, and $v_{G}(\phi)=\lambda_{\phi}\delta+\gamma$, where $\lambda_{\phi}=\mathrm{degdom}_{G}(\phi)$, $\delta=v(c)$ and $\gamma=v\left(\frac{\phi}{\phi_{S}}(\beta)\right)$ for some pseudo-limit $\beta$ of $G$ in $K$. Let $E=\{s_{\nu}\}_{\nu\in\Lambda}\subset K$ be a pseudo-convergent sequence such that $\mathcal{L}_{E}=\beta+cV=\mathcal{L}_{G}$ (Lemma 2.6); then, $\mathrm{degdom}_{E}(\phi)=\mathrm{degdom}_{G}(\phi)$ and the gauge $\{\delta_{\nu}\}_{\nu\in\Lambda}$ of $E$ tends to $\delta$, the gauge of $G$. By Proposition 3.2,

[TABLE]

and the claim is proved. In the same way, we can find a pseudo-divergent sequence $F=\{t_{\nu}\}_{\nu\in\Lambda}\subset K$ such that $\mathcal{L}_{F}=\beta+cM$; as in the proof of Proposition 7.1, setting $\{\delta^{\prime}_{\nu}\}_{\nu\in\Lambda}$ to be the gauge of $F$, we have (for large $\nu$)

[TABLE]

where $\lambda^{\prime}=\mathrm{degdom}_{F}(\phi)$. Hence, $v(\phi(t_{\nu}))\to\lambda_{\phi}\delta+\gamma=w(\phi)$, as claimed.

Suppose now that $G$ is strictly pseudo-monotone, and let $\beta\in\mathcal{L}_{G}$. If $\mathrm{Br}(G)$ is equal to $cV$ or to $cM$ for some $c\in K$, then we can find a pseudo-stationary sequence $G^{\prime}$ with breadth ideal $cV$ and having $\beta$ as a pseudo-limit; by the discussion at the beginning of the section and by Proposition 7.1, $V_{G^{\prime}}$ would properly contain $V_{G}$, against the fact that $V_{G}$ has rank one. Therefore, $\mathrm{Br}(G)$ is both strictly divisorial and nonprincipal; by Lemma 2.6 we can find a pseudo-convergent sequence $E$ and a pseudo-divergent sequence $F$ in $K$ such that $\mathcal{L}_{E}=\mathcal{L}_{F}=\mathcal{L}_{G}=\beta+\mathrm{Br}(G)$ (note that one between $E$ and $F$ could be taken equal to $G$). In particular, $\mathrm{Br}(E)=\mathrm{Br}(F)=\mathrm{Br}(G)$ and so $\delta_{E}=\delta_{F}=\delta$.

Since $W=V_{E}$ has rank 1, by [19, Theorem 4.9(c)] the valuation relative to $V_{E}$ is exactly the one mapping $\phi\in K(X)$ to

[TABLE]

where $\lambda=\mathrm{degdom}_{E}(\phi)=\mathrm{degdom}_{F}(\phi)$ and $\gamma=v\left(\frac{\phi}{\phi_{S}}(\beta)\right)$. Since $\delta$ is also the limit of $\delta^{\prime}_{\nu}$, the claim is proved. ∎