Definable valuations on ordered fields

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T
M
M T
Model Theory
no. 1 vol. 2 2023
Definable valuations on ordered fields
Philip Dittmann
,
Franziska Jahnke
,
Lothar Sebastian Krapp
and
Salma Kuhlmann
msp

msp
Model Theory
Vol. 2, No. 1, 2023
https://doi.org/10.2140/mt.2023.2.101
Definable valuations on ordered fields
Philip Dittmann
,
Franziska Jahnke
,
Lothar Sebastian Krapp
and
Salma Kuhlmann
We study the definability of convex valuations on ordered fields, with a particular
focus on the distinguished subclass of henselian valuations. In the setting of
ordered fields, one can consider definability both in the language of rings
Lr
and
in the richer language of ordered rings
Lor
. We analyse and compare definability
in both languages and show the following contrary results: while there are convex
valuations that are definable in the language
Lor
but not in the language
Lr
, any
Lor
-definable henselian valuation is already
Lr
-definable. To prove the latter, we
show that the value group and the ordered residue field of an ordered henselian
valued field are stably embedded (as an ordered abelian group and an ordered
field, respectively). Moreover, we show that in almost real closed fields any
Lor-definable valuation is henselian.
1. Introduction
One of the main objectives in the model-theoretic study of fields is the analysis of
first-order definable
1
sets and substructures. Given a field, it is a natural question
to ask whether a given valuation ring is a definable subset in some expansion of
the language
Lr= {+,−,·,
0
,
1
}
of rings. A key reason to study definability of
valuation rings is to transfer questions of decidability and existential decidability
(i.e., the question whether Hilbert’s tenth problem has a positive solution) between
different rings and fields. However, there is also a more recent motivation stemming
from classifying fields within Shelah’s classification hierarchy: whereas stable (or,
Jahnke was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)
under Germany’s Excellence Strategy EXC 2044-390685587, Mathematics Münster: Dynamics-
Geometry-Structure, as well as by a Fellowship from the Daimler and Benz Foundation.
Krapp was partially supported by Werner und Erika Messmer Stiftung.
Kuhlmann wishes to acknowledge support from the AFF Universität Konstanz for the Project TRAG-
VAL: Topics in Real Algebraic Geometry and Valuation Theory.
Part of this work was carried out while Krapp and Kuhlmann were generously hosted by the Fields
Institute, in Toronto, during the month of June 2022.
MSC2020: primary 03C64, 12J20; secondary 12L12, 13A18, 13F25, 13J30.
Keywords: definable valuations, ordered fields, convex valuations, henselian valuations, stably
embedded, almost real closed fields.
1Throughout this work, “definable” always means “definable with parameters”.
© 2023 The Authors, under license to MSP (Mathematical Sciences Publishers).

102 P. DITTMANN, F. JAHNKE, L. S. KRAPP AND S. KUHLMANN
more generally, simple) fields do not admit any nontrivial Lr-definable valuations,
a conjecture going back to Shelah predicts that infinite NIP fields which are neither
real closed nor separably closed admit a nontrivial
Lr
-definable henselian valuation.
In a recent series of spectacular papers, this was shown to hold in the “finite-
dimensional” (i.e., dp-finite) case by Johnson [2020]. For a survey on definability
of henselian valuations, mostly in the language of rings, see [Fehm and Jahnke
2017].
In this work, we primarily study valuations on ordered fields. This allows us to
also consider their definability in the language of ordered rings
Lor =Lr∪{<}
. We
focus on convex valuations, i.e., valuations whose valuation ring is convex with
respect to the ordering, as these naturally induce an ordering on the residue field
(see [Engler and Prestel 2005, Proposition 2.2.4]). Note that due to [Engler and
Prestel 2005, Lemma 4.3.6], every henselian valuation on an ordered field is already
convex.
By considering the expanded language
Lor
rather than
Lr
, one may expect further
definability results. Indeed, we present examples of ordered fields with convex
valuations that are
Lor
-definable but not
Lr
-definable (see Examples 3.5 and 3.6).
Rather surprisingly, for henselian valuations the language
Lor
does not produce
any further definability results, that is, every
Lor
-definable henselian valuation is
already
Lr
-definable (see Theorem 6.4). In the particular case of almost real closed
fields,
Lor
-definability even suffices to ensure both henselianity (thus convexity)
and Lr-definability (see Theorem 5.2).
The structure of this paper is as follows. After introducing preliminary notions
and results in Section 2, we first turn to the definability of convex valuations in
Section 3. We establish conditions on the value group and the residue field ensuring
the definability (with and without parameters) of a given convex valuation (see
Theorem 3.1 and Corollary 3.2). Subsequently we compare these results to other
known definability conditions in the literature (see Remark 3.3) and construct our
main examples — Examples 3.5 and 3.6 — to show that there are convex valuations
that are
∅
-
Lor
-definable but not
Lr
-definable. Lastly, we answer [Krapp et al. 2021,
Question 7.1]positively by presenting in Example 3.7 an ordered valued field that
is dense in its real closure but still admits a nontrivial
∅
-
Lor
-definable convex
valuation. In Section 4 we turn to ordered henselian valued fields and establish in
Theorem 4.2 that their value group (as an ordered abelian group) and their residue
field (as an ordered field) are always stably embedded as well as orthogonal. As a
result, we obtain that within ordered fields, every
Lor
-definable coarsening of an
Lr
-definable henselian valuation is already
Lr
-definable (see Corollary 4.4) and that
any
Lor
-definable (not necessarily convex) valuation is comparable to any henselian
valuation (see Proposition 4.5). The special class of almost real closed fields, which
are ordered fields admitting a henselian valuation with real closed residue field, is

DEFINABLE VALUATIONS ON ORDERED FIELDS 103
studied in Section 5. We show in Theorem 5.2 that within almost real closed fields
any
Lor
-definable valuation (which a priori does not have to be convex) is already
henselian and
Lr
-definable, thereby giving a negative answer to [Krapp et al. 2021,
Question 7.3]. Building on the results of the previous sections, we finally prove in
Section 6 the main theorem of this paper stating as follows:2
Theorem A (main theorem). Let
(K, <)
be an ordered field and let
v
be a henselian
valuation on K . If vis Lor-definable,then it is already Lr-definable.
2. Preliminaries
We denote by
N
the set of natural numbers without 0 and by
ω
the set of natural
numbers with 0.
We mostly follow the valuation-theoretic notation of [Engler and Prestel 2005].
For a valuation
v
on a field
K
, we write
Ov
for its valuation ring,
Mv
for the
maximal ideal of
Ov
,
Kv=Ov/Mv
for the residue field and
vK
for its value group
(written additively). For an element
x∈Ov
, its residue
x+Mv∈Kv
is denoted
x
,
where the valuation vin question will always be clear from context.
Given an ordering
<
on
K
(always compatible with the field structure), a valuation
v
is called convex (with respect to
<
) if
Ov
is a convex set in the usual sense. See
[Engler and Prestel 2005, Section 2.2.2]for a number of equivalent conditions. In
particular,
v
is convex if and only if the ordering
<
induces an ordering on the
residue field
Kv
in the natural way. This residue ordering is then also denoted by
<
,
which should not lead to confusion.
For a given field
K
and ordered abelian group
G
, we write
K((G))
for the Hahn
field consisting of those formal sums
Pg∈Gagtg
with coefficients
ag∈K
,
t
a
formal variable, whose support is well-ordered. See for instance [van den Dries
2014, Section 3.1]for details. We generally endow
K((G))
with its natural valuation
with value group
G
, assigning to each element of
K((G))×
the order of the lowest
nonzero coefficient. An ordering
<
on
K
can naturally be extended to
K((G))
by stipulating that an element of
K((G))×
is positive if and only if its nonzero
coefficient of lowest order is positive. We denote this ordering on
K((G))
again
by <.
For background on the model theory of valued fields, see [van den Dries 2014],
or [Marker 2002] for model theory more generally. We consider fields as structures
in the language of rings
Lr={+,−,·,
0
,
1
}
and ordered abelian groups as structures
in the language
Log = {+,−,
0
, <}
in the natural way. Given an ordering
<
on
a field
K
, we consider the ordered field
(K, <)
as a structure in the language
Lor =Lr∪{<}
. Given a valuation
v
on a field
K
, we usually work in the one-sorted
language
Lvf =Lr∪{O}
, where the unary predicate
O
is to be interpreted as the
2This result will be restated as Theorem 6.4.

104 P. DITTMANN, F. JAHNKE, L. S. KRAPP AND S. KUHLMANN
valuation ring
Ov⊆K
. In Section 4, we also work in a three-sorted language. For a
field
K
with an ordering
<
and a valuation
v
, we use the language
Lovf =Lvf ∪{<}
for (K, <, v).
If
a
is an element of an ordered abelian group (or an ordered field), we denote
its absolute value max{a,−a}by |a|.
A set is
L
-definable if it is definable in the first-order language
L
. If we wish to
specify that the parameters can be chosen to come from a specific set
A
, we write
A-L-definable.
3. Convex valuations
We start by giving sufficient conditions on the value group or the residue field
of a convex valuation
v
such that
v
is
Lor
-definable. By this, we strengthen all
of the three cases given in [Krapp et al. 2021, Theorem 5.3]. Subsequently, we
present several cases in which the given valuation is already
Lor
-definable without
parameters and discuss how these cases generalise other known definability results
in the literature.
Theorem 3.1. Let
(K, <)
be an ordered field and let
v
be a convex valuation on
K
.
Suppose that at least one of the following holds:
(i) vK is discretely ordered,i.e.,admits a least positive element.
(ii) vK is not closed in its divisible hull.
(iii) Kvis not closed in its real closure.
Then
v
is
Lor
-definable. Moreover
,
in the cases (i) and (ii)
,v
is definable by a
formula using only one parameter.
Proof. We may assume that vis nontrivial.
(i) Since
vK
is discretely ordered, we can choose
b∈K×
such that
v(b)
is the
minimal positive element of vK. Note that for every x∈Mv, we have v(x2/b)=
2
v(x)−v(b) >
0. Since every element
y∈Mv
satisfies
|y|<
1, we deduce that
Mv={x∈K| |x2/b|<
1
}
. Hence,
Mv
is
{b}
-
Lor
-definable, and
Ov
can be defined
in terms of Mv.
(ii) Since
vK
is not closed in its divisible hull, we can take
γ∈vK
and
n>
1 such
that
γ /∈n·vK
but every open interval in
vK
containing
γ
contains an element
of n·vK. Let b∈Kwith b>0 and v(b)=γ, and set
Sb:= {x∈K|x≥0 and xn/b<1}={x∈K|x≥0 and nv(x)>γ}(3-1)
(where the equality uses that we cannot have
v(xn/b)=
0 since
γ /∈n·vK
). It now
suffices to prove that
Ov= {y∈K|y4Sb⊆Sb},(3-2)

DEFINABLE VALUATIONS ON ORDERED FIELDS 105
since the set on the right-hand side is
{b}
-
Lor
-definable. The inclusion
⊆
is clear
since the condition
nv(x)>γ
in (3
-
1) is stable under multiplying
x
by an element
of Ov.
For the inclusion
⊇
, suppose that
y∈K\Ov
, so
v(y) <
0. By the choice of
γ
,
we can take
z∈K
with
z>
0 and
γ+v(y) < nv(z) < γ −v(y)
. Now
z/y2∈Sb
since
nv(z/y2)=nv(z)−
2
nv(y) > nv(z)−v( y)>γ
, but
y4(z/y2) /∈Sb
since
nv(y4(z/y2)) =nv(z)+
2
nv(y) < nv(z)+v( y)<γ
. This proves that
Sb
is not
stable under multiplication by y4, completing the proof of (3-2).
(iii) Let
f∈Kv[X]
be the minimal polynomial of an element
x0∈R\Kv
, where
R
denotes the real closure of
(Kv, <)
, such that
x0
can be arbitrarily approximated by
elements of
Kv
. Then there are
a,b∈Kv
with
a<x0<b
such that the following
hold:
(1)
The polynomial
f
has exactly one zero in
{x∈R|a≤x≤b}
. In particular,
fchanges sign precisely once in this interval.
(2)
For any
ε∈Kv
with 0
< ε < b−a
, there exists
x∈Kv
with
a<x<x+ε < b
such that f(x)f(x+ε) < 0.
Passing to
−f
if necessary, we may assume that
f(a) <
0
<f(b)
. Let
F∈K[X]
be a lift of
f
, let
a0,b0∈K
be lifts of
a
and
b
, and consider the
Lor
-definable set
Sgiven by
{x∈K|a0≤x≤b0and F(x) < 0}={x∈K|a0≤x≤b0and f(x) < 0},
where the equality uses that fhas no zero in Kv.
It now suffices to prove that for any y∈Kwe have
y+S⊆S⇐⇒ y∈Mvand y≥0,
since then
Mv
and therefore
Ov
are
Lor
-definable. For the implication
⇐
, let
y∈Mv
be nonnegative and
x∈S
. Then we have
x+y≥x≥a0
and
f(x+y)=f(x) <
0.
In particular x+y<band thus x+y<b0. Hence y+x∈S, as desired.
For the implication
⇒
, let
y∈K
with
y+S⊆S
. Since
a0∈S
, we must have
a0+y∈S
. Thus,
a0≤a0+y≤b0
, implying that 0
≤y≤b0−a0
and
y∈Ov
.
Note that
y= b−a
, as otherwise
f(y+a)=f(b) >
0, contradicting the fact that
y+a0∈S. Hence, y<b−a.
In order to show
y∈Mv
, suppose for a contradiction that
v(y)=
0, so 0
<y<b−a
.
By choice of
f
, we can find
z∈Ov
with
a<z<b
and
f(z) <
0
<f(z+y)
. Then
we have z∈Sbut z+y/∈S, in contradiction to our assumption y+S⊆S.□
In the following corollary, we point out several distinguished cases in which we
obtain Lor-definability without parameters.

106 P. DITTMANN, F. JAHNKE, L. S. KRAPP AND S. KUHLMANN
Corollary 3.2. Let
(K, <)
be an ordered field and let
v
be a convex valuation on
K
.
Suppose that at least one of the following holds:
(i) vK is p-regular but not p-divisible for some prime p ∈N.3
(ii) Kvis dense in its real closure but not real closed.
Then vis ∅-Lor-definable.
Proof. In both cases, at least one of the three conditions in Theorem 3.1 is satisfied:
if
vK
is
p
-regular but not
p
-divisible, then it is either discrete or not closed in
its divisible hull (see [Krapp et al. 2022, Proposition 3.3]). Thus, there exists an
Lor-formula ψ (x,z)and a parameter tuple b∈Ksuch that ψ(x,b)defines Ov.
(i) For any nontrivial convex subgroup
C≤vK
we have that
vK/C
is
p
-divisible
(see [Hong 2014, page 14]). Thus, any strict coarsening of
v
has a
p
-divisible value
group. As
vK
is not
p
-divisible,
Ov
is defined by the
Lor
-formula
ϕ( y)
expressing
the following:
∃z(“ψ(x,z)defines a nontrivial convex valuation ring whose
value group contains an element that is not p-divisible” ∧ψ( y,z)).
(ii) For any strict refinement
w
of
v
we have that
Kw
is real closed. Indeed, since
Kv
is dense in its real closure and the induced valuation
w
on
Kv
is nontrivial
and convex, we have that
Kw=(Kv)w
is real closed (see [Krapp et al. 2021,
Corollary 4.9]). Let
θ
be an
Lor
-sentence that is true in the theory of real closed
fields but does not hold in
Kv
.
4
Then
Ov
is defined by the
Lor
-formula
ϕ( y)
expressing the following:
∀z(“ψ(x,z)defines a nontrivial convex valuation ring whose
residue field does not satisfy θ”→ψ(y,z)). □
Remark 3.3. (i) The cases (i) and (ii) of Theorem 3.1 are optimal in the sense
that, in general, one cannot obtain parameter-free definability. More precisely, in
[Krapp et al. 2022, Examples 4.9 and 4.10]two ordered valued fields
(L1, <, v1)
and (L2, <, v2)are presented such that the following hold:
•v1and v2are henselian and thus convex;
•neither v1nor v2is ∅-Lor-definable;
•v1L1is discrete, and v2L2is not closed in its divisible hull.
3
Equivalently
,vK
contains a rank 1 convex subgroup
H
that is not
p
-divisible but
vK/H
is
p
-divisible. See [Hong 2013, Section
2
.
2
]for further characterisations of
p
-regular ordered abelian
groups.
4
For instance,
θ
may express that there exists a polynomial of a certain degree that does not have a
zero.

DEFINABLE VALUATIONS ON ORDERED FIELDS 107
(ii) The results for
Lr
-definability of henselian (rather than convex) valuations
corresponding to Theorem 3.1(i) and (ii) as well as Corollary 3.2(i) are proven
in [Hong 2014, Corollary 2],[Krapp et al. 2022, Theorem A]and [Hong 2013,
Lemmas 2.3.6 and 2.3.7], respectively.
(iii) Corollary 3.2 applies in particular if
vK
is of rank 1 (i.e.,
v
is the coarsest
nontrivial convex valuation on
K
) but nondivisible, or if
Kv
is archimedean (i.e.,
v
is the finest convex valuation) but not real closed.
We now apply the
Lor
-definability results above in order to obtain convex non-
henselian valuations that are definable in the language
Lor
but not in the language
Lr
.
Lemma 3.4. Let
K=Q(si|i∈ω),
where
{si|i∈ω}
is algebraically independent
over
Q
. Suppose that
v
is any valuation on
K
with
v(si)≥
0for any
i∈N
and v(s0) < 0. Then vis not Lr-definable in K .
Proof. First note that
K
is the
Lr
-definable closure of
S:= {si|i∈ω}
in
K
. Hence,
any Lr-definable subset of Kis S-Lr-definable.
Assume, for a contradiction, that some
Lr
-formula
ϕ(x,s)
defines
Ov
, where
s=(s0,s1,...,sn)for some n∈N. Since also the set
{s0,s1,...,sn,sn+1+s0,sn+2, . . .}
is algebraically independent over
Q
, we can set
α
to be the uniquely determined
Lr-automorphism on Kwith
α(si):= sifor i∈ω\{n+1},
sn+1+s0for i=n+1.
Now since v(sn+1)≥0, we have
K|H ϕ(sn+1,s).
As α(sn+1)=sn+1+s0and α(s)=s, we obtain
K|H ϕ(sn+1+s0,s),
i.e., sn+1+s0∈Ov. However, v(sn+1+s0)=v(s0) < 0, a contradiction. □
Example 3.5. We construct an ordered valued field
(K, <, v)
such that
K
is a sub-
field of the Laurent series field
R((Z))
,
vK
is discretely ordered,
Kv
is archimedean
and vis ∅-Lor-definable but not Lr-definable.
Let
k=Q(s1,s2, . . .) ⊆R
for some set
{si|i∈N} ⊆ R
that is algebraically
independent over
Q
. Consider the field
K=k(t)
, which we endow with the valuation
v
and ordering
<
given as the restriction of the valuation and ordering on the Hahn
field
k((t)) =k((Z))
. Then
vK=Z
and
Kv=k
, which is archimedean. Corollary 3.2
shows that
v
is
∅
-
Lor
-definable. Setting
s0=t−1
, we obtain
K=Q(s0,s1, . . .)
and

108 P. DITTMANN, F. JAHNKE, L. S. KRAPP AND S. KUHLMANN
v(s0)= −
1
<
0 as well as
v(si)=
0 for any
i∈N
. Hence, Lemma 3.4 implies that
vis not Lr-definable. □
Example 3.6. We construct an ordered valued field
(K, <, v)
such that
K
is a
subfield of the Puiseux series field
Sn∈NR((t1/n)) ⊆R((Q))
,
vK
is densely ordered,
Kvis archimedean and vis ∅-Lor-definable but not Lr-definable.
Let {ri|i∈N} ⊆ Rbe an algebraically independent set over Q. We set
s0=t−1∈R((Q)) and si=rit1/i∈R((Q))
for any
i∈N
. Let
K=Q(s0,s1, . . .) ⊆R((Q))
, and endow
K
with the order-
ing
<
and valuation
v
given as the restriction of the Hahn field ordering and
valuation on
R((Q))
. The archimedean residue field of
K
is not real closed, as
Kv⊆Q(r1,r2, . . .)
and thus, for instance,
√2/∈Kv
. Hence, Corollary 3.2(ii) shows
that
v
is
∅
-
Lor
-definable. Since
v(si)=1
i
for any
i∈N
, we have
vK=Q
. Finally,
we can apply Lemma 3.4 to show that
v
is not
Lr
-definable, as
v(s0)= −
1
<
0 and
v(si)=1
i>0 for any i∈N.□
To complete this section, we relate the property of an ordered field to admit a
nontrivial
Lor
-definable convex valuation to the property of being dense in the real
closure. It is known that an ordered field is either dense in its real closure or admits
a nontrivial
Lor
-definable convex valuation (see [Jahnke et al. 2017, Proposition 6.5;
Krapp et al. 2021, Fact 5.1]). However, the question whether these two cases are
nonexclusive (see [Krapp et al. 2021, Question 7.1]) has so far been open. In the
following, we answer this question positively by presenting an ordered field that is
dense in its real closure and whose natural valuation is ∅-Lor-definable.
Example 3.7. Let K=R(t1/n|n∈N)⊆R((Q)),
and endow
K
with the ordering
<
and valuation
v
given by restricting the ordering
and the valuation of the Hahn field
R((Q))
. Since
Kv=R
is real closed and
vK=Q
is divisible and of rank 1, [Viswanathan 1977, Proposition 9]implies that
K
is
dense in its real closure.
We first claim that the subset
R⊆K
is defined by the parameter-free
Lr
-formula
ϕ(x)given by
∃y1+x4=y4.
Clearly
R⊆ϕ( K)
, since for any
a∈R
we have
4
√1+a4∈R
. For the other inclusion,
take
a,y∈K
with 1
+a4=y4
. There exists
N∈N
such that
a,y∈R(t1/N)
. Letting
s=t1/N
, we have
R(s)|H ϕ(a)
, and therefore
a∈R
by [Maltsev 1960, Lemma 2].
This shows R=ϕ(K), as desired.
Therefore
Ov
, the convex hull of
R
in
K
, is defined by the parameter-free
Lor-formula
∃x1,x2(ϕ(x1)∧ϕ(x2)∧x1≤z≤x2). □

DEFINABLE VALUATIONS ON ORDERED FIELDS 109
Note that Example 3.7 presents an ordered valued field with real closed residue
field and divisible value group. Hence, none of the cases in Corollary 3.2 can be
applied, but we still obtain ∅-Lor-definability of the valuation.
4. Stable embeddedness
In this section, we establish that the ordered value group and the ordered residue
field of an ordered henselian valued field are stably embedded and orthogonal (see
Theorem 4.2). Stable embeddedness and orthogonality are best known to hold in
the unordered situation of henselian valued fields of equicharacteristic 0 (see, e.g.,
[van den Dries 2014, Section 5]), and various other well-behaved settings (see also
[Aschenbrenner et al. 2017, Section 8.3; Jahnke and Simon 2020, pages 171–172]).
As the key technical tool in the ordered context, we use Farré’s embedding lemma
[Farré 1993, Theorem 3.4].
We consider the three-sorted language L′
ovf given by
L′
ovf =(Lor,Lor,Log;·, v).
The three sorts are denoted by
f
(field sort),
r
(residue field sort) and
v
(value group
sort). The two unary function symbols ·and vhave sorts · : f→rand v:f→v.
Let
(K, <, v)
be an ordered valued field with convex valuation
v
. Then
(K, <, v)
induces an L′
ovf-structure
K=((K, <), ( Kv, <), vK;v, ·),
where the domains of the valuation and the residue map are extended to
K
by
setting
v(
0
)=
0 and
a=
0 for any
a∈K\Ov
. When considering definability in
K
,
we allow parameters from all sorts as usual. An
Lor
-formula
ϕ( y)
with parameters
from
Kv
may be considered as an
L′
ovf
-formula with parameters from
K
, where the
variables
y
become
r
-variables. Similarly, an
Log
-formula
ϕ(z)
with parameters
from
vK
may be considered as an
L′
ovf
-formula with parameters from
K
, where the
variables zbecome v-variables.
In the following, we prove a weak version of relative quantifier elimination; we
only consider formulas whose variables are varying over residue field and value
group.
Lemma 4.1. Let
(K, <, v)
be an ordered henselian valued field and let
T
be the
diagram of the
L′
ovf
-structure
K
as above
,
i.e.
,
the complete theory of
K
in the
language
L′
ovf
expanded by constants for all elements of
K
. Further
,
let
y
and
z
be
tuples of distinct
r
- and
v
-variables
,
respectively. Then any
L′
ovf
-formula
ϕ( y,z)
with parameters from Kis T -equivalent to an L′
ovf-formula of the form
(ψ1(y)∧θ1(z)) ∨···∨(ψN(y)∧θN(z)) (4-1)

110 P. DITTMANN, F. JAHNKE, L. S. KRAPP AND S. KUHLMANN
for some
N∈N,
where all
ψi
are
Lor
-formulas with parameters from
Kv
and all
θi
are Log-formulas with parameters from vK .
Proof. Let
2
be the set of all
L′
ovf
-formulas of the form (4
-
1), i.e., of all finite
disjunctions of conjunctions of an
Lor
-formula and an
Log
-formula with parameters
from
Kv
and
vK
, respectively. Modulo logical equivalence,
2
contains
⊤
as well as
⊥
and is closed under finite disjunctions, finite conjunctions and negation. Hence by
[Aschenbrenner et al. 2017, Corollary B.9.3], we only have to verify the following:
Let
p
and
q
be any two complete
T
-realisable
(y,z)
-types with
p∩2=
q∩2. Then p =q.
Let
p
and
q
be as described above and let
M
be a sufficiently saturated elementary
extension of
K
in which
p
and
q
are realised. Denote by
(M, <, v)
the ordered
henselian valued field inducing
M
, and let
r,r∗∈Mv
and
g,g∗∈vM
with
M|H p(r,g)∧q(r∗,g∗). By construction of the set 2, we have
tpvM(g/vK)=tpvM(g∗/vK)and tp(Mv,<) (r/Kv) =tp(Mv ,<)(r∗/Kv). (4-2)
Now let
K⪯M0⪯M
, with
M0
smaller than the saturation of
M
, such
that
r∈M0v
and
g∈vM0
, where
(M0, <, v)
denotes the ordered henselian val-
ued field inducing
M0
. Due to (4
-
2), we can fix an
Lor
-elementary embedding
σ:(M0v, <) →(Mv, <)
over
Kv
with
σ (r)=r∗
and an
Log
-elementary embedding
ρ:vM0→vMover vKwith ρ(g)=g∗(see [Marker 2002, Proposition 4.1.5]).
The quotient
vM0/vK
is torsion free, as
vK⪯vM0
. We can thus apply [Farré
1993, Theorem 3.4](with all appearing levels equal to 1) in order to obtain an em-
bedding
ι:(M0, <, v) →(M, <, v )
over
K
inducing both
σ
and
ρ
. Moreover, since
both
σ
and
ρ
are elementary embeddings, [Farré 1993, Corollary 4.2(ii)]implies that
(ι(M0), <, v) ⪯(M, <, v)
. Let
M′
0
be the
L′
ovf
-structure induced by
(ι(M0), <, v)
and denote by
h
the isomorphism
(ι, σ, ρ ) :M0→M′
0
over
K
. For any
ϕ( y,z)∈p
we have
M0|H ϕ(r,g)
. By applying
h
, we obtain
M′
0|H ϕ(σ (r), ρ (g))
and hence
M|H ϕ(r∗,g∗). This establishes p⊆q. The other inclusion follows likewise. □
Theorem 4.2. Let
(K, <, v)
be an ordered henselian valued field inducing the
L′
ovf-structure K. Then for any m,n∈ωthe following hold:
(i)
Any subset of
(Kv)m×(vK)n
definable in
K
is a finite union of rectangles of
the form
Y×Z,
where
Y⊆(Kv)m
is
Lor
-definable in
(Kv, <)
and
Z⊆(vK)n
is Log-definable in vK .
(ii) For any set B ⊆Om
vthat is Lovf-definable in (K, <, v),the set
B:= {(b1,...,bm)|(b1,...,bm)∈B} ⊆ (Kv)m
is Lor-definable in (Kv, <).

DEFINABLE VALUATIONS ON ORDERED FIELDS 111
(iii) For any set C ⊆(K×)nthat is Lovf-definable in (K, <, v),the set
v(C):= {(v(c1), . . . , v(cn)) |(c1,...,cn)∈C} ⊆ (vK)n
is Log-definable in vK .
Proof. By Lemma 4.1, any subset of
(Kv)m×(vK)n
definable in
K
can be defined
by a formula of the form (4
-
1). This immediately implies (i). In order to obtain (ii)
and (iii), it remains to notice that
B×v(C)⊆(Kv)m×(vK)n
is definable in
K
.
□
Corollary 4.3. Let
(K, <, v)
be an ordered henselian valued field and let
w
be an
Lovf-definable valuation on K. Then the following hold:
(i)
If
w
is a refinement of
v,
then the valuation
w
induced by
w
on
Kv
is
Lor
-
definable in (Kv, <).
(ii) If wis a coarsening of v,then v(O×
w)is Log-definable in vK .
Proof. Both
O×
w
and
Ow
are
Lovf
-definable in
K
. It remains to apply Theorem 4.2(ii)
to
B=Ow⊆Ov
in order to obtain (i) and Theorem 4.2(iii) to
C=O×
w⊆K×
in
order to obtain (ii).□
Corollary 4.4. Let
(K, <)
be an ordered field
,
let
v
be an
Lr
-definable henselian
valuation on
K
and let
w
be an
Lor
-definable coarsening of
v
. Then
w
is already
Lr-definable.
Proof. Corollary 4.3(ii) shows that
H=v(O×
w)
is
Log
-definable in
vK
. Since
wK=vK/H
, for any
x∈K
we have
x∈Ow
if and only if
v(x)≥
0
∨v(x)∈H
. As
v
is
Lr
-definable in
K
, the latter can be expressed as an
Lr
-formula with parameters
from K.□
For later use, we also deduce the following.
Proposition 4.5. Let
(K, <, v)
be an ordered henselian valued field. Then any
Lor-definable valuation won K is comparable to v.
Proof. We may suppose that neither
v
nor
w
are trivial. We first claim that the
valuation ring
Ow
contains a set
U= ∅
which is open in the topology induced by
v
.
This follows from a suitable form of relative quantifier elimination for henselian
valued fields of residue characteristic zero: In the terminology of [Cluckers et al.
2022], the
Lvf
-theory of the valued field
(K, v)
is
ω
-h-minimal [Cluckers et al. 2022,
Corollary 6.2.6(1.)]. Let
P
be the unary predicate on
RV =K×/(
1
+Mv)
given by
P(a(
1
+Mv))
if and only if
a>
0 for any
a∈K×
. Since elements of 1
+Mv
are
squares and hence automatically positive, the positive cone of
(K, <)
consists of
all
a∈K×
satisfying
P(a(
1
+Mv))
. As by [Cluckers et al. 2022, Theorem 4.1.19]
ω
-h-minimality is preserved under expansions by additional predicates on
RV
, we
obtain that the
Lovf
-theory of
(K, <, v)
is
ω
-h-minimal. In models of
ω
-h-minimal

112 P. DITTMANN, F. JAHNKE, L. S. KRAPP AND S. KUHLMANN
theories, any infinite definable set contains a nonempty
v
-open ball [Cluckers et al.
2022, Lemma 2.5.2], proving our claim about Ow.
It follows that
w
cannot be independent from
v
, since otherwise weak approxima-
tion [Engler and Prestel 2005, Theorem 2.4.1]would imply that the set
U∩(K\Ow)
is nonempty as the intersection of a v-open and a w-open set.
Let us now suppose for a contradiction that
w
and
v
are incomparable. Let
v0
be
the finest common coarsening of
v
and
w
, and
v
,
w
the induced valuations on the
residue field
Kv0
, which are nontrivial and independent. Writing
<
for the induced
ordering on
Kv0
, we now have an ordered henselian valued field
(Kv0, <, v)
with a
valuation
w
independent from
v
. By Corollary 4.3(i),
w
is
Lor
-definable in
(Kv0, <)
.
Since
w
is independent from the henselian valuation
v
on
Kv0
, this contradicts the
first part of the proof. □
5. Almost real closed fields
Following the terminology of [Delon and Farré 1996], we call a field
K
almost real
closed if it admits a henselian valuation
v
such that
Kv
is real closed. Almost real
closed fields arise in many valuation-theoretic contexts, and they have been studied
extensively (under varying names) both algebraically and model-theoretically (see,
e.g., [Brown 1988;Becker et al. 1999;Delon and Farré 1996]). Due to the Baer–
Krull representation theorem (see [Engler and Prestel 2005, pages 37–38]), any
almost real closed field admits at least one ordering. In this section, we consider
Lor
- and
Lr
-definability of valuations (which are a priori not necessarily convex)
in almost real closed fields. We establish in Theorem 5.2 that any
Lor
-definable
valuation is already
Lr
-definable and henselian. Thereby we give a negative answer
to [Krapp et al. 2021, Question 7.3].
Let
K
be an almost real closed field. Then for any prime
p∈N
there exists a
coarsest henselian valuation on K, denoted by vp, with the property that
Kvp=(Kvp)p∪[−(Kvp)p]
(see [Delon and Farré 1996, page 1126]).
Lemma 5.1. Let
p∈N
be prime and let
(K, v)
be a henselian valued field with
real closed residue field. Then
v(O×
vp)
is the maximal
p
-divisible convex subgroup
of vK .
Proof. By [Delon and Farré 1996, Proposition 2.5(iv)],
vpK=vK/v(O×
vp)
has no
nontrivial
p
-divisible convex subgroup, so
v(O×
vp)
contains the maximal
p
-divisible
convex subgroup of
vK
. On the other hand,
v
induces a valuation on the residue
field
Kvp
with value group
v(O×
vp)
, from which it is easy to see that
v(O×
vp)
must
itself be
p
-divisible by the defining property of
vp
(or see [Delon and Farré 1996,
Lemma 2.4(iii)]). □

DEFINABLE VALUATIONS ON ORDERED FIELDS 113
With the results of the last section at our disposal, we can now imitate the proof
of [Delon and Farré 1996, Theorem 4.4]to obtain the following.
Theorem 5.2. Let
K
be an almost real closed field and let
<
be any ordering on
K
.
Then any Lor-definable valuation on (K, <) is henselian and Lr-definable.
Proof. Let us denote by
vK
the canonical henselian valuation on
K
. Since
K
is
almost real closed,
vK
coincides with the natural, i.e., finest convex valuation
vnat
on
(K, <)
for any ordering
<
on
K
, as
vnat
is henselian [Delon and Farré 1996,
Proposition 2.1(iv)].
Let
v
be an
Lor
-definable valuation on
K
. By Proposition 4.5,
v
and
vK
are
comparable. If
v
is a coarsening of
vK
, then it is also henselian. Otherwise,
vK
is
a strict coarsening of
v
. Thus, by Corollary 4.3(i) the nontrivial valuation that
v
induces on
KvK
is
Lor
-definable in
(KvK, <)
, contradicting that
KvK
is real closed.
Hence, vis henselian.5
In order to show that
v
is
Lr
-definable, by [Delon and Farré 1996, Theorem 4.4]
it suffices to verify that
Gv:= vK(O×
v)
is
Log
-definable in
vKK
and that
Ovp⊆Ov
for some prime
p∈N
. The first condition follows from Corollary 4.3(ii). For the
other condition, we distinguish between two cases.
Case 1:
v= vK
.Then
Gv= {
0
}
and by [Delon and Farré 1996, Corollary 4.3]we
have Gp≤Gvfor some prime p∈N, where Gpdenotes the maximal p-divisible
convex subgroup of
vKK
. Now
Gp=vK(O×
vp)
by Lemma 5.1. Hence,
Ovp⊆Ov
,
as required. This establishes that vis Lr-definable in K.
Case 2: v=vK.Consider the set of formulas
p(x)= {x>n∧v(x)=0|n∈N}.
This set is finitely satisfiable in
(K, <, v)
, i.e., a type. Hence, for some elementary
extension
(L, <, v∗)
of
(K, <, v)
, there is some
x∈L
with
v∗(x)=
0 and
x>n
for all
n∈N
. In particular,
v∗
is a strict coarsening of the natural valuation
vL
on
(L, <)
. By Case 1,
v∗
is
Lr
-definable in
L
. Thus, there exists an
Lr
-formula
ϕ(x,y)such that
(L, <, v∗)|H ∃y∀x(v∗(x)≥0↔ϕ(x,y)).
By elementary equivalence, there exists
b∈K
such that
ϕ(x,b)
defines
v
in
K
.
□
6. Henselian valuations
We now consider definability in general ordered henselian valued fields. Throughout
this section, we freely use Farré’s Ax–Kochen–Ershov principles [Farré 1993,
5
An alternative argument is the following: If
K
is almost real closed, then one can show that
(K, <)
has NIP. In this case, any
Lor
-definable valuation on
K
is henselian by [Halevi et al. 2020,
Corollary 5.8].

114 P. DITTMANN, F. JAHNKE, L. S. KRAPP AND S. KUHLMANN
Corollary 4.2](with all levels equal to 1 in the notation there), stating that two
ordered henselian valued fields are elementarily equivalent in
Lovf
if and only if
the ordered residue fields and the value groups are so, and similarly for elementary
extensions.
Our first step is to show that a henselian valuation that is “slippery” in a precise
sense involving residue field and value group cannot be Lor-definable.
Lemma 6.1. Let (K, <, v) be an ordered henselian valued field satisfying
(Kv, <) ≡(L((Q)), <) and vK≡0⊕Q
for some ordered field
L
and some ordered abelian group
0
. Then
v
is not
Lor
-
definable.
Proof. Assume, for a contradiction, that
v
were
Lor
-definable. Fix an
Lor
-formula
ϕ(x,y)
such that for some
b∈K
the valuation ring
Ov
is defined by
ϕ(x,b)
. Since
some instance of ϕ(x,y)also defines the valuation win any
(M, <, w) ≡(K, <, v),
we may assume that (K, <, v) is in fact equal to
(L((Q)) ((Q))((0))
| {z }
w
, <, w),
where
w
denotes the power series valuation with value group
0⊕Q
ordered
lexicographically. Let
v0
denote the power series valuation on
K
with value group
0and residue field L((Q))((Q)).
Applying Corollary 4.3(i) to the ordered residue field of
(K, <, v0)
, the
Lor
-
definability of
v
implies that
v
(i.e., the valuation induced by
v
on the residue field
of its coarsening v0) is also Lor-definable on (L((Q))((Q)), <).
Applying Corollary 4.3(ii) to the value group of
(L((Q))((Q)), <, v)
, the convex
subgroup
Q
corresponding to
v
is already
Log
-definable in
Q⊕Q
. This is a
contradiction, as
Q⊕Q
is divisible, and divisible ordered abelian groups admit no
nontrivial proper Log-definable subgroups. □
We now prove a lemma used to define coarsenings of a valuation that is essentially
already shown in [Jahnke and Koenigsmann 2017]. It states that although in an
ordered abelian group
G
, the smallest convex subgroup containing a given element
γ∈G
need not be definable (e.g., the convex subgroup generated by
(
0
,
1
)
in the
lexicographic sum
Q⊕Z
is not definable), it is definable up to
p
-divisible “noise”.
Lemma 6.2. Let
p∈N
be prime. There exists an
Log
-formula
ϕ(x,y)
such that
the following holds
:
Let
G
be an ordered abelian group and
γ∈G>0,
and let
⟨γ⟩
denote the smallest convex subgroup of
G
that contains
γ
. Then the set
1γ⊆G

DEFINABLE VALUATIONS ON ORDERED FIELDS 115
defined by
ϕ(x, γ )
in
G
is the maximal convex subgroup of
G
containing
γ
such
that 1γ/⟨γ⟩is p-divisible.
Proof. We set ϕ(x, γ ) to express
[0,p|x|] ⊆ [0,pγ]+ pG.
By [Jahnke and Koenigsmann 2017, Lemma 4.1],
1γ
is a convex subgroup of
G
with
γ∈1γ
such that no nontrivial convex subgroup of
G/1γ
is
p
-divisible. In
particular, for every convex subgroup
1
of
G
properly containing
1γ
, the group
1/⟨γ⟩is not p-divisible, since it has 1/1γ≤G/1γas a quotient.
On the other hand, every positive element
δ∈1γ
can by definition be written as
the sum of an element of
[
0
,pγ]⊆⟨γ⟩
and an element of
pG
, which implies that
1γ/⟨γ⟩is p-divisible. □
We extract the following consequence of the definability results of [Jahnke
and Koenigsmann 2015]. See the introduction of that paper for the notion of
p-henselianity used in the proof.
Proposition 6.3. Let
(K, v)
be a henselian valued field such that the residue field
Kv
is neither separably closed nor real closed. Then there exists an
Lr
-definable
(not necessarily henselian)refinement wof v.
Proof. Note that
Kv
either has a Galois extension of degree divisible by some prime
p= 2 or it only has Galois extensions of 2-power degree.
In the latter case, since
Kv
is neither separably closed nor real closed, it has
Galois extensions of degree 2 but is not Euclidean. Since
v
is henselian and thus,
in particular, 2-henselian, we can now apply [Jahnke and Koenigsmann 2015,
Corollary 3.3]to obtain that it admits an
Lr
-definable refinement. Indeed, let
v2
K
denote the canonical 2-henselian valuation (see [Jahnke and Koenigsmann 2015,
page 743]). Then by [Jahnke and Koenigsmann 2015, Corollary 3.3],
v2
K
is
Lr
-
definable if
Kv2
K
is non-Euclidean, otherwise the coarsest 2-henselian valuation
with Euclidean residue field v2∗
Kis Lr-definable. Either is a refinement of v.
Thus, we now assume that there is a prime
p=
2 such that
Kv
has a finite
Galois extension
L
of degree divisible by
p
. Then there is a finite separable
extension
M/Kv
such that
L/M
is a finite Galois extension of degree
pn
for some
n>
0 (e.g., take
M
to be the fixed field of the
p
-Sylow subgroup of
Gal(L/Kv)
inside
L
). Let
F0/K
be a finite separable extension such that the (by henselianity
unique) prolongation of
v
to
F0
has residue field
M
(see [Engler and Prestel 2005,
Theorem 5.2.7(2)]for the existence of
F0
). Consider
F=F0
if the characteristic of
K
is
p
, and
F=F0(ζp)
otherwise, where
ζp
is a primitive
p
-th root of unity. Then
F/K
is a finite separable extension, and the residue field of the unique prolongation
u
of
v
to
F
is a finite extension of
M
. In particular,
Fu
admits Galois extensions of
p
-power degree, e.g., the compositum of
L
and
Fu
. Therefore
u
is a henselian (and

116 P. DITTMANN, F. JAHNKE, L. S. KRAPP AND S. KUHLMANN
thus in particular
p
-henselian) valuation with
Fu = F u(p)
, and hence coarsens
the canonical
p
-henselian valuation
vp
F
of
F
, which is
∅
-
Lr
-definable in
F
by the
main theorem of [Jahnke and Koenigsmann 2015].
Now
F
is interpretable in
K
as the splitting field of a separable polynomial
(see [Marker 2002, page 31]). Hence
w=vp
F|K
is
Lr
-definable in
K
. Lastly,
v=u|Kis a coarsening of w, as uis a coarsening of vp
Fin F.□
We can now state our main theorem about definability of henselian valuations
on ordered fields. In the proof, we need the following notion: for
n∈N
, we say
that a valuation
v
on a field
K
is
n≤
-henselian if Hensel’s lemma holds in
(K, v)
for all polynomials of degree at most
n
. Note that for a fixed
n
, the property of
n≤-henselianity is elementary in the language Lvf.
Theorem 6.4. Let
(K, <, v)
be an ordered henselian valued field. If
v
is
Lor
-
definable,then it is Lr-definable.
Proof. If
v
is trivial, then the proof is clear; thus we assume that
v
is nontrivial from
now on. If
K
is almost real closed, then the result follows from Theorem 5.2, and
hence we also assume that
K
is not almost real closed. Let
vK
denote the canonical
henselian valuation on
K
, i.e., the finest henselian valuation on
K
. In particular,
vK
is a (not necessarily proper) refinement of
v
. The residue field
KvK
carries an
ordering induced by the ordering on
K
, but is not real closed since
K
is not almost
real closed by assumption. By Proposition 6.3, we may thus fix an
Lr
-definable
refinement wof vK(and hence of v).
Now, by Lemma 6.1, we can make the following case distinction.
Case 1: vK≡ 0⊕Qfor any ordered abelian group 0.
Let
1v≤wK
be the convex subgroup such that
wK/1v=vK
. We first show
that there is a prime
p
such that
vK
contains no nontrivial
p
-divisible convex
subgroup. Assume for a contradiction that
vK
contains a nontrivial
p
-divisible
convex subgroup for every prime
p
. Then any sufficiently saturated elementary
extension
G∗
of
vK
contains a nontrivial convex divisible subgroup
Q
. Now,
[Schmitt 1982, Lemma 1.11]implies
vK≡G∗≡G∗/Q⊕Q≡G∗/Q⊕Q,
contradicting that we are in Case 1.
Hence, we can fix some prime
p
such that
vK
does not contain any nontrivial
p
-divisible convex subgroup. By Lemma 6.2, there exists an
Log
-formula
ϕ(x,y)
such that for any positive
γ∈wK
, the subgroup
1γ
of
wK
defined by
ϕ(x, γ )
is the maximal convex subgroup of
wK
containing
⟨γ⟩
and such that
1γ/⟨γ⟩
is
p-divisible. In case we choose γ∈1v, we have 1γ≤1v: otherwise
⟨γ⟩ ≤ 1v⪇1γ

DEFINABLE VALUATIONS ON ORDERED FIELDS 117
implies that
1γ/1v≤vK
is a nontrivial convex subgroup which is
p
-divisible
since it is a quotient of the p-divisible group 1γ/⟨γ⟩.
For every
γ∈wK
, let
uγ
be the
Lr
-definable coarsening of
w
on
K
with value
group
wK/1γ
. Since
1γ
is uniformly
Log
-definable in
wK
, also
uγ
is uniformly
Lr
-definable in
K
, i.e., there exists an
Lr
-formula
ψ(x,y,z)
and a parameter tuple
b∈Ksuch that for every a∈K×the formula ψ(x,b,a)defines uw(a).
If
uγ
is already henselian for some
γ∈1v
, then
v
is an
Lor
-definable coarsening
of an
Lr
-definable henselian valuation and hence
v
is
Lr
-definable by Corollary 4.4.
Thus, we assume that for every γ∈1vthe valuation uγis not henselian.
First suppose that there is some
n∈N
such that for every
γ∈1v
we have
that
uγ
is not
n≤
-henselian. Let
B
be the
Lr
-definable subset of
K
consisting of
all
a∈K×
such that
uw(a)
is not
n≤
-henselian. We claim that
w(B)=1v
holds.
Let
a∈K×
and set
γ=w(a)
. First suppose that
γ∈1v
. Then
uγ=uw(a)
is
not
n≤
-henselian. Thus,
a∈B
and
γ∈w(B)
. Conversely, suppose that
γ /∈1v
.
Then
1v⪇1γ
and thus
uγ
is a strict coarsening of
v
. Since
v
is henselian,
uγ
is
n≤
-henselian. Hence,
a/∈B
and
γ /∈w(B)
, as required. Thus, in this case
v
is
Lr
-definable as
Ov
consists exactly of all
x∈K
with
w(x)≥
0
∨w(x)∈w(B)
,
which is an Lr-definable condition as wis Lr-definable.
Now suppose that for every
n∈N
there exists
γn∈1v
such that
uγn
is
n≤
-
henselian. Then for every
n∈N
, there is some
an∈K
(with
w(an)=γn
) such
that
ψ(x,b,an)
defines an
n≤
-henselian refinement of
v
in
(K, <, v)
. Hence, in
some sufficiently saturated elementary extension
(K∗, <, v∗)
of
(K, <, v)
, there
exists
a∈K∗
such that
ψ(x,b,a)
defines a henselian refinement
u∗
of
v∗
. Since
v∗
is
Lor
-definable by the same formula in
K∗
as
v
in
K
, it is an
Lor
-definable
coarsening of the
Lr
-definable henselian valuation
u∗
. Hence,
v∗
is
Lr
-definable in
K∗by Corollary 4.4 and thus also vis Lr-definable in K.
Case 2: (Kv , <) ≡ (L((Q)), <) for any ordered field (L, <).
First suppose that
v= vK
. Then
vK
is strictly finer than
v
. Assume that
vK(Kv)
is divisible, where
vK
denotes the valuation induced by
vK
on
Kv
. Then
vK(Kv)
is
elementarily equivalent to Qas an ordered abelian group, and thus we obtain
(Kv, <) ≡(KvK((vK(Kv))), <) ≡(KvK((Q)), <),
in contradiction to the assumption of Case 2.
Therefore we can assume that
vK(Kv) =1
is nondivisible. We show that there
exists some Lr-definable henselian refinement uγof v.
We may assume that
1
does not have a rank 1 quotient: otherwise we could con-
sider a sufficiently saturated elementary extension
(K∗, <∗, v∗, w∗)
of
(K, <, v, w)
in which — by the definability of the refinement
w
of
vK
—
w∗(K∗v∗)
(and hence
also
vK∗(K∗v∗)
) has no rank 1 quotient. Just as in Case 1, an
Lr
-definition of
v∗

118 P. DITTMANN, F. JAHNKE, L. S. KRAPP AND S. KUHLMANN
would give rise to an
Lr
-definition of
v
. We claim that there is a prime
p
such that
1
has no nontrivial
p
-divisible quotient. If not, then some saturated elementary
extension
1∗
of
1
has a divisible nontrivial quotient
1∗/ 0
, where
0
is a convex
proper subgroup. Then, as before, we have
(Kv, <) ≡(KvK((vK(Kv))), <) =(KvK((1)), <)
≡(KvK((1∗)), <) ≡(KvK((0))((1∗/ 0)), <) ≡(KvK((0))((Q)), <),
contradicting that we are in Case 2. Hence, there is a prime
p
such that
1
is
p
-antiregular, i.e.,
1
has no nontrivial
p
-divisible quotient and no rank 1 quotient
(see [Jahnke and Koenigsmann 2017, page 670]).
Recall that
w
is an
Lr
-definable refinement of
vK
. Then, there are convex
subgroups
1vK⪇1v≤wK
with vKK=wK/1vKand vK=wK/1v.
Let
γ∈1v\1vK
be positive, and let
⟨γ⟩ ≤ wK
denote the smallest convex
subgroup containing
γ
. Since the convex subgroups of
wK
are ordered by inclusion,
we have
1vK⪇⟨γ⟩ ≤ 1v.
Note that
⟨γ⟩
need not be
Log
-definable in
wK
. However, by Lemma 6.2, the
maximal convex subgroup
1γ≤wK
that contains
⟨γ⟩
and such that
1γ/⟨γ⟩
is
p-divisible is Log-definable in wK.
We claim that
1γ≤1v
, i.e., that
1γ
corresponds to an
Lr
-definable refinement
of
v
. Assume for a contradiction that we have
1v⪇1γ
. Since
1v
contains
⟨γ⟩
,
the choice of
1γ
implies that
1v/⟨γ⟩
is
p
-divisible. If
⟨γ⟩ = 1v
, then
1v/⟨γ⟩
is a nontrivial
p
-divisible quotient of
1=1v/1vK
, a contradiction. Otherwise,
⟨γ⟩= 1v
and the quotient of
⟨γ⟩
by its maximal convex subgroup not containing
γ
is of rank 1, also a contradiction. Thus, we have found an
Lr
-definable refinement
of v. Since we have
1vK⪇1γ,
this refinement is a coarsening of vKand thus henselian.
Now suppose that
v=vK
. If
Kv
is not
t
-henselian, then
v
is
Lr
-definable by
[Fehm and Jahnke 2015, Proposition 5.5]. Hence, suppose that
Kv
is
t
-henselian.
Then for a sufficiently saturated elementary extension
(K∗, <∗, v∗)⪰(K, <, v)
the
residue field
K∗v∗⪰Kv
is itself henselian. Since
v
is
Lor
-definable in
K
, also
v∗
is
Lor
-definable in
K∗
. However, since
K∗v∗
is henselian,
v∗
is not the canonical
henselian valuation of
K∗
, and therefore by the arguments above
v∗
is already
Lr-definable in K∗. Hence, also vis Lr-definable in K.□

DEFINABLE VALUATIONS ON ORDERED FIELDS 119
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Received 11 Jul 2022.
PHILIP DITTMANN:
philip.dittmann@tu-dresden.de
Institut für Algebra, Technische Universität Dresden, Dresden, Germany
FRANZISKA JAHNKE:
franziska.jahnke@uni-muenster.de
Mathematisches Institut, Fachbereich Mathematik und Informatik, Westfälische Wilhelms-Universität
Münster, Münster, Germany
LOTH AR SE BA ST IA N KRA PP:
sebastian.krapp@uni-konstanz.de
Fachbereich Mathematik und Statistik, Universität Konstanz, Konstanz, Germany
SAL MA KUHLMANN:
salma.kuhlmann@uni-konstanz.de
Fachbereich Mathematik und Statistik, Universität Konstanz, Konstanz, Germany
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T
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Model Theory
no. 1 vol. 2 2023
1Keisler measures in the wild
GABRIEL CONAN T, KYL E GANNON and JAME S HANSON
69Quasi groupes de Frobenius dimensionnels
SAMUEL ZAMOUR
101Definable valuations on ordered fields
PHILIP DITTMANN, FRANZISKA JAHNKE, LOTHAR
SEBAS TIAN KR APP and SALMA KUHLMANN
121A note on geometric theories of fields
WIL L JOHN S ON and JI NHE YE
Model Theory 2023 vol. 2 no. 1