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Strongly NIP almost real closed fields

September 2021
Mathematical logic quarterly MLQ 67(3)

DOI:10.1002/malq.202000060

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University of Zurich

The following conjecture is due to Shelah–Hasson: Any infinite strongly NIP field is either real closed, algebraically closed, or admits a non‐trivial definable henselian valuation, in the language of rings. We specialise this conjecture to ordered fields in the language of ordered rings, which leads towards a systematic study of the class of strongly NIP almost real closed fields. As a result, we obtain a complete characterisation of this class.

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Math. Log. Quart. 67, No. 3, 321–328 (2021) / DOI 10.1002/malq.202000060

Strongly NIP almost real closed elds

Lothar Sebastian Krapp1,∗, Salma Kuhlmann1,∗∗,and Gabriel Lehéricy1, 2,∗∗∗

1Fachbereich Mathematik und Statistik, Universität Konstanz, 78457 Konstanz, Germany

2École supérieure d’ingénieurs Léonard-de-Vinci, Pôle Universitaire Léonard de Vinci, 92 916 Paris La

Défense Cedex, France

Received 13 August 2020, revised 22 December 2020, accepted 25 January 2021

Published online 28 September 2021

The following conjecture is due to Shelah–Hasson: Any innite strongly NIP eld is either real closed, alge-

braically closed, or admits a non-trivial denable henselian valuation, in the language of rings. We specialise

this conjecture to ordered elds in the language of ordered rings, which leads towards a systematic study of the

class of strongly NIP almost real closed elds. As a result, we obtain a complete characterisation of this class.

1 Introduction

The study of tame ordered algebraic structures has received a considerable amount of attention since the notion

of o-minimality was introduced in [23]. Frequently, the goal is to give a complete characterisation of such model-

theoretically well-behaved structures in terms of their algebraic properties. For instance, a (totally) ordered group

is o-minimal if and only if it is abelian and divisible (cf. [23, Proposition 1.4, Theorem 2.1]) and an ordered eld

is o-minimal if and only if it is real closed (cf. [23, Proposition 1.4, Theorem 2.3]). In fact, these characterisations

hold true under the more general tameness condition of weak o-minimality (cf. [4, p. 117] and [21, Theorem 5.1,

Theorem 5.3]). While o-minimality and weak o-minimality are comparatively strong tameness conditions, the

property NIP (‘not the independence property’), introduced in [26], is at the other end of the spectrum; indeed,

any o-minimal structure is also NIP (cf. [23, Corollary 3.10] and [24]).

A strategy to examine the class of NIP ordered algebraic structures is to rst consider renements of the property

NIP. In this regard, we are mainly concerned with dp-minimal as well as strongly NIP1ordered groups and elds.

Since any weakly o-minimal theory is dp-minimal (cf. [5, Corollary 4.3]), we obtain the following hierarchy:

o-minimal →weakly o-minimal →dp-minimal →strongly NIP →NIP.

In particular, any divisible ordered abelian group and any real closed eld are strongly NIP. A full algebraic

characterisation of dp-minimal ordered elds follows from [13, Theorem 6.2] (cf. Proposition 4.4), but so far

there has not been a systematic study of the strongly NIP ordered eld case.

With this paper, we contribute to the analysis of strongly NIP ordered elds, in light of a conjecture suggested by

Shelah in [28, Conjecture 5.34(c)]. This conjecture was veried by Johnson for dp-minimal elds in [15, Theorem

1.6] and more recently for dp-nite2elds in [16, Theorem 1.2]. Shelah’s conjecture was reformulated as follows

in [7, p. 820], [10, p. 2214], [11, p. 720] and [12, p. 183].

Shelah–Hasson Conjecture. Let K be an innite strongly NIP eld. Then K is either real closed, or alge-

braically closed, or admits a non-trivial Lr-denable3henselian valuation.

The Shelah–Hasson Conjecture specialised to ordered elds reads as follows.

∗Corresponding author; e-mail: sebastian.krapp@uni-konstanz.de

∗∗ E-mail: salma.kuhlmann@uni-konstanz.de

∗∗∗ E-mail: gabriel.lehericy@uni-konstanz.de

1A strongly NIP theory is usually said to be strongly dependent.

2Dp-niteness can be classed between dp-minimality and strong NIP in the picture above.

3Throughout this work ‘denable’ always means ‘denable with parameters’.

This is an open access article under the terms of the CreativeCommonsAttribution-NonCommercial License, which permits use, distribution and reproduction in any medium, provided the

original work is properly cited and is not used for commercial purposes.

Konstanzer Online-Publikations-System (KOPS)

URL: http://nbn-resolving.de/urn:nbn:de:bsz:352-2-10p592t4efjsk8

322 L. S. Krapp, S. Kuhlmann, and G. Lehéricy: Strongly NIP almost real closed elds

Conjecture 1.1 Let (K,<)be a strongly NIP ordered eld. Then K is either real closed or admits a non-trivial

Lor-denable henselian valuation.

The valuation and model theory of almost real closed elds (cf. [3] and also Denition 3.1) is well-understood.

A main achievement of our paper is to show that Conjecture 1.1 is equivalent to the following.

Conjecture 1.2 Any strongly NIP ordered eld (K,<)is almost real closed.

We highlight this result in the following theorem.4

Theorem 1.3 Conjecture 1.1 and Conjecture 1.2 are equivalent.

Dp-minimal and more generally strongly NIP ordered abelian groups have already been fully classied (cf. [13,

Proposition 5.1] and [11, Theorem 1]). Moreover, Conjecture 1.1 has already been veried for dp-minimal ordered

elds in [13, Corollary 6.6]. By a careful analysis of the results of [13], we deduce in Proposition 4.4 that also

Conjecture 1.2 holds for dp-minimal ordered elds. Actually, we prove that an ordered eld is dp-minimal if and

only if it is almost real closed with respect to some dp-minimal ordered abelian group G. This latter result raises

the question whether the analogous classication holds for strongly NIP ordered elds. We therefore address the

following:

Question 1.4 Is it true that an ordered eld (K,<) is strongly NIP if and only if it is almost real closed with

respect to some strongly NIP ordered abelian group G?

In § 4, we prove our other main result:5

Theorem 1.5 Let (K,<)be an almost real closed eld with respect to some ordered abelian group G. Then

(K,<)is strongly NIP if and only if G is strongly NIP.

This answers positively the backward direction of Question 1.4 about the classication of strongly NIP ordered

elds. Thus, only the following question remains open.

Question 1.6 Is every strongly NIP ordered eld (K,<) almost real closed with respect to some strongly NIP

ordered abelian group G?

Finally, we note that a positive answer to Question 1.6 would verify Conjecture 1.2. Conversely, if Conjec-

ture 1.2 is veried, then by Theorem 1.5 the answer to Question 1.6 is positive.

We conclude in § 6 by stating some further open questions motivated by this work.6

2 General preliminaries

The set of natural numbers with 0 is denoted by N0, the set of natural numbers without 0 by N.LetLr=

{+,−,·,0,1}be the language of rings, Lor =Lr∪{<}the language of ordered rings and Log ={+,0,<}the

language of ordered groups. Throughout this work, we abbreviate the Lr-structure of a eld (K,+,−,·,0,1) sim-

ply by K,theLor -structure of an ordered eld (K,+,−,·,0,1,<)by(K,<) and the Log -structure of an ordered

group (G,+,0,<)byG.

All notions on valued elds can be found in [8, 20]. Let Kbe a eld and va valuation on K. We denote the

valuation ring of vin Kby Ov,thevaluation ideal, i.e., the maximal ideal of Ov,byMv,theordered value group

by vKand the residue eld Ov/Mvby Kv.Fora∈Ovwe also denote a+Mvby a. For an ordered eld (K,<)

a valuation is called convex (in (K,<)) if the valuation ring Ovis a convex subset of K. In this case, the relation

a<b:⇐⇒ a= b∧a<bdenes an order relation on Kvmaking it an ordered eld. Note that in ordered elds,

henselian valuations are always convex:

Fact 2.1 ([17, Lemma 2.1]) Let (K,<)be an ordered eld and let vbe a henselian valuation on K. Then vis

convex on (K,<).

4This will be restated as Theorem 5.4.

5This will be restated as Theorem 4.12.

6A preliminary version of this work is contained in our arXiv preprint [18], which contains also a systematic study of Lor-denable

henselian valuations in ordered elds as well as of the class of ordered elds which are dense in their real closure. This systematic study, of

independent interest, will be the subject of a separate publication [19].

Math. Log. Quart. 67, No. 3 (2021) / www.mlq-journal.org 323

Let Lvf =Lr∪{Ov}be the language of valued elds, where Ovstands for a unary predicate. Let (K,Ov)be

a valued eld. An atomic formula of the form v(t1)≥v(t2), where t1and t2are Lr-terms, stands for the Lvf-

formula t1=t2=0∨(t2= 0∧Ov(t1/t2)). Thus, by abuse of notation, we also denote the Lvf-structure (K,Ov)

by (K,v). Similarly, we also call (K,<,v) an ordered valued eld. We say that a valuation vis L-denable for

some language L∈{Lr,Lor }if its valuation ring is an L-denable subset of K.

For any ordered abelian groups G1and G2, we denote the lexicographic sum of G1and G2by G1⊕G2.This

is the abelian group G1×G2with the lexicographic ordering (a,b)<(c,d)ifa<c,ora=cand b<d.

Let Kbe a eld and let vand wbe valuations on K. We write v≤wif and only if Ov⊇Ow. In this case we say

that wis ner than vand vis coarser than v. Note that ≤denes an order relation on the set of convex valuations

of an ordered eld. We call two elements a,b∈K archimedean equivalent (in symbols a∼b) if there is some

n∈Nsuch that |a|<n|b|and |b|<n|a|.LetG={[a]|a∈K×}, the set of archimedean equivalence classes of

K×. Equipped with addition [a]+[b]=[ab] and the ordering [a]<[b] dened by a∼ b∧|b|<|a|,thesetG

becomes an ordered abelian group. Then K×→G,a→ [a] denes a convex valuation on K. This is called the

natural valuation on Kand denoted by vnat.7

Let (k,<) be an ordered eld and let Gbe an ordered abelian group. We denote the Hahn eld with coefcients

in kand exponents in Gby k(( G)). The underlying set of k(( G)) consists of all elements in the group product g∈Gk

with well-ordered support, where the support of an element sis given by supps={g∈G|s(g)= 0}. We denote

an element s∈k(( G)) by s=g∈Gsgtg, where sg=s(g) and tgis the characteristic function on Gmapping gto

1 and everything else to 0. The ordering on k(( G)) i s give n b y s>0:⇐⇒ s(min supps)>0. Let vmin be the

valuation on k(( G)) g ive n b y vmin (s)=min suppsfor s= 0. Note that vmin is convex and henselian. Note further

that if kis archimedean, then vmin coincides with vnat.

We repeatedly use the Ax–Kochen–Ershov Principle for ordered elds. This follows from [9, Corollary 4.2(iii)],

where all appearing levels in the premise equal 1 (cf. [9, p. 916]).

Fact 2.2 (Ax–Kochen–Ershov Principle) Let (K,<,v)and (L,<,w)be two ordered henselian valued elds.

Then (Kv,<)≡(Lw,<)and vK≡wL if and only if (K,<,v)≡(L,<,w).

Since we do not use explicitly the denitions of the independence property (IP), ‘not the independence prop-

erty’ (NIP), strong NIP and dp-minimality, we refer the reader to [29] for all denitions in this regard. For a

structure N, we say that Nis NIP (respectively, strongly NIP and dp-minimal) if its complete theory Th(N)is

NIP (respectively, strongly NIP and dp-minimal). A well-known example of an IP theory is the complete theory of

the Lr-structure (Z,+,−,·,0,1) (cf. [29, Example 2.4]). Since Zis parameter-free denable in the Lr-structure Q

(cf. [25, Theorem 3.1]), also the complete Lr-theory of Qhas IP. Any reduct of a strongly NIP structure is strongly

NIP (cf. [27, Claim 3.14 (3)]) and any reduct of a dp-minimal structure is dp-minimal (cf. [22, Observation 3.7]).

3 Almost real closed elds

Algebraic and model theoretic properties of the class of almost real closed elds in the language Lrhave been stud-

ied in [3]; in particular, [3, Theorem 4.4] gives a complete characterisation of Lr-denable henselian valuations.

In the following, we prove some useful properties of almost real closed elds in the language Lor.

Denition 3.1 Let (K,<) be an ordered eld, Gan ordered abelian group and va henselian valuation on K.

We call Kan almost real closed eld (with respect to vand G)ifKvis real closed and vK=G.

Depending on the context, we may simply say that (K,<) is an almost real closed eld without specifying the

henselian valuation vor the ordered abelian group G=vK.

Remark 3.2 In [3], almost real closed elds are dened as pure elds which admit a henselian valuation with

real closed residue eld. However, any such eld admits an ordering, which is due to the Baer–Krull Representation

Theorem (cf. [8, p. 37f.]). We consider almost real closed elds as ordered elds with a xed order.

Due to Fact 2.1 and the following fact, we do not need to make a distinction between convex and henselian

valuations in almost real closed elds.

7Note that the ordered residue eld (Kvnat,<) is always archimedean. Note further that vnat is trivial if and only if (K,<) is archimedean.

324 L. S. Krapp, S. Kuhlmann, and G. Lehéricy: Strongly NIP almost real closed elds

Fact 3.3 ([3, Proposition 2.9]) Let (K,<)be an almost real closed eld. Then any convex valuation on (K,<)

is henselian.

[3, Proposition 2.8] implies that the class of almost real closed elds in the language Lris closed under ele-

mentary equivalence. We can easily deduce that this also holds in the language Lor.

Proposition 3.4 Let (K,<)be an almost real closed eld and let (L,<)≡(K,<). Then (L,<)is an almost

real closed eld.

Proof. SinceL≡K, we obtain by [3, Proposition 2.8] that Ladmits a henselian valuation vsuch that Lvis

real closed. Hence, (L,<) is almost real closed. 

Corollary 3.5 Let (K,<)be an ordered eld. Then (K,<)is almost real closed if and only if (K,<)≡

(R(( G)),<)for some ordered abelian group G.

P r o o f . The forward direction follows from Fact 2.2. The backward direction is a consequence of Proposi-

tion 3.4. 

Corollary 3.6 Let (K,<)be an almost real closed eld and let G be an ordered abelian group. Then (K(( G)),<)

is almost real closed.

Proof. Letvbe a henselian valuation on Ksuch that Kis almost real closed with respect to v. Since vmin

is henselian on K(( G)), we can compose the two henselian valuations vmin and vin order to obtain a henselian

valuation on K(( G)) with real closed residue eld (cf. [8, Corollary 4.1.4]). 

4 Strongly NIP ordered elds

In this section we study the class of strongly NIP ordered elds in light of Conjectures 1.1 & 1.2. A special class

of strongly NIP ordered elds are dp-minimal ordered elds. These are fully classied in [13]. In Proposition 4.4

below we show that our query (cf. Question 1.4) holds for dp-minimal ordered elds. An ordered group Gis called

non-singular if G/pG is nite for all prime numbers p.

Fact 4.1 ([13, Proposition 5.1]) An ordered abelian group G is dp-minimal if and only if it is non-singular.8

Fact 4.2 ([13, Theorem 6.2]) An ordered eld (K,<)is dp-minimal if and only if there exists a non-singular

ordered abelian group G such that (K,<)≡(R(( G)),<).

Lemma 4.3 Let (K,<)be a dp-minimal almost real closed eld with respect to some henselian valuation v.

Then vK is dp-minimal.

Proof. SinceKvis real closed, it is not separably closed. Thus, by [14, Theorem A], vis denable in the

Shelah expansion (K,<)Sh (cf. [14, § 2]) of (K,<). By [22, Observation 3.8], also (K,<)Sh is dp-minimal, whence

the reduct (K,v) is dp-minimal. By [28, Observation 1.4(2)]9, any structure which is rst-order interpretable in

(K,v) is dp-minimal (cf. also [2, 13]). Hence, also vKis dp-minimal. 

Proposition 4.4 Let (K,<)be an ordered eld. Then (K,<)is dp-minimal if and only if it is almost real closed

with respect to a dp-minimal ordered abelian group.

P r o o f . Suppose that (K,<) is almost real closed with respect to a dp-minimal ordered abelian group G.By

Fact 4.1, Gis non-singular. By Fact 2.2, we have (K,<)≡(R(( G)),<), which is dp-minimal by Fact 4.2. Hence,

(K,<) is dp-minimal.

Conversely, suppose that (K,<) is dp-minimal. By Fact 4.2, we have (K,<)≡(R(( G)),<) for some non-

singular ordered abelian group G. Since (R(( G)) ,<) is almost real closed, by Proposition 3.4 also (K,<)isalmost

real closed with respect to some henselian valuation v. By Lemma 4.3, also vKis dp-minimal, as required. 

As a result, we obtain a characterisation of dp-minimal archimedean ordered elds.

8The saturation condition in [13] can be dropped, as non-singularity of groups transfers via elementary equivalence.

9We thank Yatir Halevi for pointing out this reference to us.

Math. Log. Quart. 67, No. 3 (2021) / www.mlq-journal.org 325

Corollary 4.5 Let (K,<)be a dp-minimal archimedean ordered eld. Then K is real closed.

P r o o f . The only archimedean almost real closed elds are the archimedean real closed elds. This is due

to the fact that any henselian valuation won an archimedean eld Lis convex and thus trivial, whence the residue

eld of Lwis equal to L. Thus, by Proposition 4.4, any archimedean dp-minimal ordered eld is real closed. 

We now turn to strongly NIP almost real closed elds, aiming for a characterisation of these (cf. Theorem 4.12).

We have seen in Proposition 4.4 that every almost real closed eld with respect to a dp-minimal ordered abelian

group is dp-minimal. We obtain a similar result for almost real closed elds with respect to a strongly NIP ordered

abelian group. The following two results will be exploited.

Fact 4.6 ([11, Theorem 1]) Let G be an ordered abelian group. Then the following are equivalent:

(1) G is strongly NIP.

(2) G is elementarily equivalent to a lexicographic sum of ordered abelian groups i∈IGi, where for every

prime p, we have |{i∈I|pGi= Gi}| <∞, and for any i ∈I, we have |{pprime |[Gi:pGi]=∞}|<∞.

Details on angular component maps are given in [6, § 5.4f.]. Recall from § 2 that any henselian valuation

on an ordered eld is convex (cf. Fact 2.1) and thus naturally induces an ordering on the residue eld given by

a<b:⇐⇒ a= b∧a<b.

Observation 4.7 Let (K,<,v)be an ordered henselian valued eld and let ac : K×→Kv×be an angular

component map. Suppose that the induced ordering of K on Kvis Lr-denable. Then the ordering <is denable

in (K,v,ac).

Proof. Letϕ(x)beanLr-formula such that for any a∈Kvwe have a≥0 if and only if Kv|= ϕ(a). Then

the formula x= 0→ϕ(ac(x)) denes the positive cone of the ordering <on K.

Lemma 4.8 Let G be a strongly NIP ordered abelian group. Then the ordered Hahn eld (R(( G)) ,<)is

strongly NIP.

Proof. IfK=R((G)) is real closed, then we are done.

Otherwise let v=vmin. Then (K,v) is ac-valued with angular component map ac : K→Rgiven by ac(s)=

s(v(s)) for s= 0 and ac(0) =0. Following a similar argument as [12, p. 188], we obtain that (K,v,ac) is a strongly

NIP ac-valued eld; more precisely, (K,v,ac) eliminates eld quantiers in the generalised Denef–Pas language

(cf. [6, § 5.6], noting that both Kand Kvhave characteristic 0), whence by [12, Fact 3.5] we obtain that (K,v,ac)

is strongly NIP. Since Ris closed under square roots for positive elements, for any a∈Kwe have a≥0 if and

only if the following holds in K:

∃yy

2=ac(a)

(cf. Observation 4.7). Hence, the order relation <is denable in (K,v,ac). We obtain that (K,<) is strongly

NIP. 

Proposition 4.9 Let (K,<)be an almost real closed eld with respect to a strongly NIP ordered abelian group

and let G be a strongly NIP ordered abelian group. Then (K(( G)) ,<)is a strongly NIP ordered eld.

Proof. LetHbe a strongly NIP ordered abelian group such that (K,<) is almost real closed with respect

to Hand let wbe a henselian valution on Kwith wK=H. As in the proof of Corollary 3.6, we can compose the

valuation vmin on K(( G)) with won Kto obtain a henselian valuation on K(( G)) with real closed residue eld and

value group isomorphic to G⊕H. Hence, (K(( G)) ,<)≡(R(( G⊕H)) ,<). Since Gand Hare strongly NIP, also

G⊕His strongly NIP by Fact 4.6. Hence, by Lemma 4.8, also (K(( G)),<) is strongly NIP. 

Corollary 4.10 Let (K,<)be an almost real closed with respect to a henselian valuation vsuch that vKis

strongly NIP. Then (K,<)is strongly NIP.

P r o o f . This follows immediately from Proposition 4.9 by setting G={0}and H=vK.

For the proof of Theorem 4.12, we need one further result on general strongly NIP ordered elds, which will

also be used for the proof of Theorem 5.4.

326 L. S. Krapp, S. Kuhlmann, and G. Lehéricy: Strongly NIP almost real closed elds

Proposition 4.11 Let (K,<)be a strongly NIP ordered eld and let vbe a henselian valuation on K. Then

also (Kv,<)and vK are strongly NIP.

P r o o f . Arguing as in the proof of Lemma 4.3, we obtain that vis denable in (K,<)Sh.Now(K,<)Sh is

also strongly NIP (cf. [22, Observation 3.8]), whence (K,<,v) is strongly NIP. By [28, Observation 1.4(2)], both

(Kv,<) and vKare strongly NIP, as they are rst-order interpretable in (K,<,v). 

We obtain from Corollary 4.10 and Proposition 4.11 the following characterisation of strongly NIP almost real

closed elds.

Theorem 4.12 Let (K,<)be an almost real closed eld with respect to some ordered abelian group G. Then

(K,<)is strongly NIP if and only if G is strongly NIP.

Remark 4.13 Fact 4.6 and Theorem 4.12 give us the following complete characterisation of strongly NIP al-

most real closed elds: An almost real closed eld (K,<) is strongly NIP if and only if it is elementarily equivalent

to some ordered Hahn eld (R(( G)) ,<) where Gis a lexicographic sum as in Fact 4.6(2).

5 Equivalence of conjectures

Recall our two main conjectures.

Conjecture 1.1 Let (K,<)be a strongly NIP ordered eld. Then K is either real closed or admits a non-trivial

Lor-denable henselian valuation.

Conjecture 1.2 Any strongly NIP ordered eld is almost real closed.

In this section, we show that Conjectures 1.1 & 1.2 are equivalent (cf. Theorem 5.4).

Remark 5.1 (1) An ordered eld is real closed if and only if it is o-minimal (cf. [23, Proposition 1.4, The-

orem 2.3]). Hence, for any real closed eld K,ifO⊆Kis a denable convex ring, its endpoints must lie in

K∪{±∞}. This implies that any denable convex valuation ring must already contain K, i.e., is trivial. Thus, the

two cases in the consequence of Conjecture 1.1 are exclusive.

(2) Recall from § 2 that the eld Qis not NIP. By [1], the henselian valuation vmin is Lr-denable in Q(( Z)).

Hence, Proposition 4.11 yields that (Q(( Z)) ,<) is an example of an ordered eld which is not real closed, admits

a non-trivial Lor-denable henselian valuation but is not strongly NIP.

Lemmas 5.2 & 5.3 below are used in the proof of Theorem 5.4. For the rst result, we adapt the proof of [12,

Lemma 3.7] to the context of ordered elds.

Lemma 5.2 Assume that any strongly NIP ordered eld is either real closed or admits a non-trivial henselian

valuation10. Let (K,<)be a strongly NIP ordered eld. Then (K,<)is almost real closed with respect to the

canonical valuation, i.e., the nest henselian valuation on K.

Proof. Let(K,<) be a strongly NIP ordered eld. If Kis real closed, then we can take the natural valuation.

Otherwise, by assumption, the set of non-trivial henselian valuations on Kis non-empty. Let vbe the canonical

valuation on K. By Proposition 4.11, we have that (Kv,<) is strongly NIP. Note that Kvcannot admit a non-trivial

henselian valuation, as otherwise this would induce a non-trivial henselian valuation on Kner than v. Hence, by

assumption, Kvmust be real closed. 

The next result is obtained from an application of [11, Proposition 5.5].

Lemma 5.3 Let (K,<)be a strongly NIP ordered eld which is not real closed but is almost real closed with

respect to a henselian valuation v. Then there exists a non-trivial Lr-denable henselian coarsening of v.

P r o o f . By Proposition 4.11, we have that vK=Gis strongly NIP. Since Kis not real closed, Gis non-

divisible (cf. [8, Theorem 4.3.7]). By [11, Proposition 5.5], any henselian valuation with non-divisible value

group on a strongly NIP eld has a non-trivial Lr-denable henselian coarsening. Hence, there is a non-trivial

Lr-denable henselian coarsening uof v.

10 Note that this valuation does not necessarily have to be Lor-denable.

Math. Log. Quart. 67, No. 3 (2021) / www.mlq-journal.org 327

Theorem 5.4 Conjecture 1.1 and Conjecture 1.2 are equivalent.

P r o o f . Assume Conjecture 1.2, and let (K,<) be a strongly NIP ordered eld which is not real closed.

Then (K,<) admits a non-trivial henselian valuation v. By Lemma 5.3, it also admits a non-trivial Lr-denable

henselian valuation. Now assume Conjecture 1.1. Let (K,<) be a strongly NIP ordered eld. By Lemma 5.2, we

obtain that Kis almost real closed with respect to the canonical valuation v.

As a nal observation, we give twofurther equivalent formulations of Conjecture 1.2 which follow from results

throughout this work.

Observation 5.5 The following are equivalent:

(1) Any strongly NIP ordered eld (K,<)is almost real closed.

(2) For any strongly NIP ordered eld (K,<), the natural valuation vnat on K is henselian.

(3) For any strongly NIP ordered valued eld (K,<,v), whenever vis convex, it is already henselian.

P r o o f . (1) implies (3) by Fact 3.3. Suppose that (3) holds and let (K,<) be strongly NIP. Now vnat is

denable in the Shelah expansion (K,<)Sh, as it is the convex closure of Zin K. Hence, (K,<,vnat) is a strongly

NIP ordered valued eld. By assumption, vnat is henselian on K, which implies (2). Finally, suppose that (2) holds.

Let (K,<) be a strongly NIP ordered eld and (K1,<)anℵ1-saturated elementary extension of (K,<). Then

K1vnat =R, as any Dedekind cut on the rational numbers in K1is realised in K1.

More precisely, let a∈Rand set L={q∈Q|q<a}and R={q∈Q|a<q}. Then any nite subset of the

1-type p(x)={q<x|q∈L}∪{x<q|q∈R}is realised in Qand thus also in K1.Asp(x) is countable, the

ℵ1-saturation of K1implies that p(x) is realised in K1by some α∈K1. Since K1vnat is archimedean, it embeds

as an ordered eld into R, i.e., (Q,<)⊆(K1vnat,<)⊆(R,<). Finally, by application of the residue map, for

any q1,q2∈Qwith q1<a<q2we obtain q1≤α≤q2. Hence, α=a. Since awas chosen arbitrary, we obtain

K1vnat =R.

By assumption, vnat is henselian on K1, whence (K1,<) is almost real closed. By Proposition 3.4, also (K,<)

is almost real closed. 

6 Open questions

We conclude with open questions connected to results throughout this work. Conjecture 1.2 for archimedean elds

states that any strongly NIP archimedean ordered eld is real closed, as the only archimedean almost real closed

elds are the real closed ones. Corollary 4.5 shows that any dp-minimal archimedean ordered eld is real closed.

We can ask whether the same holds for all strongly NIP ordered elds.

Question 6.1 Let (K,<) be a strongly NIP archimedean ordered eld. Is Knecessarily real closed?

It is shown in [19] that any almost real closed eld which is not real closed cannot be dense in its real closure.

Thus, any dp-minimal ordered eld which is dense in its real closure is real closed. Moreover, if Conjecture 1.2

is true, then, in particular, a strongly NIP ordered eld which is not real closed cannot be dense in its real closure.

Question 6.2 Let (K,<) be a strongly NIP ordered eld which is dense in its real closure. Is (K,<) real closed?

Note that Question 6.2 is more general than Question 6.1, as a positive answer to Question 6.2 would automat-

ically tell us that any archimedean ordered eld is real closed (since every archimedean eld is dense in its real

closure).

Acknowledgement We started this research at the Model Theory, Combinatorics and Valued elds Trimester at the Institut

Henri Poincaré in March 2018. All three authors wish to thank the IHP for its hospitality.

The rst author was supported by a doctoral scholarship of Studienstiftung des deutschen Volkes as well as of Carl-Zeiss-

Stiftung, and by Werner und Erika Messmer-Stiftung.

328 L. S. Krapp, S. Kuhlmann, and G. Lehéricy: Strongly NIP almost real closed elds

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We give an explicit algebraic characterisation of all definable henselian valuations on a dp-minimal real field. Additionally we characterise all dp-minimal real fields that admit a definable henselian valuation with real closed residue field. We do so by first proving this for the more general setting of almost real closed fields.

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In this paper, we undertake a systematic model and valuation theoretic study of the class of ordered fields which are dense in their real closure. We apply this study to determine definable henselian valuations on ordered fields, in the language of ordered rings. In light of our results, we re-examine the Shelah-Hasson Conjecture (specialised to ordered fields) and provide an example limiting its valuation theoretic conclusions.

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Lothar Sebastian Krapp

An exponential exp on an ordered field (K, +, −, ·, 0, 1, <) is an order-preserving isomorphism from the ordered additive group (K, +, 0, <) to the ordered multiplicative group of positive elements(K^(>0), ·, 1, <). The structure (K, +, −, ·, 0, 1, <, exp) is then called an ordered exponential field. A linearly ordered structure (M, <, ...) is called o-minimal if every parametrically definable subset ofMis a finite union of points and open intervals of M.The main subject of this thesis is the algebraic and model theoretic examinationof o-minimal exponential fields (K, +, −, ·, 0, 1, <, exp) whose exponential satisfiesthe differential equation exp′ = exp with initial condition exp(0) = 1. This study is mainly motivated by the Transfer Conjecture, which states as follows: Any o-minimal exponential field (K, +, −, ·, 0, 1, <, exp) whose exponential satisfies the differential equation exp′ = exp with initial condition exp(0) = 1is elementarily equivalent to R_exp. Here, R_exp denotes the real exponential field (R, +, −, ·, 0, 1, <, exp), where exp denotes the standard exponential on R. Moreover, elementary equivalence means that any first-order sentence in the language L_exp = {+, −, ·, 0, 1, <, exp} holds for (K, +, −, ·, 0, 1, <, exp) if and only if it holds for R_exp. The Transfer Conjecture, and thus the study of o-minimal exponentialfields, is of particular interest in the light of the decidability of R_exp. To the date, it is not known if R_exp is decidable, i.e. whether there exists a procedure determining for agiven first-order L_exp-sentence whether it is true or false in R_exp. However, underthe assumption of Schanuel’s Conjecture – a famous open conjecture from Transcendental Number Theory – a decision procedure for R_exp exists. Also a positive answerto the Transfer Conjecture would result in the decidability of R_exp. Thus, we study o-minimal exponential fields with regard to the Transfer Conjecture, Schanuel’s Conjecture and the decidability question of R_exp.Overall, we shed light on the valuation theoretic invariants of o-minimal exponential fields – the residue field and the value group – with additional induced structure. Moreover, we explore elementary substructures and extensions of o-minimal exponential fields to the maximal ends – the smallest elementary substructures being prime models and the maximal elementary extensions being contained in thesurreal numbers. Further, we draw connections to models of Peano Arithmetic, integer parts, density in real closure, definable henselian valuations and strongly NIP ordered fields.

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Lothar Sebastian Krapp

An exponential

\exp

on an ordered field

(K,+,-,\cdot ,0,1, is an order-preserving isomorphism from the ordered additive group

(K,+,0, to the ordered multiplicative group of positive elements

(K^{>0},\cdot ,1, . The structure

(K,+,-,\cdot ,0,1, is then called an ordered exponential field (cf. [6]). A linearly ordered structure

(M, is called o-minimal if every parametrically definable subset of M is a finite union of points and open intervals of M . The main subject of this thesis is the algebraic and model theoretic examination of o-minimal exponential fields

(K,+,-,\cdot ,0,1, whose exponential satisfies the differential equation

\exp ' = \exp

with initial condition

\exp (0) = 1

. This study is mainly motivated by the Transfer Conjecture, which states as follows: Any o-minimal exponential field

(K,+,-,\cdot ,0,1, whose exponential satisfies the differential equation

\exp ' = \exp

with initial condition

\exp (0)=1

is elementarily equivalent to

\mathbb {R}_{\exp }

. Here,

\mathbb {R}_{\exp }

denotes the real exponential field

(\mathbb {R},+,-,\cdot ,0,1, , where

\exp

denotes the standard exponential

x \mapsto \mathrm {e}^x

\mathbb {R}

. Moreover, elementary equivalence means that any first-order sentence in the language

\mathcal {L}_{\exp } = \{+,-,\cdot ,0,1, holds for

(K,+,-,\cdot ,0,1, if and only if it holds for

\mathbb {R}_{\exp }

. The Transfer Conjecture, and thus the study of o-minimal exponential fields, is of particular interest in the light of the decidability of

\mathbb {R}_{\exp }

. To the date, it is not known if

\mathbb {R}_{\exp }

is decidable, i.e., whether there exists a procedure determining for a given first-order

\mathcal {L}_{\exp }

-sentence whether it is true or false in

\mathbb {R}_{\exp }

. However, under the assumption of Schanuel’s Conjecture—a famous open conjecture from Transcendental Number Theory—a decision procedure for

\mathbb {R}_{\exp }

exists (cf. [7]). Also a positive answer to the Transfer Conjecture would result in the decidability of

\mathbb {R}_{\exp }

(cf. [1]). Thus, we study o-minimal exponential fields with regard to the Transfer Conjecture, Schanuel’s Conjecture, and the decidability question of

\mathbb {R}_{\exp }

. Overall, we shed light on the valuation theoretic invariants of o-minimal exponential fields—the residue field and the value group—with additional induced structure. Moreover, we explore elementary substructures and extensions of o-minimal exponential fields to the maximal ends—the smallest elementary substructures being prime models and the maximal elementary extensions being contained in the surreal numbers. Further, we draw connections to models of Peano Arithmetic, integer parts, density in real closure, definable Henselian valuations, and strongly NIP ordered fields. Parts of this thesis were published in [2–5]. Abstract prepared by Lothar Sebastian Krapp E-mail : sebastian.krapp@uni-konstanz.de URL : https://d-nb.info/1202012558/34

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Given a Henselian valuation, we study its definability (with and without parameters) by examining conditions on the value group. We show that any Henselian valuation whose value group is not closed in its divisible hull is definable in the language of rings, using one parameter. Thereby we strengthen known definability results. Moreover, we show that in this case, one parameter is optimal in the sense that one cannot obtain definability without parameters. To this end, we present a construction method for a t -Henselian non-Henselian ordered field elementarily equivalent to a Henselian field with a specified value group.

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Will Johnson

We construct a nontrivial definable type V field topology on any dp-minimal field (Formula presented.) that is not strongly minimal, and prove that definable subsets of (Formula presented.) have small boundary. Using this topology and its properties, we show that in any dp-minimal field (Formula presented.), dp-rank of definable sets varies definably in families, dp-rank of complete types is characterized in terms of algebraic closure, and (Formula presented.) is finite for all (Formula presented.). Additionally, by combining the existence of the topology with results of Jahnke, Simon and Walsberg [Dp-minimal valued fields, J. Symbolic Logic 82(1) (2017) 151–165], it follows that dp-minimal fields that are neither algebraically closed nor real closed admit nontrivial definable Henselian valuations. These results are a key stepping stone toward the classification of dp-minimal fields in [Fun with fields, Ph.D. thesis, University of California, Berkeley (2016)].

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Strongly dependent ordered abelian groups have finite dp-rank. They are precisely those groups with finite spines and

|\{p\text{ prime}:[G:pG]=\infty\}|<\infty

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Definable Valuations induced by multiplicative subgroups and NIP Fields

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ARCH MATH LOGIC

We study the algebraic implications of the non-independence property (NIP) and variants thereof (dp-minimality) on infinite fields, motivated by the conjecture that all such fields which are neither real closed nor separably closed admit a definable henselian valuation. Our results mainly focus on Hahn fields and build up on Will Johnson's preprint "dp-minimal fields", arXiv: 1507.02745v1, July 2015.

A Guide to NIP Theories

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Pierre Simon

The study of NIP theories has received much attention from model theorists in the last decade, fuelled by applications to o-minimal structures and valued fields. This book, the first to be written on NIP theories, is an introduction to the subject that will appeal to anyone interested in model theory: graduate students and researchers in the field, as well as those in nearby areas such as combinatorics and algebraic geometry. Without dwelling on any one particular topic, it covers all of the basic notions and gives the reader the tools needed to pursue research in this area. An effort has been made in each chapter to give a concise and elegant path to the main results and to stress the most useful ideas. Particular emphasis is put on honest definitions, handling of indiscernible sequences and measures. The relevant material from other fields of mathematics is made accessible to the logician.

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We prove the elimination of field quantifiers for strongly dependent henselian fields in the Denef-Pas language. This is achieved by proving the result for a class of fields generalizing algebraically maximal Kaplansky fields. We deduce that if (K, v) is strongly dependent, then so is its henselization.

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