Distality in Valued Fields and Related Structures
Matthias Aschenbrenner, Artem Chernikov, Allen Gehret, Martin Ziegler
aa r X i v : . [ m a t h . L O ] A ug DISTALITY IN VALUED FIELDS AND RELATED STRUCTURES
MATTHIAS ASCHENBRENNER, ARTEM CHERNIKOV, ALLEN GEHRET, AND MARTIN ZIEGLER
Abstract.
We investigate distality and existence of distal expansions in valued fields and relatedstructures. In particular, we characterize distality in a large class of ordered abelian groups, providean AKE-style characterization for henselian valued fields, and demonstrate that certain expansionsof fields, e.g., the differential field of logarithmic-exponential transseries, are distal. As a new toolfor analyzing valued fields we employ a relative quantifier elimination for pure short exact sequencesof abelian groups.
Contents
Introduction 11. Preliminaries on Distality 42. Distal Fields and Rings 133. Distality in Ordered Abelian Groups 194. Distality and Short Exact Sequences of Abelian Groups 245. Eliminating Field Quantifiers in Henselian Valued Fields 366. Distality in Henselian Valued Fields 467. Distality in Expansions of Fields by Operators 52Acknowledgements 56References 56
Introduction
Distal theories were introduced in [62] as a way to distinguish those NIP theories in which no stablebehavior of any kind occurs. Examples include all (weakly) o-minimal theories (e.g., the theory of theexponential ordered field of reals) and all P -minimal theories (such as the theory of the field of p -adicnumbers and its analytic expansion from [24]); see the introduction of [18] for a detailed discussion.Distality has been investigated both from the point of view of pure model theory [6, 7, 14, 49] andin connection to the extremal combinatorics of restricted families of graphs. Indeed, as demonstratedin [18], distality of a theory is equivalent to a definable version of the strong Erd˝os-Hajnal Property .Further results in [11, 17] show that many of the combinatorial consequences of distality, including thestrong Erd˝os-Hajnal Property, improved regularity lemmas and various generalized incidence bounds,continue to hold for structures which are merely interpretable in distal structures. Curiously, finding adistal expansion also appears to be the easiest way of establishing these combinatorial results in a givenstructure. This motivates the question: which NIP structures admit distal expansions? Currently,the only known reason for not having a distal expansion comes from interpreting an infinite field ofpositive characteristic; see Section 2 below, where we also point out that more generally, every infinitedistal unital ring without zero-divisors has characteristic zero.
Date : August 21, 2020.2010
Mathematics Subject Classification.
Primary 03C45, 03C60; Secondary 12L12, 12J25.
The aim of this paper is to investigate both issues—distality and existence of distal expansions—in the setting of valued fields and various related structures: ordered abelian groups, short exactsequences of abelian group, valued fields with operators. This provides new examples in which theaforementioned combinatorial results hold, and along the way yields some general tools to addressthese problems in similar settings. The question of classifying NIP (valued) fields is currently anactive area of research motivated by various versions of Shelah’s Conjecture. (See [27, 37, 42, 48] andreferences therein for some recent results.) In particular, good understanding has been achieved in thedp-minimal case [45, 46]; see Section 6.6 for more details. (We recall the definition of dp-minimalityin Section 1.1.) Our results demonstrate that some of the issues in this program simplify in the distalcase, where infinite fields of positive characteristic are ruled out, while new complications arise dueto the fact that distality is not preserved under taking reducts.As a practical matter, we will not in general set out to prove from scratch that the structures weare interested in are distal (or not distal). Instead, whenever possible we will view structures as mildexpansions of certain distal reducts, and then study how distality passes from the reduct up to theoriginal structure. For instance, in Section 7 we show that certain expansions of valued fields by unaryoperators are distal by reducing the problem to the reduct of said valued field without the additionaloperators. For this reason, we will often rely on abstract criteria which (under certain circumstances)show how the distality of a structure can be deduced from the distality of a suitably chosen reduct.In Section 1 we recall basic results and notions around distality, as well as prove some auxiliary lemmasfor verifying that certain expansions in an abstract model-theoretic setting are distal. In Section 2 webriefly discuss distal fields and rings. Using Hahn products we give an example of an infinite unitalring of prime characteristic which has a distal expansion.In Section 3 we then study distality in the class of ordered abelian groups. While every orderedabelian group G is NIP by [35], distality may fail due to the presence of infinite stable quotients of theform G/nG . Theorem 3.13 makes this precise by characterizing distality in a large class of orderedabelian groups. To properly state this result requires the many-sorted language L qe of Cluckers andHalupczok [19], so we only mention here a consequence and save the discussion of L qe and the fullstatement of Theorem 3.13 for Section 3. Corollary.
Let G be a strongly dependent ordered abelian group; then G is distal ⇐⇒ G is dp -minimal ⇐⇒ G is non-singular ( i.e., G/pG finite for every prime p ) . In Section 4 we consider distality in short exact sequences of abelian groups with extra structure.That is, we consider short exact sequences of abelian groups 0 → A → B → C → A and C . In Section 4.1 we give a generalquantifier elimination result for pure short exact sequences, i.e., where the image of A is assumedto be a pure subgroup of B . (This applies when C is torsion-free.) In this case only sorts forthe quotients A/nA and certain induced maps B → A/nA have to be added in order to eliminatequantification over B ; see Corollary 4.3 for the precise statement. This generalizes a result in [15],where all of the quotients A/nA ( n ≥
1) were assumed to be finite. Using this quantifier elimination,we show in Section 4.2 that such a pure short exact sequence is distal (has a distal expansion) ifand only if both A and C are distal (have distal expansions, respectively). Note that the theory of apure short exact sequence is interpretable in the theory of the direct product A × C , as explained atthe beginning of Section 4.1; however in general, distality is not preserved under passing to reducts,thus a precise description of the definable sets is necessary for our purpose. In Sections 4.3, 4.4,and 4.5 we consider variants and extensions of our quantifier elimination theorem. We expect theseelimination theorems for short exact sequences to have many uses. As an illustration, we employ some ISTALITY IN VALUED FIELDS AND RELATED STRUCTURES 3 of these variants in Section 5 to prove some quantifier elimination theorems for henselian valued fieldsof characteristic zero.In Section 6 we consider distality in henselian valued fields. Relying on the results of the previoussections, in Sections 6.1 and 6.2 we prove the following Ax-Kochen-Erˇsov (AKE) type characterization.Recall that a valued field K with valuation v : K × = K \ { } → Γ = v ( K × ) is said to be finitelyramified if for each n ≥ γ ∈ Γ such that 0 ≤ γ ≤ v ( n ). If Γ = { } , thenthis clearly implies that the field K has characteristic zero; if K has equicharacteristic zero, then K is always finitely ramified. Main Theorem.
Let K be a henselian valued field, viewed as a structure in the language of ringsaugmented by a predicate for the valuation ring, with value group Γ and residue field k . Then K isdistal ( has a distal expansion ) if and only if (1) K is finitely ramified, and (2) both Γ and k are distal ( respectively, have distal expansions ) .In this case k is either finite or of characteristic zero. For example, this theorem implies that a finitely ramified henselian valued field K with regularnon-singular value group is distal if and only if the residue field of K is distal; this generalizes thewell-known facts that each p -adically closed field is distal, and that a real closed valued field is distaliff its residue field is real closed.In Section 6.3 we consider Jahnke’s results [42] on naming a henselian valuation in the distal case. InSection 6.5 we formulate a conjectural classification of fields admitting a distal expansion: a (pure)NIP field does not have a distal expansion if and only if it interprets an infinite field of positivecharacteristic. We show that this statements holds modulo Shelah’s conjecture on NIP fields anda conjecture on distal expansions of ordered abelian groups from Section 3. For this, we rely ondefinability theorems of Koenigsmann-Jahnke [43], in a similar way as Johnson [47, Chapter 9]. InSection 6.6 we concentrate on the dp-minimal case; based on Johnson’s results [46], we observe thatour conjecture does hold unconditionally for dp-minimal fields.Finally, in Section 7 we show that a certain “forgetful functor” argument preserves distality. Utilizingthis, we exhibit expansions of (valued) fields with additional operators (e.g., derivations) which aredistal. Examples include the differential field of transseries [2] and certain topological fields with ageneric derivation in the sense of [36, 67]. This also implies that the theory of differentially closedfields of characteristic zero admits a distal expansion (Corollary 7.7). These techniques also yield thatanalytic expansions of distal valued fields of characteristic zero are distal (Corollary 7.10). Conventions and notations.
Throughout, m and n (possibly with decorations) range over theset N = { , , , . . . } . In general we adopt the model theoretic conventions of Appendix B of [2].In particular, L can be a many-sorted language. Given a complete L -theory T , we will sometimesconsider a model M | = T and a cardinal κ ( M ) > |L| such that M is κ ( M )-saturated and every reductof M is strongly κ ( M )-homogeneous. Such a model is called a monster model of T . Then every modelof T of size ≤ κ ( M ) can be elementarily embedded into M . “Small” will mean “of size < κ ( M )”.We use x , y , z (sometimes with decorations) to denote multivariables. Unless otherwise specified, allmultivariables are assumed to have finite size, and the size of such a multivariable x is denoted by | x | .We shall write “ | = θ ” to indicate that θ is an L M -formula and M | = θ . Likewise, “Φ( x ) | = Θ( x )” willmean that Φ( x ) and Θ( x ) are small sets of L M -formulas such that every a ∈ M x realizing Φ( x ) alsorealizes Θ( x ). We write “ ϕ ( x ) | = Θ( x )” to abbreviate { ϕ ( x ) } | = Θ( x ), etc.Given linearly ordered sets I and J we denote by I ⌢ J the concatenation of I and J , that is, theset K := I ∪ J (disjoint union) equipped with the linear ordering extending both the orderings of I and J such that I < J . If, say, I = { i } is a singleton, we also write I ⌢ J = i ⌢ J . Similarly, given ASCHENBRENNER, CHERNIKOV, GEHRET, AND ZIEGLER sequences a = ( a i ) i ∈ I and b = ( b j ) j ∈ J in M x , where I , J are linearly ordered sets, we let a ⌢ b denotethe sequence ( c k ) k ∈ K where K = I ⌢ J and c i = a i for i ∈ I , c j = b j for j ∈ J . We extend thisnotation to the concatenation of several (finitely many) linearly ordered sets respectively sequencesin the natural way. If a = ( a i ) i ∈ I is a sequence and J ⊆ I , we let a J := ( a j ) j ∈ J . By convention“indiscernible sequence” means “ ∅ -indiscernible sequence”.1. Preliminaries on Distality
Throughout this section L is a language and T is a complete L -theory. We also fix a monster model M of T . The definitions below do not depend on the choice of this monster model.1.1. Two ways of defining distality.
Distality has many facets, and can be introduced in a numberof equivalent ways. In this subsection we present two of them: by means of indiscernible sequences, and via honest definitions.
Definition 1.1.
We say that T is distal if for every small parameter set B ⊆ M , every indiscerniblesequence a = ( a i ) i ∈ I in M x , and every i ∈ I , the following holds: if(1) I < = I = I >i := { j ∈ I : i < j } are infinite, and(2) a I \{ i } is B -indiscernible,then a is B -indiscernible. We say that an L -structure is distal if its theory is distal.While the definition of distality given above involves checking a certain condition for all infinite linearlyordered sets I < and I > , standard arguments show that this definition is equivalent to the variantwhere I < and I > are fixed infinite linearly ordered sets. More precisely, fix a linearly ordered set I = I <⌢ i ⌢ I > where I < , I > are infinite; then the theory T is distal if for every small parameter set B ⊆ M ,an indiscernible sequence ( a i ) i ∈ I in M x is B -indiscernible provided ( a i ) i ∈ I \{ i } is B -indiscernible. Forthis reason, in practice we can (and often will) assume that I < and I > are “nice” infinite linearlyordered sets such as Q or [0 , Definition 1.2 ([62, Definition 2.1]) . Let a = ( a i ) i ∈ I be an indiscernible sequence in M x . Then a is distal if for every indiscernible sequence a ′ = ( a ′ i ) i ∈ I ′ in M x with the same EM-type as a and I ′ = I ⌢ I ⌢ I where I , I , I are dense without endpoints, and all c, d ∈ M x , the following holds:if the sequences a ′ I ⌢ c ⌢ a ′ I ⌢ a ′ I and a ′ I ⌢ a ′ I ⌢ d ⌢ a ′ I are indiscernible, then so is a ′ I ⌢ c ⌢ a ′ I ⌢ d ⌢ a ′ I .Definitions 1.1 and 1.2 are connected by the following fact. Fact 1.3 ([62, Lemma 2.7]) . Suppose T is NIP, and let a = ( a i ) i ∈ I be an indiscernible sequencein M x ; then the following are equivalent: (1) a is distal; (2) for every small parameter set B ⊆ M , b ∈ M x , and B -indiscernible sequence a ′ = ( a ′ i ) i ∈ I ′ in M x with I ′ = I ⌢ I , I and I without endpoints, having the same EM-type as a , if a ′ I ⌢ b ⌢ a ′ I is indiscernible, then it is also B -indiscernible.In particular, T is distal if and only if every infinite indiscernible sequence is distal. It is well-known that if T is distal, then T is NIP; for instance, see [34, Proposition 2.8]. Distality canbe thought of as a notion of pure instability among NIP theories. The following fact (which followsfrom [62, Corollary 2.15]) is evidence for this point of view.
Fact 1.4. If T is distal then no infinite non-constant indiscernible sequence is totally indiscernible. ISTALITY IN VALUED FIELDS AND RELATED STRUCTURES 5
In the dp-minimal case we also have a converse. We first recall the definition of dp-minimality. Recallthat a cut in a linearly ordered set I is a downward closed subset of I ; such a cut c is trivial if c = ∅ or c = I . We let I be the set of nontrivial cuts in I , totally ordered by inclusion; if I does not have alargest element, then the map which sends i ∈ I to the cut { j ∈ I : j ≤ i } is an embedding I → I ofordered sets, and we then identify I with its image under this embedding. Now the theory T is calleddp -minimal if for each indiscernible sequence a = ( a i ) i ∈ I in M x indexed by a dense linearly orderedset I and each c ∈ M y there is a cut i ∈ I such that the sequences ( a i ) i< i and ( a i ) i> i are c -indiscernible.(This is not the original definition from [54], but equivalent to it thanks to [61, Lemma 1.4].) Fact 1.5 ([62, Lemma 2.10]) . If T is dp -minimal and every non-constant indiscernible sequence ofsingletons is not totally indiscernible, then T is distal. In particular, if T is dp -minimal and everysort of M expands a linearly ordered set, then T is distal. Linear orders in distal theories also occur on indiscernible sequences:
Corollary 1.6.
Suppose T is distal, and let a = ( a i ) i ∈ I be a non-constant indiscernible sequencein M x . Then there is an L -formula θ ( u, x, y, w ) and some n such that for all I , I ⊆ I of size n andall i, j ∈ I such that I < i, j < I we have i < j ⇐⇒ | = θ ( a I , a i , a j , a I ) . Proof.
By 1.4, a is not totally indiscernible, and for every indiscernible sequence which is not totallyindiscernible there are such θ and n ; see, e.g., the explanation after [13, Fact 3.1]. (cid:3) In the following we sometimes employ L -formulas whose free variables have been separated into multi-variables x , y thought of as object and parameter variables, respectively. We use the notation ϕ ( x ; y )to indicate that the free variables of the L -formula ϕ are contained among the components of the mul-tivariables x , y (which we also assume to be disjoint). We refer to ϕ ( x ; y ) as a partitioned L -formula. Given a ∈ M x and B ⊆ M y we lettp ϕ ( a | B ) := (cid:8) ϕ ( x ; b ) : b ∈ B, | = ϕ ( a ; b ) (cid:9) ∪ (cid:8) ¬ ϕ ( x ; b ) : b ∈ B, | = ¬ ϕ ( a ; b ) (cid:9) be the ϕ -type of a over B . Definition 1.7.
Let ϕ ( x ; y ) be a partitioned L -formula, and let y , y , . . . be disjoint multivariables ofthe same sort as y . A partitioned L -formula ψ ( x ; y , . . . , y n ) is a (uniform) strong honest definition for ϕ ( x ; y ) (in T ) if for every a ∈ M x and finite B ⊆ M y with | B | ≥
2, there are b , . . . , b n ∈ B suchthat | = ψ ( a ; b , . . . , b n ) and ψ ( x ; b , . . . , b n ) | = tp ϕ ( a | B ) . Remark.
A strong honest definition for ϕ ( x ; y ) remains a strong honest definition for ¬ ϕ ( x ; y ).Moreover, if ψ ( x ; y , . . . , y m ), ψ ′ ( x ; y ′ , . . . , y ′ n ) are strong honest definitions for the partitioned L -formulas ϕ ( x ; y ), ϕ ′ ( x ; y ), respectively, with all multivariables y i , y ′ j disjoint, then ψ ∧ ψ ′ is a stronghonest definitions for ϕ ∧ ϕ ′ .By [14, Theorem 21] we have: Fact 1.8.
The following are equivalent: (1) T is distal; (2) every partitioned L -formula ϕ ( x ; y ) has a strong honest definition in T . When proving distality of a particular structure, Definition 1.1 is typically easier to verify. Onthe other hand, occasionally 1.8(2) is more useful since it ultimately gives more information aboutdefinable sets, and obtaining bounds on the complexity of strong honest definitions is important forcombinatorial applications.
ASCHENBRENNER, CHERNIKOV, GEHRET, AND ZIEGLER
Reduction to singletons.
In order to verify that a theory is distal, it is enough to checkdistality for “singletons”. There are two ways to interpret this claim. First, we observe that existenceof strong honest definitions for all formulas reduces to formulas in a single free variable.
Proposition 1.9.
Suppose every partitioned L -formula ϕ ( x ; y ) with | x | = 1 has a strong honestdefinition in T . Then every partitioned L -formula ϕ ( x ; y ) with | x | arbitrary has a strong honestdefinition in T , so T is distal.Proof. We argue by induction on the size | x | of x , with the base case | x | = 1 given by the assumption.Assume that x = ( x , x ), and let a partitioned L -formula ϕ ( x , x ; y ) be given. By the inductiveassumption, take a strong honest definition ψ ( x ; z , . . . , z n ) for the partitioned L -formula ϕ ( x ; x , y ),where z i = ( x i , y i ) for i = 1 , . . . , n . Set χ ( x ; x , ~y ) := ψ (cid:0) x ; ( x , y ) , . . . , ( x , y n ) (cid:1) where ~y := ( y , . . . , y n ),let χ + ( x ; y, ~y ) := ∀ x (cid:0) χ ( x ; x , ~y ) → ϕ ( x ; x , y ) (cid:1) ,χ − ( x ; y, ~y ) := ∀ x (cid:0) χ ( x ; x , ~y ) → ¬ ϕ ( x ; x , y ) (cid:1) , and by inductive assumption, let ρ + ( x ; ~y + ) and ρ − ( x ; ~y − ) be strong honest definitions for χ + and χ − , respectively; here ~y + = ( ~y +1 , . . . , ~y + n + ) for some n + , and similarly with − in place of +. Weclaim that γ ( x , x ; ~y, ~y + , ~y − ) := χ ( x ; x , ~y ) ∧ ρ + ( x ; ~y + ) ∧ ρ − ( x ; ~y − )is a strong honest definition for ϕ ( x ; y ). To see this let a i ∈ M x i ( i = 0 ,
1) and a finite B ⊆ M y with | B | ≥ ψ to a and the set of parameters { a } × B , we obtain some ~b ∈ B n such that | = χ ( a ; a ,~b ) and χ ( x ; a ,~b ) | = tp ϕ (cid:0) a (cid:12)(cid:12) { a } × B (cid:1) . Next choose ~b + ∈ (cid:0) B × { ~b } (cid:1) n + such that | = ρ + ( a ; ~b + ) and ρ + ( x ; ~b + ) | = tp χ + (cid:0) a (cid:12)(cid:12) B × { ~b } (cid:1) . Then for any a ′ | = ρ + ( x ,~b + ) and b ∈ B we have | = χ ( x , a ′ ,~b ) → ϕ ( x , a ′ , b ) ⇐⇒ | = χ ( x , a ,~b ) → ϕ ( x , a , b ) ⇐⇒ | = ϕ ( a , a , b ) . Similarly, we find ~b − ∈ (cid:0) B × { ~b } (cid:1) n − such that for any a ′ | = ρ − ( x ,~b − ) and b ∈ B we have | = χ ( x , a ′ ,~b ) → ¬ ϕ ( x , a ′ , b ) ⇐⇒ | = ¬ ϕ ( a , a , b ) . Combining, we see that for all a ′ | = ρ + ( x ,~b + ) ∧ ρ − ( x ,~b − ) and a ′ | = χ ( x , a ′ ,~b ) and each b ∈ B wehave | = ϕ ( a ′ , a ′ , b ) ↔ ϕ ( a , a , b ). Thus γ ( x , x ; ~b,~b + ,~b − ) | = tp ϕ ( a a | B ) and | = γ ( a , a ; ~b,~b + ,~b − )hold, as wanted. (cid:3) Remark.
Let f ( m ) be the smallest possible number of parameters n in a strong honest defini-tion ψ ( x ; y , . . . , y n ) for partitioned L -formulas ϕ ( x ; y ) with | x | ≤ m . It follows from the proofthat if f (1) is finite, then f ( m ) ≤ f (1) + f ( m −
1) for m ≥
1; so f ( m ) ≤ (2 m − f (1) for all m ≥ P -minimal theories. ISTALITY IN VALUED FIELDS AND RELATED STRUCTURES 7
Secondly, in terms of indiscernible sequences we have the following equivalence.
Proposition 1.10.
The following are equivalent: (1) T is distal; (2) for every indiscernible sequence a = ( a i ) i ∈ I in M x , i ∈ I such that I i are infinite,and b ∈ M y with | y | = 1 , if a I \{ i } is b -indiscernible, then so is a ; (3) for every indiscernible sequence a = ( a i ) i ∈ I in M x where | x | = 1 , i ∈ I such that I i are infinite, and b ∈ M y , if a I \{ i } is b -indiscernible, then so is a .Proof. It is not hard to see that the condition in (2) can be iterated to obtain the same conclu-sion with y an arbitrary multivariable, which is sufficient to satisfy Definition 1.1. (Alternatively,Proposition 1.9 provides a more explicit version of this argument.) The equivalence of (1) and (3) isestablished in [62, Theorem 2.28]. (See also Proposition 1.17 below for a discussion.) (cid:3) Corollary 1.11.
The following are equivalent: (1) T is not distal; (2) there is an indiscernible sequence a = ( a i ) i ∈ Q in M x and some b ∈ M y such that a Q \{ } is b -indiscernible, and some partitioned L -formula ϕ ( x ; y ) such that | = ϕ ( a i ; b ) ⇐⇒ i = 0;(3) the same statement as in (2) with | x | = 1 .Proof. To show (1) ⇒ (3), assume that the condition in Proposition 1.10(3) fails. Then we can takesome indiscernible sequence a = ( a i ) i ∈ Q in M x where | x | = 1 and some b ∈ M y such that a Q \{ } is b -indiscernible, but a is not. Thus we can take an L -formula ψ ( x , . . . , x n ; y ), where x , . . . , x n aresingle variables of the same sort as x , as well as finite subsets I , I of Q with | I | + | I | = n − I < < I , such that(1) | = ¬ ψ ( a I , a , a I ; b );(2) | = ψ ( a J , a j , a J ; b ) for all J , J ⊆ Q \ { } and j ∈ Q \ { } with | J | + | J | = n − J < j < J .Let y ′ := ( y, y , y ) where y = ( x , . . . , x m ), y = ( x m +2 , . . . , x n ), m = | I | , and set ϕ ( x ; y ′ ) := ψ ( y , x, y , y ) , b ′ := ( b, a I , a I ) ∈ M y ′ . Choose ε ∈ Q with I < − ε < < ε < I and set I ′ := { i ∈ Q : − ε < i < ε } . Then the sequence a I ′ is indiscernible and a I ′ \{ } is b ′ -indiscernible; moreover, for i ∈ I ′ we have | = ϕ ( a i ; b ′ ) ⇐⇒ i = 0 . It follows that (3) holds. Finally, (3) ⇒ (2) and (2) ⇒ (1) are obvious. (cid:3) Remark . Let a = ( a i ) i ∈ Q be an indiscernible sequence in M x and b ∈ M y such that a Q \{ } is b -indiscernible. It is easy to see that the set of L -formulas ϕ ( x ; y ) violating the conclusion of (2) inCorollary 1.11 (that is, such that | = ϕ ( a ; b ) or | = ¬ ϕ ( a i ; b ) for some, or equivalently, all i = 0) isclosed under positive boolean combinations. Remark . Let Q ∞ = Q ∪ {∞} where ∞ / ∈ Q is a new symbol and the usual ordering of Q isextended to a total ordering of Q ∞ with Q < ∞ . Then Corollary 1.11 and Remark 1.12 remain truewith the linearly ordered set Q replaced by Q ∞ . (This is used in the proof of Theorem 4.6 below.) ASCHENBRENNER, CHERNIKOV, GEHRET, AND ZIEGLER
Induced structure and mild expansions.
From [62] we record the following. (For part (2)use [62, Corollary 2.9] along with Fact 1.3.)
Fact 1.14. (1) If T is distal, then so is every complete theory bi-interpretable with T . (2) Naming a small set of constants does not affect distality: if M is distal, then for each small A ⊆ M , the L A -structure M A is also distal. In what follows, we will often be in a situation when T is NIP and we have a definable set D ⊆ M x (often, a sort) such that the induced structure on D is distal. More precisely, denote the full inducedstructure on D by D ind ; that is, we introduce the one-sorted language L ind which contains, for each L -formula ϕ ( y , . . . , y n ) where each y i is a multivariable of the same sort as x , an n -ary relationsymbol R ϕ ; then D ind is the L ind -structure with underlying set D where each relation symbol R ϕ isinterpreted by ϕ M ∩ D n . The following is then straightforward by Definition 1.1. Lemma 1.15. If T is distal, then D ind is also distal. We have the following lemmas in the converse direction.
In the rest of this subsection we assumethat T is NIP, and we let D be an ∅ -definable set such that D ind is distal. Our goal is to concludethat under suitable circumstances, T itself is distal. Lemma 1.16.
Let B ⊆ M be small and b ∈ M y , and let ( a i ) i ∈ Q be a B -indiscernible sequence ofelements from D . If ( a i ) i ∈ Q \{ } is Bb -indiscernible, then so is ( a i ) i ∈ Q .Proof. If a fails the conclusion of the lemma, then using distality of a (in the sense of Definition 1.2),following the proof of [62, Lemma 2.7] gives a contradiction to T being NIP. (cid:3) We also have a dual fact, where the sequence may be anywhere in M , but the new parameters arecoming from our distal set D . (A similar observation is stated in [29, Remark 4.26].) Proposition 1.17.
Let a = ( a i ) i ∈ Q be an indiscernible sequence in M x and b ∈ D N , where N ∈ N .If ( a i ) i ∈ Q \{ } is b -indiscernible, then so is ( a i ) i ∈ Q . This proposition can be shown along the same lines as the proof of [62, Theorem 2.28]; we providethe details for the sake of completeness and correcting some inaccuracies there. First we recall someterminology and facts from [62].A nontrivial cut c in a linearly ordered set I is dedekind if c does not have a largest and I \ c does nothave a smallest element. Let a = ( a i ) i ∈ I be an ( ∅ -) indiscernible sequence in M x where I is endless,and B ⊆ M is an arbitrary parameter set. Recall that since T is NIP, the L B -formulas ϕ ( x ) with theproperty that the set of i ∈ I with | = ϕ ( a i ) is cofinal in I form a complete x -type lim( a | B ) over B .(See, e.g., [63, Proposition 2.8].) Given a dedekind cut c in I , letting c + denote the complement I \ c of c ordered by the reverse ordering, we setlim − ( c | B ) := lim( a c | B ) , lim + ( c | B ) := lim( a c + | B ) . (Here a is understood from the context.) We say that b ∈ M x fills c in a if the sequence a c ⌢ b ⌢ a I \ c is indiscernible. Fact 1.18 (Strong base change, [62, Lemma 2.8]) . Let a = ( a i ) i ∈ I be an indiscernible sequence in M x and A ⊆ M x be a small parameter set containing all a i . Let also ( c λ ) λ ∈ Λ be a family of pairwisedistinct dedekind cuts in I , and for each λ ∈ Λ , let a λ fill the cut c λ in a . Then there exists afamily ( a ′ λ ) λ ∈ Λ in M x such that ( a ′ λ ) ≡ a ( a λ ) and tp( a ′ λ | A ) = lim + ( c λ | A ) for all λ ∈ Λ . Let a = ( a i ) i ∈ I and b = ( b j ) j ∈ J be sequences in M x and M y , respectively, indexed by linearly orderedsets I , J . We say that a is b -indiscernible if a is B -indiscernible where B := { b j : j ∈ J } . If a is b -indiscernible and b is a -indiscernible, then a , b are said to be mutually indiscernible . ISTALITY IN VALUED FIELDS AND RELATED STRUCTURES 9
Definition 1.19. ([62, Definition 2.12]) Indiscernible sequences a = ( a i ) i ∈ I and b = ( b i ) i ∈ I are weakly linked if for all disjoint subsets I , I ⊆ I , the sequences a I and b I are mutually indis-cernible.The following is [62, Lemma 2.14(1)]. It is stated there with the additional assumption that thesequence of pairs ( a i , b i ) i ∈ I is indiscernible; however, this assumption is not needed, and this point isimportant in the proof of Proposition 1.17 given below. Lemma 1.20.
Let a = ( a i ) i ∈ I and b = ( b i ) i ∈ I be weakly linked indiscernible sequences, where a isdistal; then a and b are mutually indiscernible.Proof. We may arrange that I is dense. To show that a is indiscernible over b , let I ′ ⊆ I be an arbitraryfinite set; it is enough to show that a is b I ′ -indiscernible. Now a I \ I ′ is b I ′ -indiscernible as a , b areweakly linked. Since a is distal, repeatedly applying Fact 1.3 we conclude that a is b I ′ -indiscernible.Towards a contradiction assume that b is not a -indiscernible. This yields finite subsets I , I of I such that b I is not a I -indiscernible. But then by indiscernibility of a over b I , there exists someset I ′ disjoint from I such that a I ′ ≡ b I a I ; in particular, b I is not a I ′ -indiscernible, contradictingthat a , b are weakly linked. (cid:3) Proof of Proposition 1.17.
Toward a contradiction assume that the sequence ( a i ) i ∈ Q \{ } is b -indiscer-nible but ( a i ) i ∈ Q is not. We will show that then there is an indiscernible sequence ( b n ) with b n ≡ b which is not distal (in the sense of Definition 1.2); since b n ∈ D N , this will contradict distalityof D ind . We proceed by establishing a sequence of claims. In Claims 1.21–1.23 below we let I be adense linearly ordered set without endpoints and c be a dedekind cut in I . Claim 1.21.
There is a b -indiscernible sequence ( a ′ i ) i ∈ I and some a ′ filling the cut c in ( a ′ i ) such that tp( a ′ , b ) = tp( a ′ i , b ) for all i ∈ I .Proof. By assumption a := ( a i ) i ∈ Q is not b -indiscernible, so we find finite subsets J , J of Q and anonzero rational number j such that J < , j < J and(1.1) a J ⌢ a ⌢ a J b a J ⌢ a j⌢ a J . We may assume J , J = ∅ ; let j := max J , j := min J , and set a ′ j := a J ⌢ a j⌢ a J for j ∈ J := ( j , j ) ⊆ Q .Then (1.1) holds for all j ∈ J \ { } , the sequence ( a ′ j ) j ∈ J is still indiscernible, ( a ′ j ) j ∈ J \{ } is b -indiscernible, and tp( a ′ , b ) = tp( a ′ j , b ) for j ∈ J \ { } . Using compactness, this yields the claim. (cid:3) Let now ( a ′ i ) and a ′ be as in Claim 1.21; to simplify notation (and since we have no use of our originalsequence ( a i ) i ∈ Q anymore), we now rename ( a ′ i ) i ∈ I , a ′ as ( a i ) i ∈ I , a , respectively. Thus • ( a i ) i ∈ I is b -indiscernible, and • a fills the cut c in ( a i ) and satisfies tp( a, b ) = tp( a i , b ) for all i ∈ I .We also fix an L -formula θ ( x, y ) such that | = ¬ θ ( a, b ) ∧ θ ( a i , b ) for all i ∈ I . Claim 1.22.
Let c ′ be a dedekind cut in I with c ⊆ c ′ . Then there exists an a ′ ∈ M x such that (1) a ′ fills the cut c ′ in ( a i ) , (2) tp( a, b ) = tp( a ′ , b ) , so in particular | = ¬ θ ( a ′ , b ) .Proof. As ( a i ) i ∈ I is b -indiscernible, we can choose a ′ satisfying (1) and (2) by compactness: givenfinite subsets I ⊆ c α and I ⊆ I \ c α there is a b -automorphism of M which sends a I , a I to a J , a J ,respectively, where J ⊆ c , J ⊆ I \ c . (cid:3) In the next claim we let α , β be ordinals, and let r , s (also with decorations) range over α respectively β .We also assume that we have a strictly increasing sequence ( c r ) of dedekind cuts in I with c = c . Claim 1.23.
There exists an array ( a r,s ) and a sequence ( b s ) such that: (1) if s < s ′ , then | = θ ( a r,s ′ , b s ) ; (2) | = ¬ θ ( a r,s , b s ) ; (3) for all r < · · · < r n and pairwise distinct s , . . . , s n , we have ( a r ,s , . . . , a r n ,s n ) ≡ ( a i , . . . , a i n ) for some ( equivalently, all ) i < · · · < i n in I ; (4) b s ≡ b .Proof. By Claim 1.22 we obtain a sequence a ′ = ( a ′ r ) such that for all r , • a ′ r fills the cut c r in ( a i ), and • | = ¬ θ ( a ′ r , b ).Let a := ( a i ). By induction on β we choose sequences ( a s ) and tuples ( b s ), with a s = ( a r,s ), such that(a) a r,s | = lim + ( c r | aa
There exists an array ( a m,n ) and a sequence ( b n ) satisfying (1)–(4) of Claim 1.23 for α = β = ω such that additionally (5) ( a n , b n ) is indiscernible, where a n = ( a m,n ) , and (6) (cid:0) ( a m,n ) n (cid:1) is B -indiscernible where B = { b , b , . . . } .Proof. We take an ordinal α sufficiently large compared to | T | (how large will become clear during thecourse of the rest of the proof), and then an ordinal β ≥ α and sufficiently large compared to α (alsoto be determined). Next, we take a linearly ordered set I which has more than | α | many dedekindcuts, so that we can choose a strictly increasing sequence ( c r ) of dedekind cuts in I . Then Claim 1.23applies and yields ( a r,s ) and ( b s ) having properties (1)–(4) in that claim. Set a s = ( a r,s ). ISTALITY IN VALUED FIELDS AND RELATED STRUCTURES 11
Assuming that β is large enough compared to α , Erd˝os-Rado and compactness (see, e.g., [63, Pro-position 1.1]) give us an indiscernible sequence ( a ′ n , b ′ n ) such that for every l ∈ ω there exist some s < · · · < s l such that ( a ′ k , b ′ k ) k ≤ l ≡ ( a s k , b s k ) k ≤ l . In particular, ( a ′ r,n ) and ( b ′ n ) satisfy (1)–(4)for β = ω , and (5) holds as well.Assuming α is large enough compared to | T | , we can similarly find a B ′ -indiscernible sequence (cid:0) ( a ′′ m,n ) n (cid:1) , where B ′ = { b ′ , b ′ , . . . } , such that for every l ∈ ω there exist some r < · · · < r l such that (cid:0) ( a ′′ k,n ) (cid:1) k ≤ l ≡ B (cid:0) ( a ′ r k ,n ) (cid:1) k ≤ l . In particular, ( a ′′ m,n ), ( b ′ n ) still satisfy (1)–(5), and (6) holds as well. (cid:3) Let now ( a m,n ) and ( b n ) be as in Claim 1.24; so (1)–(6) in Claims 1.23 and 1.24 hold. Claim 1.25.
The sequences ( a n,n ) and ( b n ) are weakly linked, but not mutually indiscernible.Proof. First note that ( a n,n ) is indiscernible by (3) applied with η given by η ( n ) = n for each n ,and ( b n ) is indiscernible by (5). Clearly, the sequences are not mutually indiscernible because wehave | = θ ( a n,n , b m ) for all m < n by (1), but | = ¬ θ ( a n,n , b n ) for all n by (2).Given a finite tuple i = ( i , . . . , i n − ) ∈ N n , we write a i := ( a i ,i , . . . , a i n − ,i n − ) and b i :=( b i , . . . , b i n − ). We say that such a tuple is strictly increasing if i < · · · < i n − . To show that ( a n,n )and ( b n ) are weakly linked, it is enough to show that for all strictly increasing i , i ′ , j , j ′ ∈ N n we have:( ∗ ) ( i ∪ i ′ ) ∩ ( j ∪ j ′ ) = ∅ = ⇒ a i b j ≡ a i ′ b j ′ . (Here in the antecedent we identify the tuples i , i ′ , j , j ′ with the corresponding subsets of N .) Firstnote that by (5) and (6), for strictly increasing i , i ′ , j , j ′ ∈ N n we easily have( ∗ ) ij ≡ qf < i ′ j ′ = ⇒ a i b j ≡ a i ′ b j ′ , where ≡ qf < indicates the equality of quantifier-free types in the language of ordered sets. Hence inorder to prove ( ∗ ), it is enough to show that for any finite tuples i , j , i ′ , j ′ of natural numberswith i ∩ j = ∅ and i ′ ∩ j ′ = ∅ and i , i , j ∈ N we have( ∗ ) i , j < i < j < i < i ′ , j ′ = ⇒ a ( i ) ≡ a i a i ′ b j b j b j ′ a ( i ) . Indeed, suppose i , i ′ , j , j ′ ∈ N n are strictly increasing with ( i ∪ i ′ ) ∩ ( j ∪ j ′ ) = ∅ as in ( ∗ ). Weclaim that we can use ( ∗ ) and ( ∗ ) to arrange that ij ≡ qf < i ′ j ′ . To see this let i = ( i , . . . , i n − )and j = ( j , . . . , j n − ), and suppose we have k, l ∈ { , . . . , n − } with i k < j l whereas i ′ k > j ′ l . If k = n −
1, then we take any integer e i k > j l ; otherwise, using ( ∗ ) we first arrange that i k +1 − j l is aslarge as necessary so that we may take an integer e i k / ∈ j ∪ j ′ with j l < e i k < i k +1 . In both cases set e i m := i m for m = k and consider the strictly increasing tuple e i := ( e i , . . . , e i n − ) ∈ N n ; then by ( ∗ )we have a i b j ≡ a e i b j . Thus by induction on the number of pairs ( k, l ) with i k < j l and i ′ k > j ′ l , wearrive at the case ij ≡ qf < i ′ j ′ , and then a i b j ≡ a i ′ b j ′ follows from ( ∗ ).To show ( ∗ ), let now i , j , i ′ , j ′ be finite tuples of natural numbers with i , j < i < j < i < i ′ , j ′ , andtowards a contradiction assume that we have an L -formula ψ ( x, y, z ) (for suitable disjoint multivari-ables x , y , z ), such that with ϕ ( x, y ) := ψ ( x, y, a i a i ′ b j b j ′ ), we have | = ϕ ( a ( i ) , b j ), but | = ¬ ϕ ( a ( i ) , b j ).Recall that T is NIP, so we may let m be the alternation number of the partitioned L -formula ϕ ( x ; y, z ).(See [63, Section 2.1].) In view of ( ∗ ), we can arrange: i < j − m < j < j + m < i , ( ∗ ) | = ϕ ( a i,n , b j ) for all i , n with j − m < n < j and j − m < i < j + m, ( ∗ ) | = ¬ ϕ ( a i,n , b j ) for all i , n with j < n < j + m and j − m < i < j + m. ( ∗ ) To see this first replace the tuple ( i , j , i , j, i , i ′ , j ′ ) by a tuple with the same order type so that i + m < j < i − m ; modifying ϕ accordingly, we then still have | = ϕ ( a ( i ) , b j ) ∧ ¬ ϕ ( a ( i ) , b j ) by ( ∗ ),and ( ∗ ) holds. Next, note that if i < n < j , then the tuple ( i , j , i , j, i , i ′ , j ′ ) has the same ordertype as the tuple ( i , j , n, j, i , i ′ , j ′ ), so | = ϕ ( a ( n ) , b j ) by ( ∗ ). Similarly we see that | = ¬ ϕ ( a ( n ) , b j ) for j < n < i . Property (6) then implies ( ∗ ) and ( ∗ ).Now let η : ω → ω be an injective function such that • η ( n ) = n for | n − j | ≥ m , • η ( n ) < j for even n with | n − j | < m , and • η ( n ) > j for odd n with | n − j | < m .Then the sequence ( a n,η ( n ) ) is indiscernible by (4), and the truth value of the formula ϕ ( x ; b j ) alter-nates > m times on it by the choice of η and ( ∗ ) and ( ∗ ), a contradiction. (cid:3) By Lemma 1.20 and Claim 1.25, we conclude that the indiscernible sequence ( b n ) is not distal, and b n ≡ b for all n by (4), as promised. (cid:3) Corollary 1.26.
Suppose M ⊆ acl( D ) . Then T is distal.Proof. We verify that T satisfies Definition 1.1. Let a = ( a i ) i ∈ Q be an indiscernible sequence, and letsome tuple b such that a Q \{ } is b -indiscernible be given. By assumption, there is some d ∈ D n suchthat b ⊆ acl( d ). By Ramsey and compactness, moving d by an automorphism over b , we may assumethat a Q \{ } is d -indiscernible. By Proposition 1.17, a is d -indiscernible, hence it is also b -indiscernibleas desired. (cid:3) Corollary 1.27. T is distal if and only if T eq is distal.Proof. If T eq is distal then so is T , by Lemma 1.15. For the converse note that since T is NIP, sois T eq , and M eq ⊆ acl( M ), where acl is taken in the structure M eq . Hence the previous corollaryapplies to T eq in place of T . (cid:3) Distal expansions.
We say that T has a distal expansion if there is an expansion L ∗ of L and a complete distal L ∗ -theory T ∗ which contains T . We also say that an L -structure has a distalexpansion if it can be expanded to a distal structure (in some language expanding L ). Clearly, if an L -structure M has a distal expansion, then so does its complete theory; the converse holds if M issufficiently saturated. Lemma 1.28.
Suppose T is interpretable in a complete distal L ∗ -theory T ∗ ( for some language L ∗ ) .Then T has a distal expansion.Proof. The theory T is definable in ( T ∗ ) eq , which is distal by Corollary 1.27. Hence we may replace T ∗ by ( T ∗ ) eq and assume that T is definable in T ∗ . Now Lemma 1.15 yields a distal expansion of T . (cid:3) So for example, the theory ACF of algebraically closed fields of characteristic zero has a distalexpansion, since it is interpretable (in fact, definable) in the theory RCF of real closed ordered fields:if K is a real closed ordered field then its algebraic closure is K [ i ] (where i = − K [ i ]is 0-definable in K .1.5. Distality and the Shelah expansion.
Let M be an L -structure. Recall that the Shelahexpansion of M is the structure M Sh in the language L Sh obtained from M by naming all externallydefinable subsets of M , i.e., sets of the form φ ( x, b ) N ∩ M x = (cid:8) a ∈ M x : N | = φ ( a, b ) (cid:9) with φ ( x, y ) an L -formula and b ∈ N y for some elementary extension N (cid:23) M . (Here we can replace N by an elementary extension if necessary and thus always assume N is sufficiently saturated.) ISTALITY IN VALUED FIELDS AND RELATED STRUCTURES 13
Fact 1.29. (1) M is NIP if and only if M Sh is NIP ( Shelah [60] , see also [13]) ; (2) M is distal if and only if M Sh is distal ( Boxall-Kestner [7]) . This implies the following remark on how the operations of taking Shelah expansions and reductsinteract with distality:
Lemma 1.30.
Let L ′ be an expansion of the language L and let M ′ be an L ′ -structure whose L -reductis M . If M ′ is distal, then M Sh has a distal expansion, namely ( M ′ ) Sh .Proof. We first note that ( M ′ ) Sh is indeed an expansion of M Sh , since every sufficiently satu-rated N (cid:23) M can be expanded to an L ′ -structure N ′ such that N ′ (cid:23) M ′ . Hence M Sh is areduct of ( M ′ ) Sh , and the latter is distal by Fact 1.29(2). (cid:3) Distal Fields and Rings
We emphasize the following important fact:
Fact 2.1 ([18, Corollary 6.3]) . No distal structure interprets an infinite field of positive characteristic.
We first observe that this generalizes from fields to rings without zero-divisors.
In the rest of thissection we let R be a ring; here and in the rest of this paper, all rings are assumed to be unital. Fact 2.2 (Jacobson, see e.g., [52, Theorem 12.10]) . Assume that for every r ∈ R there is some n ≥ such that r n = r . Then R is commutative. Recall that the characteristic char( R ) of R is the smallest n ≥ n · n exists, and char( R ) := 0 otherwise. For a ∈ R we let C ( a ) := { b ∈ R : ab = ba } , a subring of R . Wealso let Z ( R ) := T a ∈ R C ( a ), a commutative subring of R , the center of R . Proposition 2.3.
Suppose R is infinite without zero-divisors and interpretable in a distal structure.Then R has characteristic zero.Proof. Note that R having no zero-divisors implies that the only nilpotent element of R is 0. Firstassume that R is commutative. Then R is an integral domain, and interprets its fraction field F .But F is of characteristic 0 by Fact 2.1, and hence so is R . Now suppose R is not commutative. Inthis case, Fact 2.2 yields some r ∈ R such that r n = r for all n ≥
2. Then the powers r n of r arepairwise distinct, so the definable commutative subring R ′ = Z ( C ( r )) of R is infinite. By what wejust showed, char( R ′ ) = 0, hence char( R ) = 0. (cid:3) Here is a slight strengthening of this proposition. An idempotent e of R is said to be central if e ∈ Z ( R ), and centrally primitive if e is central, e = 0, and e cannot be written as a sum e = a + b of twononzero central idempotents a, b ∈ R with ab = 0. For every central idempotent e of R , the ideal Re of R is a ring with multiplicative identity e ; we have a surjective ring morphism r re : R → Re ,and if R has no zero-divisors, then so does Re . Corollary 2.4.
Suppose R is infinite and interpretable in a distal structure, and that for everycentrally primitive idempotent e of R , the ring Re is finite or has no zero-divisors. Then R hascharacteristic zero.Proof. Let B ( R ) be the set of central idempotents of R forms a boolean subring of R . Since R has NIP, B ( R ) is finite. Thus there are some n ≥ e , . . . , e n of R suchthat R = Re ⊕ · · · ⊕ Re n (internal direct sum of ideals of R ); see [52, § i ∈ { , . . . , n } ,the ring Re i is infinite, and hence has no zero-divisors; by Proposition 2.3 we have char( Re i ) = 0 andthus char( R ) = 0. (cid:3) In the next three subsections we show that the hypothesis of not having zero-divisors cannot bedropped in Proposition 2.3. To produce an example, we employ a certain valued F p -vector space; hereand below, we fix a prime p .2.1. Hahn spaces over F p . We first define a language L and an L -theory T whose intended modelis the Hahn product H = H ( Q , F p ), that is, the abelian group of all sequences h = ( h q ) q ∈ Q in F p withwell-ordered support supp h := (cid:8) q ∈ Q : h q = 0 (cid:9) ⊆ Q , equipped with the valuation v : H → Q ∞ satisfying v ( h ) = min(supp h ) for 0 = h ∈ H ,which makes H into a valued abelian group. (See, e.g., [2, p. 74].) Let L be the two-sorted languagewith sorts s g (for the underlying abelian group) and s v (for the value set), and the following primitives:a copy { , − , + } of the language of abelian groups on the sort s g ; a copy {≤ , ∞} of the language ofordered sets with an additional constant symbol ∞ on the sort s v , as well as a a function symbol v of sort s g s v . Next we define T − to be the (universal) L -theory whose models ( G, S ; . . . ) satisfy:(4) ( S ; ≤ ) is a linearly ordered set with largest element ∞ ,(5) ( G ; 0 , − , +) is an abelian group with pG = { } (and hence is an F p -vector space in a naturalway),(6) v : G → S is a (not necessarily surjective) F p -vector space valuation: for every g, h ∈ G ,(a) v ( g ) = ∞ iff h = 0,(b) v ( g + h ) ≥ min (cid:0) v ( g ) , v ( h ) (cid:1) ,(c) v ( kg ) = v ( g ) for every k ∈ Z \ p Z .(7) for all g, h ∈ G with vg = vh = ∞ there is k ∈ { , . . . , p − } such that v ( g − kh ) > vg (theHahn space property [2, p. 94]).Finally, we define T to be the L -theory containing T − whose models ( G, S ; . . . ) satisfy in addition:(8) the ordered set ( S ; ≤ ) is dense without smallest element, and(9) the map v : G → S is surjective.Note that if ( G, S ; . . . ) is a model of T − which satisfies (9), then ( G, S, v ) is a Hahn space over F p inthe sense of [2, Section 2.3]. All structures in the following two subsections will be models of T − ; wewill denote them by ( G, S ), ( G ′ , S ′ ), ( G ∗ , S ∗ ), and their valuation indiscriminately by v .2.2. Quantifier elimination.
There are three relevant extension lemmas for models of T − : Lemma 2.5.
Let s ∈ S \ v ( G ) . Then there are an extension ( G ′ , S ′ ) of ( G, S ) and g ′ ∈ G ′ such that (1) v ( g ′ ) = s , and (2) given any embedding i : ( G, S ) → ( G ∗ , S ∗ ) and an element g ∗ ∈ G ∗ such that v ( g ∗ ) = i ( s ) ,there is an embedding i ′ : ( G ′ , S ′ ) → ( G ∗ , S ∗ ) which extends i such that i ′ ( g ′ ) = g ∗ .Furthermore, given any ( G ′ , S ′ ) and g ′ ∈ G ′ which satisfy (1) and (2) , we have G ′ = G ⊕ F p g ′ ( internaldirect sum of F p -vector spaces ) , S ′ = S , v ( G ′ ) = v ( G ) ∪ { s } , and the embedding i ′ in (2) is unique.Proof. Let g ′ be an element of an F p -vector space extension of G with g ′ / ∈ G , and set G ′ := G ⊕ F p g ′ ,and extend v : G → S to a map G ′ → S , also denoted by v , such that v ( g + kg ′ ) = min( vg, s ) for g ∈ G , k ∈ F × p . One verifies easily that then ( G ′ , S ) is a model of T − and (1), (2) hold. (cid:3) Lemma 2.6.
Let P be a cut in S with P = S . Then there is an extension ( G ′ , S ′ ) of ( G, S ) andsome s ′ ∈ S ′ such that (1) s ′ realizes P , that is, P < s ′ < S \ P , (2) given any embedding i : ( G, S ) → ( G ∗ , S ∗ ) and an element s ∗ ∈ S ∗ such that i ( P ) < s ∗
Furthermore, given any ( G ′ , S ′ ) and s ′ ∈ S ′ which satisfy (1) and (2) , we have G = G ′ , S ′ = P ⌢ s ′ ⌢ ( S \ P ) , and the embedding i ′ in (2) is unique. The easy proof of this lemma is left to the reader. Iterating the previous two lemmas routinely implies:
Corollary 2.7.
Every model ( G, S ) of T − has a T -closure, that is, an extension ( G ′ , S ′ ) to amodel of T such that every embedding ( G, S ) → ( G ∗ , S ∗ ) into a model of T extends to an embed-ding ( G ′ , S ′ ) → ( G ∗ , S ∗ ) . We recall some basic definitions about pseudoconvergence in valued abelian groups; our reference forthis material is [2, Section 2.2]. Let ( g ρ ) be a sequence in G indexed by elements of an infinite well-ordered set without largest element. Then ( g ρ ) is said to be a pseudocauchy sequence (abbreviated:a pc-sequence ) if there is some index ρ such that for all indices τ > σ > ρ > ρ we have v ( g τ − g σ ) >v ( g σ − g ρ ). Given g ∈ G , we write g ρ ; g if the sequence (cid:0) v ( g − g ρ ) (cid:1) in S is eventually strictlyincreasing. We say that a pc-sequence ( g ρ ) in G is divergent if there is no g ∈ G with g ρ ; g . Thenext lemma is immediate from [2, Lemma 2.3.1]. Lemma 2.8.
Let ( g ρ ) be a divergent pc-sequence in G . Then there is an extension ( G ′ , S ′ ) of ( G, S ) and some g ′ ∈ G ′ such that: (1) g ρ ; g ′ , and (2) given any embedding i : ( G, S ) → ( G ∗ , S ∗ ) and an element g ∗ ∈ G ∗ such that i ( g ρ ) ; g ∗ ,there is an embedding i ′ : ( G ′ , S ′ ) → ( G ∗ , S ∗ ) which extends i such that i ′ ( g ′ ) = g ∗ .Furthermore, given any ( G ′ , S ′ ) and g ′ ∈ G ′ which satisfy (1) and (2) , we have G ′ = G ⊕ F p g ′ ( internaldirect sum of F p -vector spaces ) , S ′ = S , and the embedding i ′ in (2) is unique. We now combine the embedding lemmas above to show:
Proposition 2.9.
The L -theory T has QE.Proof. By Corollary 2.7 and one of the standard QE tests (see, e.g., [2, Corollary B.11.11]), it sufficesto show: Let (
G, S ) ( ( G , S ) be a proper extension of models of T and ( G ∗ , S ∗ ) be an | G | + -saturated elementary extension of ( G, S ); then the natural inclusion (
G, S ) → ( G ∗ , S ∗ ) extends to anembedding ( G ′ , S ′ ) → ( G ∗ , S ∗ ) of a substructure ( G ′ , S ′ ) of ( G , S ) properly extending ( G, S ).If S = S , pick an arbitrary g ∈ G with s := v ( g ) ∈ S \ S . Then | G | + -saturation of ( G ∗ , S ∗ )yields an element s ∗ of S ∗ such that for each s ∈ S we have s < s ∗ iff s < s , and by Lemma 2.6,setting G ′ := G ⊕ F p g and S ′ := S ∪ { s } gives rise to a substructure ( G ′ , S ′ ) of ( G , S ) with therequired property.Now suppose S = S . Then G = G ; pick an arbitrary g ∈ G \ G . Then [2, Lemma 2.2.18] yieldsa divergent pc-sequence ( g ρ ) in G with g ρ ; g , and | G | + -saturation of ( G ∗ , S ∗ ) yields an element g ∗ of G ∗ with g ρ ; g ∗ (see the proof of [2, Lemma 2.2.5]). In this case, setting G ′ := G ⊕ F p g and S ′ := S we obtain a substructure ( G ′ , S ′ ) of ( G , S ) with the required property. (cid:3) Corollary 2.10.
The L -theory T is complete; it is the model completion of T − . Hence if (
G, S ) | = T and G is a subgroup of G with v ( G ) = S , then ( G , S ) is an elementarysubstructure of ( G, S ). In particular, we have ( H , Q ) (cid:22) ( H, Q ) where H := (cid:8) h ∈ H : supp( h ) finite (cid:9) . Remark.
The previous proposition and its corollary can also be deduced (in a one-sorted setting) frommore general results in [51].2.3.
Indiscernible sequences.
Let (
G, S ) | = T . In the following two lemmas we prove some prop-erties of nonconstant indiscernible sequences in G . For this let ( g i ) i ∈ I be a sequence in G where I is anonempty linearly ordered set without a largest or smallest element. We let I ∗ be the set I equippedwith the reversed ordering ≥ . Lemma 2.11.
Suppose ( g i ) is nonconstant and indiscernible. Then exactly one of the following holds: (1) v ( g i − g j ) < v ( g j − g k ) for all i < j < k in I ( we say that ( g i ) is pseudocauchy) ; or (2) v ( g i − g j ) > v ( g j − g k ) for all i < j < k in I ( so the sequence ( g i ) i ∈ I ∗ is pseudocauchy ) .Proof. Choose elements 0 < < · · · < p + 1 of I and consider the p + 1 elements h := g − g p +1 , . . . , h p := g p − g p +1 of G . Let m , n range over { , . . . , p } . We have three cases to consider: Case 1: v ( h m ) = v ( h n ) for all m , n . Then by the Hahn axiom, for m ≥ k m ∈ { , . . . , p − } such that v ( h − k m b m ) > v ( h ). By the pigeonhole principle, there are 1 ≤ m < n such that k m = k n .Now note that v ( h ) < v (cid:0) ( h − k m h m ) − ( h − k n h n ) (cid:1) = v (cid:0) k m ( h n − h m ) (cid:1) = v ( h n − h m ) = v ( g n − g m )and thus v ( g n − g p +1 ) = v ( h n ) = v ( h ) < v ( g n − g m )and so we are in case (2), by indiscernibility. Case 2:
There are m < n such that v ( h m ) < v ( h n ) . Then by indiscernibility we are in case (1).
Case 3:
There are m < n such that v ( h m ) > v ( h n ) . We will actually show that this case cannothappen. Note that in this case v ( g m − g n ) = v (cid:0) h m − h n (cid:1) = v ( h n ) = v ( g n − g p +1 ) . Thus by indiscernibility, for all i < j < k < l in I we have v ( g i − g j ) = v ( g j − g k ) = v ( g k − g l )and thus taking an element i < m in I we have v ( h m ) = v ( g m − g p +1 ) = v ( g i − g m ) = v ( g m − g n ) = v ( g n − g p +1 ) = v ( h n ) , a contradiction. (cid:3) In the rest of this subsection we let A ⊆ G and B ⊆ S . Lemma 2.12.
Suppose ( g i ) is nonconstant and AB -indiscernible, and let s ∈ v ( A ) ∪ B . Then either (1) v ( g i − g j ) > s for all i = j , or (2) v ( g i − g j ) < s for all i = j .Proof. By Lemma 2.11 we have v ( g i − g j ) = v ( g k − g l ) for all i < j < k < l , and with ∈ { <, = , > } ,by s -indiscernibility of ( g i ): if v ( g i − g j ) s for some pair i < j , then v ( g i − g j ) s for all i < j . (cid:3) The two lemmas above motivate the following definition:
Definition 2.13.
We say that ( g i ) is pre- AB -indiscernible if(1) exactly one of the following is true:(a) ( g i ) i ∈ I is pseudocauchy, or(b) ( g i ) i ∈ I ∗ is pseudocauchy;(2) for each s ∈ v ( A ) ∪ B , either(a) v ( g i − g j ) > s for all i = j , or(b) v ( g i − g j ) < s for all i = j ;(3) for every a ∈ A , exactly one of the following is true:(a) (cid:0) v ( g i − a ) (cid:1) is constant,(b) (cid:0) v ( g i − a ) (cid:1) is strictly increasing,(c) (cid:0) v ( g i − a ) (cid:1) is strictly decreasing. ISTALITY IN VALUED FIELDS AND RELATED STRUCTURES 17
If ( g i ) is nonconstant and AB -indiscernible, then it is pre- AB -indiscernible, by Lemmas 2.11 and 2.12and A -indiscernibility of ( g i ). To show a converse, we first record some properties of pre- AB -indiscernible sequences. We say that ( g i ) is a “pc-sequence” if it is pseudocauchy. Lemma 2.14.
Suppose ( g i ) is a pre- AB -indiscernible pc-sequence; then for each i the value s i := v ( g i − g j ) , where j > i , does not depend on j , and (2 ′ ) for each s ∈ v ( A ) ∪ B , either (a) s i > s for all i , or (b) s i < s for all i , (3 ′ ) for each a ∈ A , either (a) (cid:0) v ( g i − a ) (cid:1) is constant, and s i > v ( g j − a ) for each i , j , or (b) s i = v ( g i − a ) for all i .Proof. The first statement is clear since ( g i ) is a pc-sequence, and implies (2 ′ ) by property (2) inDefinition 2.13. To show (3 ′ ), let a ∈ A . Suppose (3)(a) in Definition 2.13 holds, and let s be thecommon value of the v ( g i − a ); then v ( g i − g j ) ≥ s for all i < j , and since ( s i ) is strictly increasingand I does not have a smallest element, we obtain s i = v ( g i − g j ) > s for i < j . If (3)(b) holds, then s i = v ( g i − a ) for each i . Case (3)(c) does not occur: otherwise, for i < j < k we have s i = v ( g i − g j ) = v (cid:0) ( g i − a ) + ( a − g j ) (cid:1) = v ( g j − a )and similarly s i = v ( g k − a ), which is impossible. This yields (3 ′ ). (cid:3) We now arrive at our classification of nonconstant indiscernible sequences from G : Proposition 2.15.
Suppose A is a subgroup of G and ( g i ) is nonconstant. Then ( g i ) is AB -in-discernible iff ( g i ) is pre- AB -indiscernible.Proof. Suppose ( g i ) is pre- AB -indiscernible. To show that ( g i ) is AB -indiscernible we can assumethat ( g i ) is a pc-sequence; so for each i the value s i := v ( g i − g j ), where j > i , does not depend on j .For a ∈ A such that (cid:0) v ( g i − a ) (cid:1) is constant, denote by s a the common value of the v ( g i − a ). Let now t ( x , . . . , x n ) = k x + · · · + k n x n + a ( k , . . . , k n ∈ Z , a ∈ A )be an L A -term of sort s g . By quantifier elimination (Proposition 2.9) and Lemma 2.14 it is enoughto show that • v (cid:0) t ( g i , . . . , g i n ) (cid:1) is constant and contained in v ( A ) for i < · · · < i n , or • there is g ∈ A such that v ( g i − a ) is constant and v (cid:0) t ( g i , . . . , g i n ) (cid:1) = s a for i < · · · < i n , or • there is an m ∈ { , . . . , n } such that v (cid:0) t ( g i , . . . , g i n ) (cid:1) = s i m for i < · · · < i n .For this we can assume k m / ∈ p Z for some m , since otherwise t ( g ) = a for all g ∈ G , and we are done;take m minimal such that k m / ∈ p Z . Set k := k + · · · + k n . We distinguish two cases: Case 1: k ∈ p Z . Then t ( h , . . . , h n ) = k ( h − h n ) + · · · + k n − ( h n − − h n ) + a for all h , . . . , h n ∈ G .Let s := va . If (2 ′ )(a) in Lemma 2.14 holds, then v (cid:0) t ( g i , . . . , g i n ) (cid:1) = s for i < · · · < i n in I ; if (2 ′ )(b)holds, then m < n , and v (cid:0) t ( g i , . . . , g i n ) (cid:1) = s i m for i < · · · < i n in I . Case 2: k / ∈ p Z . Then we can take g ∈ A such that t ( h , . . . , h n ) = k ( h − h n ) + · · · + k n − ( h n − − h n ) + k ( h n − h ) for all h , . . . , h n ∈ G .If (3 ′ )(a) holds, then v (cid:0) t ( g i , . . . , g i n ) (cid:1) = s a for i < · · · < i n in I ; whereas if (3 ′ )(b) holds, then v (cid:0) t ( g i , . . . , g i n ) (cid:1) = s i m for i < · · · < i n in I . (cid:3) Corollary 2.16. T is distal. Proof.
By Corollary 1.26 it suffices to prove that the structure induced on the group sort s g ofmodels of T is distal. For this, suppose ( g i ) i ∈ I as above is indiscernible, the linearly ordered set I is dense, and 0 is an element of I such that ( g i ) i ∈ I = is AB -indiscernible, where I = := I \ { } ; byProposition 1.10, it is enough to show that then ( g i ) i ∈ I is AB -indiscernible. This is clear if ( g i ) i ∈ I is constant; thus we may assume that ( g i ) i ∈ I is nonconstant. Replacing A by the subgroup of G generated by A we can also arrange that A is a subgroup of G , and by Lemma 2.11, that ( g i ) i ∈ I is a pc-sequence. Let s i := v ( g i − g j ) where j > i is arbitrary. Let s ∈ v ( A ) ∪ B ; if s i > s forall i ∈ I = , then also s > s , and similarly with “ < ” in place of “ > ”. Together with Lemma 2.12applied to ( g i ) i ∈ I = , this implies that (2) in Definition 2.13 holds. Similarly, using Lemma 2.14(3 ′ )for ( g i ) i ∈ I = we see that statement (3) in Definition 2.13 holds: Let a ∈ A . Suppose (cid:0) v ( g i − a ) (cid:1) i ∈ I = is constant and s i > v ( g j − a ) for all i, j ∈ I = ; then s i > v ( g j − a ) for all i ∈ I , j ∈ I = and thus v ( g − a ) = v (cid:0) ( g − g j ) + ( g j − a ) (cid:1) = v ( g j − a ) for j = 0, hence (3)(a) holds. If s i = v ( g i − a ) for i = 0,then v ( g − a ) = v (cid:0) ( g − g j ) + ( g j − a ) (cid:1) = s for j >
0, hence (3)(b) holds. This shows that ( g i ) i ∈ I ispre- AB -indiscernible, and hence AB -indiscernible by Proposition 2.15. (cid:3) We now use the above to give our promised example of an infinite ring of positive characteristicinterpretable a distal structure.
Example.
Suppose R = F p × H , where H = H ( Q , F p ) is as in the beginning of Section 2.1, equippedwith the componentwise addition and multiplication given by( k, g ) · ( l, h ) := ( kl, kg + lh ) for k, l ∈ F p , g, h ∈ H .Then R is a commutative ring of characteristic p , with multiplicative identity 1 = (1 , R is interpretable in the L -structure ( H, Q ) | = T , which is distal by Corollary 2.16. Remark.
Distality for a more general class of valued abelian groups and certain related structures isestablished in [16], and is used there to demonstrate that in fact every abelian group (in the puregroup language) admits a distal expansion.In the remainder of this section we point out a consequence of Fact 2.1 for henselian valued fieldswith a distal expansion.2.4.
NIP in henselian valued fields.
In this subsection K is a henselian valued field with valuegroup Γ and residue field k . We view K as a model-theoretic structure ( K, O ), where O is thevaluation ring of K . We recall the following facts; the proofs below are courtesy of Franziska Jahnke. Fact 2.17.
Suppose K is finitely ramified and k is NIP and perfect; then ( K, O ) is NIP.Proof. In the case char k = 0 this follows from Delon [22] (using also [35]), and for char k > K this was shown by B´elair [5]. We reduce the finitely ramified case with char k = p > K, O ) is ℵ -saturated. Let ∆ := ∆ be the smallestconvex subgroup of Γ containing vp , and let ˙ K be the corresponding specialization of K . Then ˙ K hascharacteristic zero, cyclic value group ∆ , and residue field isomorphic to k ; saturation implies that ˙ K is complete. It is well-known (see, e.g. [69, Theorem 22.7]) that therefore ˙ K is a finite extension of acomplete unramified discretely valued subfield L with the same residue field k as ˙ K . By [5], ( L, O L )is NIP, hence so is ( ˙ K, O ˙ K ). Now the ∆-coarsening ( K, ˙ O ) of K has residue field ˙ K , and hence is NIPby [22]. The valuation ring of ˙ K is definable in the pure field ˙ K [50, Lemma 3.6]. Hence O is definablein ( K, ˙ O ), and thus ( K, O ) is NIP. (cid:3) See Corollaries 5.18 and 5.23 below for versions of the preceding fact where k and Γ are permitted tohave additional structure. Here is a partial converse of Fact 2.17: ISTALITY IN VALUED FIELDS AND RELATED STRUCTURES 19
Fact 2.18.
Suppose ( K, O ) is NIP and k is finite; then K is finitely ramified.Proof. We may assume that ( K, O ) is ℵ -saturated. This time, we let ∆ be the biggest convexsubgroup of Γ not containing vp , and let ˙ K be the corresponding specialization of K . Then ˙ K hascharacteristic p , value group ∆, and residue field isomorphic to k . The Shelah expansion of ( K, O )interprets every convex subgroup of Γ, and hence also the valued field ( ˙ K, O ˙ K ); in particular, ( ˙ K, O ˙ K )is NIP, by Fact 1.29(1). Now [48, Proposition 5.3] implies that ∆ = { } , since k is finite. Hence forevery γ > n such that nγ ≥ vp . Saturation yields some n such that for every γ > nγ ≥ vp ; hence K is finitely ramified. (cid:3) Combining 2.1 and 1.15 with 2.18 implies:
Corollary 2.19. If ( K, O ) has a distal expansion, then K is finitely ramified and k has characteristiczero or is finite.Remark . If K is finitely ramified and k is finite, p = char k , then K has a specialization whichis p -adically closed of finite p -rank. (Let ∆ = ∆ be as in the proof of Fact 2.17 and let ˙ K be the∆-specialization of K ; then ˙ K is henselian of mixed characteristic (0 , p ) with cyclic value group andfinite residue field k , hence is p -adically closed of finite p -rank [55, Theorem 3.1].)See [1, Section 5.1] for a conjectural characterization of all NIP henselian valued fields.3. Distality in Ordered Abelian Groups
In 1984, Gurevich and Schmitt [35] showed that every ordered abelian group is NIP. In this section,we investigate distality for ordered abelian groups; the main result is Theorem 3.13 below. As awarmup, in Section 3.1 we characterize distality for those ordered abelian groups which have quantifierelimination in the Presburger language (see Theorem 3.2). This already applies to a variety of familiarordered abelian groups since it includes every ordered abelian group which is elementarily equivalentto an archimedean one.
In the rest of this section we assume m, n ≥ , and we let p , q range over theset of prime numbers. An ordered abelian group G is said to be non-singular if G/pG is finite for every p . The followingfact from [45, Proposition 5.1] will be used several times: Fact 3.1.
An ordered abelian group is dp -minimal if and only if it is non-singular. The case of QE in L Pres . In this subsection we consider ordered abelian groups in the
Pres-burger language L Pres = (cid:8) , , + , − , <, ( ≡ m ) (cid:9) . We naturally construe a given ordered abelian group G as an L Pres -structure: the symbols 0, +, − , < have their usual interpretations; the constant symbol 1 is interpreted by the least positive elementof G , provided G has one, and by 0 otherwise; and for each m , the binary relation symbol ≡ m isinterpreted as equivalence modulo m , i.e., for g, h ∈ G , g ≡ m h : ⇐⇒ g − h ∈ mG. In the rest of this subsection G is an ordered abelian group, and all ordered abelian groups will beconstrued as L Pres -structures.
Recall that an ordered abelian group is regular if it is elementarilyequivalent to an archimedean ordered abelian group; moreover, G is regular if either | G/nG | = n for each n ≥
1, or nG is dense in G for each n ≥
1. In the first case, G is elementarily equivalentto ( Z ; + , < ), whereas any two dense regular ordered abelian groups G , H are elementarily equivalent ifffor each p either G/pG and
H/pH are infinite or | G/pG | = | H/pH | . (See [59, 73].) In this subsectionwe show the following. Theorem 3.2.
Suppose G is regular; then the following are equivalent: (1) G is distal; (2) G is dp -minimal; (3) G is non-singular. Theorem 3.2 applies to archimedean G , so the ordered abelian groups ( Z ; + , < ), ( Q ; + , < ), and( Z (2) ; + , < ) are distal, whereas ( Q > ; · , < ) is not.The rest of this subsection is devoted to proving Theorem 3.2. We rely on the following: Fact 3.3 (Weispfenning, [71]) . An ordered abelian group is regular if and only if it has QE in L Pres . We first note that the direction (2) ⇒ (1) in Theorem 3.2 holds by Fact 1.5. Furthermore, theequivalence (2) ⇔ (3) is Fact 3.1. Thus it suffices to establish (1) ⇒ (3). We will actually prove thecontrapositive. For the rest of the subsection we thus fix some p and assume:(1) G is regular;(2) G/pG is infinite;(3) G is sufficiently saturated.We shall prove that under these assumptions, G is not distal. By QE in L Pres , we can easily describeindiscernible sequences in a single variable:
Lemma 3.4.
A sequence ( a i ) i ∈ I in G is indiscernible if and only if for all i < · · · < i n and j < · · · < j n from I , k, k , . . . , k n ∈ Z , and m ≥ we have (1) k · P nl =1 k l a i l > ⇐⇒ k · P nl =1 k l a j l > ; (2) k · P nl =1 k l a i l = 0 ⇐⇒ k · P nl =1 k l a j l = 0 ; and (3) k · P nl =1 k l a i l ≡ m ⇐⇒ k · P nl =1 k l a j l ≡ m . We think of (1) and (2) in Lemma 3.4 as geometric conditions and of (3) as algebraic conditions . Itis easy to prescribe a certain choice of geometric conditions in a rapidly increasing sequence; here wesay that a sequence ( a i ) i ∈ I in G is rapidly increasing if for all i < j from I and m , n ,0 ≤ m < na i < a j . (That is, a i > i , and the a i and 1 lie in distinct archimedean classes.) Lemma 3.5.
Suppose ( a i ) i ∈ I is a rapidly increasing sequence in G . Then for all i < · · · < i n and j < · · · < j n from I and all k, k , . . . , k n ∈ Z , we have (1) k · P nl =1 k l a i l > ⇐⇒ k · P nl =1 k l a j l > ⇐⇒ ( k n , . . . , k , k ) > lex (0 , . . . , , and (2) k · P nl =1 k l a i l = 0 ⇐⇒ k · P nl =1 k l a j l = 0 ⇐⇒ k = k = · · · = k n = 0 . In general, it is more difficult to prescribe all of the algebraic conditions which hold in an indiscerniblesequence, but once we have an indiscernible sequence in G we can use the following: Lemma 3.6.
Suppose ( a i ) i ∈ I is an indiscernible sequence in G . Then for all distinct i , . . . , i n anddistinct j , . . . , j n from I , all k, k , . . . , k n ∈ Z and m ≥ , we have (1) k · P nl =1 k l a i l = 0 ⇐⇒ k · P nl =1 k l a j l = 0 , and (2) k · P nl =1 k l a i l ≡ m ⇐⇒ k · P nl =1 k l a j l ≡ m .Proof. The sequence ( a i ) is indiscernible in the (cid:8) , , + , − , ( ≡ m ) (cid:9) -reduct of G . However, this reductis just (an expansion by definitions and constants of) the underlying abelian group of G , which isstable. Thus the sequence ( a i ) in this reduct is totally indiscernible, which implies the conclusion ofthe lemma. (cid:3) Proposition 3.7. G is not distal. ISTALITY IN VALUED FIELDS AND RELATED STRUCTURES 21
Proof.
First, using Ramsey we obtain a rapidly increasing indiscernible sequence ( b i ) i ∈ ( − , in G suchthat b i p b j for all i < j from ( − , G/pG is infinite and that each cosetof pG is cofinal in G . We will use ( b i ) to obtain our counterexample to distality. For this, consider thecollection Φ( x ) of L Pres -formulas, with x = ( x i ) i ∈ ( − , , consisting exactly of the following formulas:(Φ1) for every i < j from ( − ,
1] and every m , n , the formula0 ≤ m < nx i < x j , (Φ2) for every i < · · · < i n from ( − ,
1) and k, k . . . , k n ∈ Z , if G | = k · P nl =1 k l b i l ≡ m
0, theformulas k · P nl =1 k l x i l ≡ m (cid:0) k · P nl =1 k l x i l ≡ m (cid:1) [ x /x ] , and otherwise the formulas k · P nl =1 k l x i l m (cid:0) k · P nl =1 k l x i l m (cid:1) [ x /x ] , where [ x /x ] denotes replacing each occurrence of x in the preceding expression by x , and(Φ3) the formula x ≡ p x .Thus Φ( x ) expresses that ( x i ) i ∈ ( − , is rapidly increasing and satisfies the same algebraic conditionsas ( b i ), x and x have the same algebraic relations with ( x i ) i ∈ ( − , \{ } , however x and x arecongruent modulo p . Claim 3.8. Φ( x ) is finitely satisfiable in G .Proof of Claim. Let Φ ⊆ Φ be finite. Set b ∗ i := b i for i ∈ ( − , b ∗ i ) i ∈ ( − , satisfiesall formulas from (Φ1), (Φ2), and (Φ3) which do not involve x . We claim that we can choose b ∗ ∈ G so that ( b ∗ i ) i ∈ ( − , satisfies Φ . To see this let N be the product of all moduli occurring in Φ , andpick b ∗ to be a sufficiently large member of the coset b + pN G . The “sufficiently large” ensuresthat all formulas in Φ coming from (Φ1) are satisfied, the choice of N ensures that b ≡ m b ∗ for allrelevant m , and thus all formulas from (Φ2) are satisfied, and clearly b ≡ p b ∗ . (cid:3) By the claim and after replacing our original sequence ( b i ) i ∈ ( − , , we can assume that we havesome b ∈ G such that ( b i ) i ∈ ( − , realizes Φ( x ). It is clear that ( b i ) i ∈ ( − , is indiscernible, andthat ( b i ) i ∈ ( − , is not b -indiscernible. It remains to establish: Claim 3.9. ( b i ) i ∈ ( − , \{ } is b -indiscernible.Proof of Claim. It is sufficient to show that ( b i ) i ∈ ( − , \{ } is indiscernible. By (Φ1) this sequenceis rapidly increasing, thus by Lemma 3.5 the geometric conditions (1) and (2) of Lemma 3.4 hold.It suffices to check (3) from Lemma 3.4. Let i < · · · < i n − < i n = 1 from ( − , ∪ (0 ,
1] and j < · · · < j n from ( − , k, k , . . . , k n ∈ Z ; it is sufficient to show that then k · P nl =1 k l b i l ≡ m ⇐⇒ k · P nl =1 k l b j l ≡ m . Now k · P nl =1 k l b i l ≡ m ⇐⇒ (cid:0) k · P nl =1 k l b i l ≡ m (cid:1) [ b /b ]by (Φ2), and (cid:0) k · P nl =1 k l b i l ≡ m (cid:1) [ b /b ] ⇐⇒ k · P nl =1 k l b j l ≡ m , by Lemma 3.6 and the fact that ( b i ) i ∈ ( − , is indiscernible. (cid:3) This concludes the proof of the proposition. (cid:3)
A review of the Cluckers-Halupczok language.
In the rest of the section, we consider or-dered abelian groups which do not in general have QE in L Pres . We use the language L qe introduced byCluckers and Halupczok [19] (see also [39]) for their (relative) quantifier elimination result for orderedabelian groups. This language is similar in spirit to one introduced by Gurevich and Schmitt [35],however it is more in line with our modern paradigm of many-sorted languages and perhaps a littlemore intuitive.The rest of the subsection is taken essentially from [19]. In what follows G is an ordered abeliangroup and we use the notation H ⋐ G to denote that H is a convex subgroup of G . We introduce L qe and at the same time describe how G is viewed as an L qe -structure G . We begin by listing the sortsof L qe : besides the main sort G whose underlying set is that of the ordered abelian group G , these arethe auxiliary sorts S p , T p , T + p (one for each p ) associated with G . Here is how they are interpretedin G : Definition 3.10. (1) For a ∈ G \ pG , let G p ( a ) be the largest convex subgroup of G such that a / ∈ G p ( a ) + pG , andfor a ∈ pG let G p ( a ) := { } ; then the underlying set of sort S p is (cid:8) G p ( a ) : a ∈ G (cid:9) ;(2) for b ∈ G , set G − p ( b ) := S (cid:8) G p ( a ) : a ∈ G, b / ∈ G a (cid:9) , where the union over the empty set isdeclared to be { } ; then the underlying set of sort T p is (cid:8) G − p ( b ) : b ∈ G (cid:9) ;(3) For b ∈ G , define G + p ( b ) := T (cid:8) G p ( a ) : a ∈ G, b ∈ G p ( a ) (cid:9) , where the intersection over theempty set is G ; then the underlying set of sort T + p is (cid:8) G + p ( b ) : b ∈ G (cid:9) .Below we don’t distinguish notationally between the sort S p and its underlying set (so we can write S p = (cid:8) G p ( a ) : a ∈ G (cid:9) ), and similar for the other auxiliary sorts. We let α range over (the underlyingsets of) the auxiliary sorts. In each case, α is a convex subgroup of G ; if we want to stress this roleof α as a convex subgroup of G (rather than as an abstract element of the underlying set of a certainsort of the structure G ), we denote it by G α , and we let π α : G ։ G/G α be the natural surjection.We let 1 α denote the minimal positive element of G/G α if the ordered abelian group G/G α is discrete,and set 1 α := 0 ∈ G/G α otherwise; for k ∈ Z we let k α := k · α . For a, b ∈ G and ⋄ denoting one ofthe relation symbols =, < , or ≡ m we also write a ⋄ α b + k α if π α ( a ) ⋄ π α ( b ) + k α holds in the orderedabelian group G/G α . We also set G [ m ] α := \ G α ( H ⋐ G ( H + mG ) and a ≡ [ m ] n,α b : ⇐⇒ a − b ∈ G [ m ] α + nG ( a, b ∈ G ) . We now describe the primitives of the L qe -structure G ; these are:(G1) on the main sort G , the usual primitives 0, +, − , ≤ of the language of ordered abelian groups;(G2) binary relations “ α ≤ α ′ ” on (cid:16) S p · ∪ T p · ∪ T + p (cid:17) × (cid:16) S q · ∪ T q · ∪ T + q (cid:17) , interpreted as G α ⊆ G α ′ (each pair ( p, q ) giving rise to nine separate binary relations);(G3) predicates for the relations a ⋄ α b + k α , where ⋄ ∈ (cid:8) = , <, ( ≡ m ) (cid:9) and k ∈ Z (each of thesebeing ternary relations on G × G × X where X ∈ {S p , T p , T + p } );(G4) for m ≥ n , the ternary relation x ≡ [ q m ] q n ,α y on G × G × S p ;(G5) a unary predicate discr of sort S p which holds of α if and only if G/G α is discrete;(G6) for d ∈ N and n , two unary predicates of sort S p defining the sets (cid:8) α ∈ S p : dim F p (cid:0) G [ p n ] α + pG (cid:1)(cid:14)(cid:0) G [ p n +1 ] α + pG (cid:1) = d (cid:9) and (cid:8) α ∈ S p : dim F p (cid:0) G [ p n ] α + pG (cid:1)(cid:14) ( G α + pG ) = d (cid:9) . We let A be the set of auxiliary sorts associated to G , and let L A qe be the sublanguage of L qe withsorts A and primitives listed in (G2), (G5), (G6). ISTALITY IN VALUED FIELDS AND RELATED STRUCTURES 23
Definition 3.11.
Let φ ( x, η ) be an L qe -formula, where x and η are multivariables of sort G and A ,respectively. We say that φ ( x, η ) is in family union form if φ ( x, η ) = n _ i =1 ∃ θ (cid:0) ξ i ( η, θ ) ∧ ψ i ( x, θ ) (cid:1) ,where θ is a multivariable of sort A , ξ i ( η, θ ) are L A qe -formulas, each ψ i ( x, θ ) is a conjunction of basicformulas (i.e., atomic or negated atomic formulas), and for each ordered abelian group G , viewed asan L qe -structure G as above, the formulas ξ i ( η, α ) ∧ ψ i ( x, α ), with i ranging over { , . . . , n } and α over tuples of the appropriate sorts in G , are pairwise inconsistent.The following is the main result from [19]: Fact 3.12.
In the theory of ordered abelian groups, each L qe -formula is equivalent to an L qe -formulain family union form. The case where all S p are finite. The main result of this section is the following.
Theorem 3.13.
Suppose that S p is finite for all p . Then G is distal if and only if G is non-singular. The hypothesis of the theorem holds if G is strongly dependent, by [12, 25, 30, 37]. The proof ofTheorem 3.13, which we now outline, is a generalization of the proof of Theorem 3.2, using Fact 3.12. For the rest of this section, G is an ordered abelian group such that for each p the underlying set ofsort S p is finite. Note that then the underlying sets of sorts T p and T + p are also finite, for each p .It suffices to show that if G/pG is infinite for some p , then G is not distal. Here we construe G asan L qe -structure, together with constants which name all of A ; since each S p is finite, the underlyingsets of auxiliary sorts will not grow when we pass to an elementary extension of G . Thus we can alsoassume that G is sufficiently saturated. In this setting, Fact 3.12 specializes as follows: Proposition 3.14. In G , each L qe -formula φ ( x ) , where x is a multivariable of sort G , is equivalentto a finite boolean combination of atomic formulas in which the only occurring predicates are thosefrom (G3) .Proof. In Fact 3.12, the quantifier “ ∃ θ ” can be replaced by a finite disjunction over all possible tuplesof constants of the same sort as θ . Upon substitution of these constants, each “ ξ i ( θ )” becomes asentence, so in the theory of G , it is equivalent to ⊥ or ⊤ . Likewise for the unary relation discr( α ),the unary “dimension” relations applied to α , and the binary relations α ≤ α ′ . Finally, as each S p isfinite, the ternary relations x ≡ [ q m ] q n ,α y from (G4) are already taken care of by the relations x ≡ q n ,α ′ y :by [19, Lemma 2.4(2)] we have G [ q m ] α = G α ′ + q m G where α ′ is the successor of α in S q m with respectto the linear ordering ≤ of S q m from (G2). (cid:3) Proposition 3.14 should be viewed as saying that G has QE in a language which is essentially a unionof countably many copies of the Presburger language, one for each of the quotient groups G/G α . Withthis point of view, it is fairly straightforward to generalize everything in subsection 3.1 by including“for every α ” in many places. For instance, we have the following generalization of Lemma 3.4: Lemma 3.15.
A sequence ( a i ) i ∈ I in G is indiscernible iff for all i < · · · < i n and j < · · · < j n from I , all k, k , . . . , k n ∈ Z , all α , and m ≥ , we have (1) P nl =1 k l a i l > α k α ⇐⇒ P nl =1 k l a j l > α k α ; (2) P nl =1 k l a i l = α k α ⇐⇒ P nl =1 k l a j l = α k α ; and (3) P nl =1 k l a i l ≡ m,α k α ⇐⇒ P nl =1 k l a j l ≡ m,α k α . Next, following the proof of Theorem 3.2, the “rapidly increasing sequence” we construct here is asequence ( a i ) i ∈ I in G such that for all i < j from I , all m , n , and all α ,0 ≤ α m · α < α n · a i < α a j . That is, the sequence ( a i ) is a rapidly increasing sequence in each of the countably many quo-tients G/G α . This gives rise to an appropriate generalization of Lemma 3.5. We also use the fact thatthe (unordered) abelian group reducts of the quotients G/G α are all stable, to get a generalization ofLemma 3.6. Finally, the proof of Proposition 3.7 generalizes to conclude our proof of Theorem 3.13.We conclude this section with the following conjecture. Conjecture 3.16.
Every ordered abelian group admits a distal expansion.
There are some partial results towards this conjecture, but the general case remains open.4.
Distality and Short Exact Sequences of Abelian Groups
In this section we prove a general quantifier elimination theorem for certain short exact sequences ofabelian groups, and analyze distality in this setting. These results are used in Sections 5 and 6 below.In Section 4.1 we show our main elimination result. The remaining subsections of this section discussan application to the preservation of distality as well as variants and refinements.4.1.
Quantifier elimination for pure short exact sequences.
Let0 → A ι −−→ B ν −−→ C → pure , which means that ι ( A ) is apure subgroup of B . (For example, this always holds if C is torsion-free.) We treat such a pure shortexact sequence as a three-sorted structure ( A, B, C ) consisting of three abelian groups, with the twomaps ι : A → B and ν : B → C added as primitives. If A is ℵ -saturated, then the short exact sequencesplits, i.e., B is the direct sum of A and C , with ι and ν being the natural embedding respectivelyprojection (see, e.g., [2, Corollary 3.3.38]). So the complete theory of ( A, B, C ) is uniquely determinedby the theory of A and the theory of C . Moreover, if ( A, C, R , R , . . . ) is an arbitrary expansion of thepair ( A, C ), then the theory of (
A, B, C, R , R , . . . ) is determined by the theory of ( A, C, R , R , . . . ).For a syntactical formulation of this observation let us fix the languages involved: • L ac = { a , + a , − a , c , + c , − c } , the language of the pair ( A, C ) of abelian groups; • L b = { b , + b , − b } , the language of abelian groups on B ; • L abc = L ac ∪ L b ∪ { ι, ν } , the language of the three-sorted structure ( A, B, C ); • L ∗ ac the language of an expansion ( A, C, R , R , . . . ) of the L ac -structure ( A, C ); • L ∗ abc = L abc ∪ L ∗ ac , the language of ( A, B, C, R , R , . . . ).Let T abc be the L abc -theory of all structures arising from pure exact sequences as above. Viewing T abc as a set of sentences in the expanded language L ∗ abc , the observation above then reads as follows: Corollary 4.1.
Every L ∗ abc -sentence is equivalent in T abc to an L ∗ ac -sentence. This is also a consequence of the quantifier elimination theorem to be proved in this section. For itsformulation we note that for each n , our short exact sequence fits into a commutative diagram of ISTALITY IN VALUED FIELDS AND RELATED STRUCTURES 25 group morphisms 0 (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / nA / / ⊆ (cid:15) (cid:15) nB / / ⊆ (cid:15) (cid:15) nC / / ⊆ (cid:15) (cid:15) / / A ι / / π n (cid:15) (cid:15) B ν / / (cid:15) (cid:15) C / / (cid:15) (cid:15) / / A/nA / / (cid:15) (cid:15) B/nB / / (cid:15) (cid:15) C/nC / / (cid:15) (cid:15)
00 0 0with exact rows and columns. We now expand (
A, B, C ) by new sorts with underlying sets
A/nA together with two unary functions: the natural surjection π n : A → A/nA and a function ρ n : B → A/nA , which, on ν − ( nC ), is the composition of the group morphisms ν − ( nC ) = nB + ι ( A ) → (cid:0) nB + ι ( A ) (cid:1) /nB ∼ −→ ι ( A ) / (cid:0) nB ∩ ι ( A ) (cid:1) ∼ −→ A/nA, and zero outside ν − ( nC ). Note that ρ : B → A agrees with the inverse of ι : A ∼ −→ ι ( A ) on ι ( A ) = ν − (0) and is zero on B \ ι ( A ). (We identify A with A/ A in the natural way.) Note also that π n = ρ n ◦ ι . Moreover, if our short exact sequence splits, and π ′ : B → A is a left inverse of ι , then ρ n agrees with π n ◦ π ′ on ν − ( nC ).We denote the language of this expansion of the L abc -structure ( A, B, C ) by L abcq = L abc ∪ { ρ , ρ , . . . , π , π , . . . } , and we let T abcq be the L abcq -theory of all these structures arising from a pure exact sequence asabove. We also let L acq = L ac ∪ { π , π , . . . } , a sublanguage of L abcq . Note that the group operations on A/nA are 0-definable in the reduct of T abcq to the two-sorted language L a ∪ { π n } , where L a = { a , + a , − a } is the language of the abelian group A .Note also that π n , ρ n are interpretable in the L abc -reduct of T abcq ; in particular, if M = ( A, B, C, . . . )and M ′ = ( A ′ , B ′ , C ′ , . . . ) are models of T abcq , then every isomorphism between the L abc -reductsof M , M ′ extends uniquely to an L abcq -isomorphism M → M ′ .Let the multivariables x a , x b , x c be of sort A , B and C , respectively. The L abcq -terms of theform ρ n (cid:0) t ( x b ) (cid:1) or ν (cid:0) t ( x b ) (cid:1) , for an L b -term t ( x b ), are called special . Theorem 4.2. In T abcq every L abc -formula φ ( x a , x b , x c ) is equivalent to a formula φ acq (cid:0) x a , σ ( x b ) , . . . , σ m ( x b ) , x c (cid:1) where the σ i are special terms and φ acq is a suitable L acq -formula. For example, the formula x b = 0 b is equivalent to ρ ( x b ) = 0 a ∧ ν ( x b ) = 0 c . Also, x b is divisible by n if and only if ρ n ( x b ) = π n (0 a ) and ν ( x b ) is divisible by n . Proof.
Let σ , σ , . . . list all special terms. Given a tuple b in a model of T abcq of the same sort as x b ,let us write σ ( b ) for the tuple σ ( b ) , σ ( b ) , . . . . Assume that we have two models M = ( A, B, C, . . . )and M ′ = ( A ′ , B ′ , C ′ , . . . ) of T abcq . We let a , b , c range over tuples in M of the same sort as x a , x b , x c , respectively, and similarly with the tuples a ′ , b ′ , c ′ in M ′ . Suppose we are given a , b , c in M and a ′ , b ′ , c ′ in M ′ such that the type of aσ ( b ) c in the L acq -reduct M acq of M is the same as the type of a ′ σ ( b ′ ) c ′ in the L acq -reduct M ′ acq of M ′ . It is enough to show that then abc and a ′ b ′ c ′ havethe same type in M and in M ′ , respectively.For this, after replacing M , M ′ by suitably saturated elementarily equivalent structures, we mayassume that there is an isomorphism M acq ∼ = −→ M ′ acq with aσ ( b ) c a ′ σ ( b ′ ) c ′ . We can then alsoassume that the short exact sequences underlying M and M ′ split, thus this isomorphism extends toan isomorphism M ∼ = −→ M ′ . Hence we may assume that M = M ′ , a = a ′ , c = c ′ and σ ( b ) = σ ( b ′ ),and it suffices to show that there is an automorphism of M which is the identity on A and C andmaps b to b ′ .Let B denote the subgroup of B generated by b and B ′ the subgroup of B ′ generated by b ′ .Since for each L b -term t ( x b ) we have t ( b ) = 0 if and only if t ( b ′ ) = 0, we obtain an isomorphism f : B → B ′ such that f ( t ( b )) = t ( b ′ ) for all L b -terms t ( x b ); in particular, we have f ( b ) = b ′ .Furthermore we have ρ n ( b ) = ρ n ( f ( b )) and ν ( b ) = ν ( f ( b )) for all b ∈ B . Set A := B ∩ ι ( A ) = B ′ ∩ ι ( A ) , C := ν ( B ) = ν ( B ′ ) . The map b ι − (cid:0) f ( b ) − b (cid:1) is a group morphism B → A . Since f fixes all elements of A ,the image of b ∈ B under this morphism only depends on ν ( b ). So f induces a group morphism h : C → A satisfying f ( b ) = b + ι (cid:0) h ( ν ( b )) (cid:1) for all b ∈ B .We show now that h is a partial morphism C → A in the sense of [74, p. 159], that is, h ( nC ∩ C ) ⊆ nA for each n : given c ∈ nC ∩ C , choose b ∈ B with ν ( b ) = c ; since ρ n is a group morphismon ν − ( nC ), we then have π n (cid:0) h ( c ) (cid:1) = ρ n (cid:0) ι ( h ( c )) (cid:1) = ρ n (cid:0) f ( b ) − b (cid:1) = ρ n (cid:0) f ( b ) (cid:1) − ρ n ( b ) = 0 , from which we conclude that h ( c ) ∈ nA .Finally we may assume that A is pure injective. Then the partial morphism h extends to a groupmorphism h : C → A [74, Corollary 3.3]. The formula b b + ι (cid:0) h ( ν ( b )) (cid:1) defines an automorphismof B which together with the identity on all other sorts is an automorphism of M which maps b to b ′ ,as required. (cid:3) The following corollary generalizes Corollary 4.1; here we view T abcq as a set of L ∗ abcq -sentences. Corollary 4.3. In T abcq every L ∗ abc -formula φ ∗ ( x a , x b , x c ) is equivalent to a formula φ ∗ acq (cid:0) x a , σ ( x b ) , . . . , σ m ( x b ) , x c (cid:1) where the σ i are special terms and φ ∗ acq is a suitable formula in the language L ∗ acq := L acq ∪ L ∗ ac .Proof. This has exactly the same proof as Theorem 4.2. We show instead that the corollary followsdirectly from the theorem itself. It is clear that the collection of all formulas equivalent in T abcq to onehaving the form in the statement of the corollary contains all atomic formulas, is closed under booleancombinations and under quantification over A and over C . It remains to show that this collectionof formulas is also closed under quantification over B . Let y b be a multivariable of sort B disjointfrom x b , and consider the formula φ ∗ ( x a , x b , x c ) = ∃ y b ψ ∗ (cid:0) x a , σ ( x b , y b ) , . . . , σ m ( x b , y b ) , x c (cid:1) with special terms σ i and a suitable L ∗ acq -formula ψ ∗ . We may assume that we have k ∈ { , . . . , m } and n , . . . , n k ∈ N such that σ i is of sort A/n i A for i = 1 , . . . , k and of sort C for i = k + 1 , . . . , m .Theorem 4.2 implies that for distinct variables z , . . . , z k of sort A and z k +1 , . . . , z m of sort C , the L abcq -formula ∃ y b k ^ i =1 π n i ( z i ) = σ i ( x b , y b ) ∧ m ^ i = k +1 z i = σ i ( x b , y b ) ! ISTALITY IN VALUED FIELDS AND RELATED STRUCTURES 27 is equivalent in T abcq to a formula χ (cid:0) z , . . . , z m , τ ( x b ) , . . . , τ n ( x b ) (cid:1) where the τ j are special terms and χ is a suitable L acq -formula. Then φ ∗ is equivalent to ∃ z · · · ∃ z m (cid:16) χ (cid:0) z , . . . , z m , τ ( x b ) , . . . , τ n ( x b ) (cid:1) ∧ ψ ∗ (cid:0) x a , π n ( z ) , . . . , π n k ( z k ) , z k +1 , . . . , z m , x c (cid:1)(cid:17) , which has the desired form. (cid:3) Remark.
Corollary 4.3 implies the quantifier elimination result in [15]: when all quotients
A/nA arefinite, the maps ρ n are quantifier-free definable in the language used there.4.2. Preservation of distality.
In this section we prove a result on preservation of distality in pureshort exact sequences. Let 0 → A ι −−→ B ν −−→ C → A and C to beequipped with arbitrary additional structure, and denote the respective languages of these expansionsby L ∗ a and L ∗ c . We also let M = (cid:0) A, ( A/nA ) n ≥ , B, C ; . . . (cid:1) be the corresponding L ∗ abcq -structure asin Section 4.1. In this situation we have: Remark . (1) The L ∗ abcq -structure M and its L ∗ abc -reduct ( A, B, C ) are bi-interpretable.(2) The collection of the sorts A and A/nA ( n ≥
0) is fully stably embedded in M (by the QEresult in the previous section), and the full structure induced on it is bi-interpretable with A .(3) Similarly, the sort C is fully stably embedded in M . Lemma 4.5. M is NIP if and only if both the L ∗ a -structure A and the L ∗ c -structure C are NIP.Proof. The forward direction is clear. Suppose A and C are NIP. To show the M is NIP we mayassume that it is a monster model of its theory. Adding a function symbol for a right-inverse of ν to thelanguage L ∗ abcq , we obtain a structure that is bi-interpretable with a two-sorted structure consistingof two sorts given by A and C with their full induced structure; this implies that M is NIP, as areduct of a NIP structure. (cid:3) Theorem 4.6. M is distal if and only if both A and C are distal.Proof. The forward implication is immediate by Lemma 1.15 and Remark 4.4; we prove the converse.Suppose A and C are distal; again, we may assume that M is a monster model of its theory, and byLemma 4.5, M is NIP. Assume towards contradiction that M is not distal; then by Remark 4.4(1),its L ∗ abc -reduct is also not distal, and hence satisfies condition (3) in Corollary 1.11. Thus, also usingRemark 1.13, we obtain a partitioned L ∗ abc -formula ϕ ( x ; y ), where | x | = 1, as well as an indiscerniblesequence ( b i ) i ∈ Q ∞ of the same sort as x and some tuple d of the same sort as the multivariable y suchthat ( b i ) i ∈ Q ∞ \{ } is d -indiscernible and M | = ϕ ( b i ; d ) ⇐⇒ i = 0. By assumption, Remark 4.4 andLemma 1.16, the variable x is necessarily of sort B . We say that a tuple is contained in d if all itscomponents appear as components of d .It is easy to see from the QE (Corollary 4.3) that the formula ϕ ( x ; d ) is equivalent to a positiveboolean combination of formulas of the form:(1) ψ ∗ (cid:0) ν ( t ( x, b ′ )) , . . . , ν ( t m ( x, b ′ )) , c (cid:1) where b ′ is a tuple of sort B , c is a tuple of sort C , bothcontained in d , the t k are L b -terms, and ψ ∗ is an L ∗ c -formula.(2) θ ∗ (cid:0) a, ρ n ( t ( x, b ′ )) , . . . , ρ n m ( t m ( x, b ′ )) (cid:1) where a is a tuple in A , b ′ is a tuple in B , bothcontained in d , the t k are L b -terms, n k ∈ N , and θ ∗ is an L ∗ aq -formula, where L ∗ aq = L ∗ a ∪ { π , π , . . . } . By Remarks 1.12 and 1.13, it is enough to show that ϕ ( x ; y ) cannot be of any of these forms. Belowwe let i , j range over Q ∞ and k over { , . . . , m } .Suppose first that (1) holds. As ν is a group morphism, ψ ∗ (cid:0) ν ( t ( x, b ′ )) , . . . , ν ( t m ( x, b ′ )) , c (cid:1) is equivalentto a formula of the form ψ ∗ (cid:0) ν ( x ) , c ′ (cid:1) where c ′ is a d -definable tuple of sort C and ψ ∗ is an L ∗ c -formula.By choice of ( b i ), the sequence (cid:0) ν ( b i ) (cid:1) in C is indiscernible, (cid:0) ν ( b i ) (cid:1) i =0 is c ′ -indiscernible, and M | = ψ ∗ ( ν ( b i ) , c ′ ) ⇐⇒ M | = ϕ ( b i , d ) ⇐⇒ i = 0 . This contradicts distality of the structure induced on C .Now suppose that we are in case (2). We may assume that for each k we have r k ∈ Z and a d -definable b ′ k ∈ B with t k ( b i , b ′ ) = r k b i − b ′ k for each i . Set B n k = ν − ( n k C ). By Case (1) applied tothe L c -formulas defining n k C and its complement, the truth value of the condition “ r k b i − b ′ k ∈ B n k ”doesn’t depend on i . If r k b i − b ′ k / ∈ B n k for some/all i , then ρ n k ( r k b i − b ′ k ) = 0 = ρ n k (0 b ) for all i bydefinition. Thus, replacing the term t k by 0 b , we still have M | = θ ∗ (cid:0) a, ρ n ( t ( b i , b ′ )) , . . . , ρ n m ( t m ( b i , b ′ )) (cid:1) ⇐⇒ i = 0 . Hence we may assume that r k b i − b ′ k ∈ B n k for all i . Repeating this argument for each k one by one,we may reduce to the case that r k b i − b ′ k ∈ B n k for all i , k . As B n k is a subgroup of B , we have r k b i − r k b j = ( r k b i − b ′ k ) − ( r k b j − b ′ k ) ∈ B n k for all i , j .Let b ki := r k b i − r k b ∞ ∈ B n k and b k := b ′ k − r k b ∞ . Note that b ki − b k = r k b i − r k b ∞ − ( b ′ k − r k b ∞ ) = r k b i − b ′ k ∈ B n k and hence b k ∈ B n k . As ρ n k restricts to a group morphism B n k → A/n k A , we have ρ n k ( r k b i − b ′ k ) = ρ n k ( b ki − b k ) = ρ n k ( b ki ) − ρ n k ( b k ) for all i .Let β i := ( β i , . . . , β mi ) and β := ( β , . . . , β m ) where β ki := ρ n k ( b ki ), β k := ρ n k ( b k ), and let x , . . . , x m be distinct variables with x k of sort A/n k A . Consider the L ∗ aq -formula θ ∗ ( x , . . . , x m , a, β ) := θ ∗ ( a, x − β , . . . , x m − β m ) . We then have: • ( β i ) i ∈ Q is indiscernible (by construction, as ( b i ) i ∈ Q is b ∞ -indiscernible), • ( β i ) i ∈ Q \{ } is aγ -indiscernible (by construction, as ( b i ) i ∈ Q \{ } is ab ∞ b ′ . . . b ′ m -indiscernible),and, unwinding, for every i ∈ Q , in M we have | = θ ∗ ( β i , a, β ) ⇐⇒ | = θ ∗ (cid:0) a, β i − β , . . . , β mi − β m (cid:1) ⇐⇒ | = θ ∗ (cid:0) a, ρ n ( b i ) − ρ n ( b ) , . . . , ρ n m ( b mi ) − ρ n m ( b m ) (cid:1) ⇐⇒ | = θ ∗ (cid:0) a, ρ n ( r b i − b ′ ) , . . . , ρ n m ( r m b i − b ′ m ) (cid:1) ⇐⇒ i = 0 . This contradicts distality of the L ∗ aq -structure A . (cid:3) Remark.
In this subsection we assumed that the L ∗ ac -structure ( A, C, R , R , . . . ) expanding the L ac -structure ( A, C ) is obtained by combining separate expansions of the L a -structure A and of the L c -structure C . Let now ( A, C ) ◦ be an arbitrary expansion of ( A, C ), and denote its language by L ◦ ac and the corresponding L ◦ abcq -structure by M ◦ . A straightforward adaption of the proofs shows thatLemma 4.5 and Theorem 4.6 remain true: M ◦ is NIP (distal) iff ( A, C ) ◦ is NIP (distal, respectively). ISTALITY IN VALUED FIELDS AND RELATED STRUCTURES 29
A variant for abelian monoids.
For later use, we now consider a slight variant of Corol-lary 4.3 for abelian groups augmented by absorbing elements. Let ( A, , +) be an abelian monoid.An element ∞ of A is said to be absorbing if ∞ + a = ∞ for all a ∈ A . (Clearly there is at most oneabsorbing element.) For example, if R is a commutative ring, then ( R, , · ) is an abelian monoid withabsorbing element 0. If A is an abelian group and ∞ / ∈ A is a new element, then A ∞ := A ∪ {∞} with the group operation + of A extended to a binary operation on A ∞ such that a + ∞ = ∞ + a = ∞ for all a ∈ A ∞ is an abelian monoid with absorbing element ∞ . In this case we also extend a
7→ − a : A → A to amap A ∞ → A ∞ by setting −∞ := ∞ . Every morphism f : A → B of abelian groups extends uniquelyto a monoid morphism f ∞ : A ∞ → B ∞ . Here is a special case of this construction: Notation.
Given a commutative ring R and a subgroup G of the multiplicative group R × of unitsof R we let R/G := ( R × /G ) ∞ . In this case we always denote the absorbing element of R/G by 0,so the residue morphism R × → R × /G extends to a surjective monoid morphism R → R/G whichmaps 0 ∈ R to 0 ∈ R/G .Let now 0 → A ι −−→ B ν −−→ C → • L ac = { a , + a , − a , ∞ a , c , + c , − c , ∞ c } , the language of the pair ( A ∞ , C ∞ ); • L b = { b , + b , − b , ∞ b } , the language of B ∞ ; • L abc = L ac ∪ L b ∪ { ι ∞ , ν ∞ } , the language of the three-sorted structure ( A ∞ , B ∞ , C ∞ ).We denote the extension of π n : A → A/nA to a morphism A ∞ → ( A/nA ) ∞ also by π n , and nowintroduce ρ n : B ∞ → ( A/nA ) ∞ by defining ρ n ( b ) ∈ A/nA for b ∈ ν − ( nC ) as before and declaring ρ n ( b ) := ∞ for b ∈ B ∞ \ ν − ( nC ). Thus ρ : B ∞ → A ∞ agrees with the inverse of ι on ι ( A ) and isconstant ∞ on B ∞ \ ι ( A ). We let L abcq = L abc ∪ { ρ , ρ , . . . , π , π , . . . } , L acq = L ac ∪ { π , π , . . . } , and we let T ∞ abcq be the theory of all L abcq -structures arising from a pure exact sequence of abeliangroups as above. The L abcq -terms of the form ρ n (cid:0) t ( x b ) (cid:1) or ν (cid:0) t ( x b ) (cid:1) , for a term t ( x b ) in the sublan-guage { b , + b , − b } of L b , are called special . Proposition 4.7. In T ∞ abcq every L abc -formula φ ( x a , x b , x c ) is equivalent to a formula φ acq (cid:0) x a , σ ( x b ) , . . . , σ m ( x b ) , x c (cid:1) where the σ i are special terms and φ acq is a suitable L acq -formula. Mutatis mutandis, the proof of this proposition is similar to that of Theorem 4.2. (Main change: B is the subgroup of B generated by those entries of b which do not equal ∞ , and similarly for B ′ .)Next, let L ∗ ac be the language of an expansion ( A ∞ , C ∞ , R , R , . . . ) of the L ac -structure ( A ∞ , C ∞ ),let L ∗ abc = L abc ∪ L ∗ ac be the language of ( A ∞ , B ∞ , C ∞ , R , R , . . . ), and L ∗ acq = L acq ∪ L ∗ ac . As in theproof of Corollary 4.3, the preceding proposition implies: Corollary 4.8. In T ∞ abcq every L ∗ abc -formula φ ∗ ( x a , x b , x c ) is equivalent to a formula φ ∗ acq (cid:0) x a , σ ( x b ) , . . . , σ m ( x b ) , x c (cid:1) where the σ i are special terms and φ ∗ acq is a suitable L ∗ acq -formula. Remark . In the previous corollary one may assume that no special terms of the form ρ (cid:0) t ( x b ) (cid:1) appear among the σ j . Since ρ n ( b − b ′ ) = ρ n (cid:0) b + ( n − b ′ (cid:1) for n ≥ b, b ′ ∈ B ∞ , we canalso arrange that the terms ρ n (cid:0) t ( x b ) (cid:1) , n ≥ σ j do not involve the functionsymbol − b . Moreover, since ν is a group morphism on its proper domain of definition, we can achievethat none of the terms of the form ν (cid:0) t ( x b ) (cid:1) appearing as some σ j involve − b .4.4. Weakly pure exact sequences.
Consider a sequence(4.1) 0 → A ι −−→ B ν −−→ C → ι is injective, ν is surjective, and ker ν ⊆ im ι , and let δ := ν ◦ ι : A → C . Note that with ν denoting the composition of ν with the natural surjection c c := c + im δ : C → C := C/ im δ, we obtain a short exact sequence 0 → A ι −−→ B ν −−→ C → , which we call the short exact sequence associated to our given sequence (4.1). Lemma 4.10.
Suppose the short exact sequence associated to (4.1) as well as the short exact sequence → ker A ⊆ −−→ A δ −−→ im δ → both split. Then with A := ker δ , B := im δ , and C := coker δ = C ,we have a commutative diagram / / A ι / / f A ∼ = (cid:15) (cid:15) B ν / / f B ∼ = (cid:15) (cid:15) C / / f C ∼ = (cid:15) (cid:15) / / A ⊕ B / / A ⊕ B ⊕ C / / B ⊕ C / / where the second arrow on the bottom row is the natural inclusion and the third arrow the naturalprojection.Proof. Take group morphisms s : B → A and t : C → B such that δ ◦ s = id B and ν ◦ t = id C .Since ν induces an isomorphism B/ im ι → C/ im δ , we have g ( b ) := b − t ( ν ( b )) ∈ im ι . One checksthat f A , f B , f C defined by f A ( a ) = (cid:0) a − s ( δ ( a )) (cid:1) + δ ( a ) , f B ( b ) = f A (cid:0) ι − ( g ( b )) (cid:1) + ν ( b ) , f C ( c ) = (cid:0) c − ν ( t ( c )) (cid:1) + c for a ∈ A , b ∈ B , c ∈ C have the required properties. (cid:3) We say that (4.1) is weakly pure exact if im ι is a pure subgroup of B and ker ν is a pure subgroupof im ι . Thus every pure short exact sequence is weakly pure exact; moreover, if (4.1) is weakly pureexact, then its associated short exact sequence is pure. Lemma 4.11.
Suppose C = C/ im δ and im δ are both torsion-free; then (4.1) is weakly pure exact.Proof. Let b ∈ B and n ≥ nb ∈ im ι . Take a ∈ A with ι ( a ) = nb ; then nν ( b ) = δ ( a ) ∈ im δ andhence ν ( b ) ∈ im δ (since C is torsion-free), so ν ( b ) = ν ( ι ( a ′ )) where a ′ ∈ A ; then b − ι ( a ′ ) ∈ ker ν ⊆ im ι and hence b ∈ im ι . This shows that im ι is a pure subgroup of B . Next, let a ∈ A and n ≥ nι ( a ) ∈ ker ν ; then nδ ( a ) = 0 and thus δ ( a ) = 0 (since im δ is torsion-free), that is, ι ( a ) ∈ ker ν .Therefore ker ν is a pure subgroup of im ι . (cid:3) A variant of Theorem 4.2 holds for weakly pure exact sequences. To make this precise, view eachweakly pure exact sequence (4.1) as an L abc -structure in the natural way. For each n let π n : A → A/nA be the natural surjection, define ρ n : B → A/nA according to the pure exact sequence associated
ISTALITY IN VALUED FIELDS AND RELATED STRUCTURES 31 to (4.1), and expand the L abc -structure (4.1) to a structure in the language L abcd := L abcq ∪ { δ } inthe natural way. Let T abcd be the theory of L abcd -structures( A, B, C, π , π , . . . , ρ , ρ , . . . , δ )which arise from a weakly pure exact sequence (4.1) in this way. Let L acd be the sublanguage L acq ∪{ δ } of L abcd . We then have: Theorem 4.12. In T abcd every L abc -formula φ ( x a , x b , x c ) is equivalent to a formula φ acd (cid:0) x a , σ ( x b ) , . . . , σ m ( x b ) , x c (cid:1) where the σ i are special terms and φ acd is a suitable L acd -formula.Proof. The proof is similar to the proof of Theorem 4.2 with the following modifications. Let M =( A, B, C, . . . ) and M ′ = ( A ′ , B ′ , C ′ , . . . ) be models of T abcd , and with the same notational conventionsas in the proof of Theorem 4.2, assume that we are given a , b , c in M and a ′ , b ′ , c ′ in M ′ such thatthe type of aσ ( b ) c in the L acd -reduct M acd = ( A, C, δ ) of M is the same as the type of a ′ σ ( b ′ ) c ′ inthe L acd -reduct M ′ acd = ( A ′ , C ′ , δ ′ ) of M ′ ; we need to show that then abc and a ′ b ′ c ′ have the sametype in M and in M ′ , respectively.Assuming, as we may, that M , M ′ are sufficiently saturated, we first show that a given isomor-phism M acd → M ′ acd extends to an isomorphism M → M ′ . For this, by Lemma 4.10 we may assumethat B = A ⊕ B ⊕ C , where A = A ⊕ B , C = B ⊕ C and ι and ν are the natural injection andthe natural projection; then δ ( a + b ) = b for a ∈ A , b ∈ B . Similarly with A ′ , B ′ , C ′ , etc. inplace of A , B , C , etc. If the isomorphisms f A : A → A ′ and f C : C → C ′ are compatible with δ , δ ′ ,then they have the form f A ( a + b ) = f ( a + b ) + g ( b ) f C ( b + c ) = (cid:0) g ( b ) + h ( c ) (cid:1) + h ( c ) ( a ∈ A , b ∈ B , c ∈ C )for group morphims f : A → A ′ , g : B → B ′ , h : C → B ′ , and h : C → C ′ . Then( a + b + c ) f ( a + b ) + (cid:0) g ( b ) + h ( c ) (cid:1) + h ( c ) ( a ∈ A , b ∈ B , c ∈ C )is a group isomorphism f B : B → B ′ , and ( f A , f B , f C ) is an isomorphism between the L abc -reductsof M and M ′ , which gives rise to an isomorphism M → M ′ of L abcd -structures as required.Therefore, as in the proof of Theorem 4.2 we can assume M = M ′ , a = a ′ , c = c ′ , σ ( b ) = σ ( b ′ ),and it suffices to show that there is an automorphism of M which is the identity on A and C andsends b to b ′ . Let B , B ′ and the group isomorphism f : B → B ′ be as in the proof of Theorem 4.2.Identifying C = coker δ with C in the natural way, the short exact sequence associated to our givenweakly pure exact sequence is0 → A = A ⊕ B ι −−→ B = A ⊕ B ⊕ C ν −−→ C → ι is the natural inclusion and ν the natural projection. In particular A = ker ν , and since ν ( b ) = ν ( f ( b )), we have f ( b ) − b ∈ A for each b ∈ B . Set A := B ∩ A = B ′ ∩ A, C := ν ( B ) = ν ( B ′ ) ⊆ C . As in the proof of Theorem 4.2 we see that we have a morphism h : C → A satisfying f ( b ) = b + h (cid:0) ν ( b ) (cid:1) for all b ∈ B .Now h is a partial morphism C → A , and thus also a partial morphism C → A since A is purein A . Extend h to a group morphism h : C → A ; then b b + h ( ν ( b )) defines an automorphismof B which, together with the identity on all other sorts, is an automorphism of M fixing A and C and mapping b to b ′ as desired. (cid:3) The theorem above yields a quantifier elimination result for arbitrary expansions of L acd just as inCorollary 4.3. We also have a version of Theorem 4.12 for abelian monoids, just like Proposition 4.7.To formulate this, redefine the languages L ac , L b , and L abc as in Section 4.3. Given a weakly pureexact sequence (4.1), denote the extension of π n : A → A/nA to a morphism A ∞ → ( A/nA ) ∞ by π n . We define ρ n : B ∞ → ( A/nA ) ∞ by defining ρ n ( b ) ∈ A/nA for b ∈ ν − ( nC ) = nB + ι ( A )as before and ρ n ( b ) := ∞ for b ∈ B ∞ \ (cid:0) nB + ι ( A ) (cid:1) . With L abcd , L acd as before, let T ∞ abcd be thetheory of all L abcd -structures which arise this way from a weakly pure exact sequence (4.1). ThenTheorem 4.12 goes through, with a similar proof, and implies a version with additional structure onthe L ac -structure ( A, C ) as in Corollary 4.8.4.5.
Connection to abelian structures.
In this subsection we generalize Theorems 4.2 and 4.12to pure exact sequences of abelian structures in the sense of Fisher [31]; for this we use a well-known generalization of the Baur-Monk quantifier simplification for modules to the case of abelianstructures. (This is not used later in the paper.) Recall that an abelian structure is an S -sortedstructure A = (cid:0) ( A s ); ( R i ) , ( f j ) (cid:1) where for each sort s ∈ S , among the primitives of A are distinguisheda constant 0 s ∈ A s , a unary function − s : A s → A s , and a binary function + s : A s × A s → A s , suchthat the (one-sorted) structure ( A s ; 0 s , − s , + s ) is an abelian group, and all other relations R i ⊆ A s × · · · × A s m are subgroups and all functions f j : A s × · · · × A s n → A s are group morphisms. Alsorecall that given a language L , the set of positive primitive (p.p.) L -formulas is the closure of the setof atomic L -formulas under conjunction and existential quantification. Let now L be the language ofan abelian structure A as above. For each p.p. L -formula φ ( x ), φ A = (cid:8) a ∈ A x : A | = φ ( a ) (cid:9) is a subgroup of A x . Given two p.p. L -formulas φ ( x ), ψ ( x ) where x is a single variable of sort s ∈ S ,we set dim ≥ nφ,ψ := ∃ x · · · ∃ x n ^ ≤ i ≤ n φ ( x i ) ∧ ^ ≤ i Proposition 4.13. Each L -formula is equivalent, in the theory of abelian L -structures, to a booleancombination of p.p. L -formulas and dimension sentences. We call a family of p.p. L -formulas fundamental (for A ) if every p.p. L -formula is equivalent in A toa conjunction of formulas φ ( t ( x )) where φ is fundamental and t is a tuple of L -terms. For example, itis well-known that if A is just an abelian group, then the formulas of the form n | x for n = 0 , , , . . . form a fundamental family [40, A.2.1].Let now A , B , C be abelian L -structures. Let ι : A → B be a morphism of L -structures. Recallthat ι is said to be an embedding if ι is injective and for each relation symbol R of L we have R A = ι − ( R B ); as a consequence, φ A ⊆ ι − ( φ B ) for each p.p. L -formula φ ( x ). We say that such anembedding ι is pure if φ A = ι − ( φ B ) for each p.p. L -formula φ ( x ). If A is a substructure of B andthe natural inclusion A → B is a pure embedding, then A is said to be a pure substructure of B . Amorphism ν : B → C is said to be a projection if ν is surjective and R C = ν ( R B ) for every relationsymbol R of L , and such a projection ν is said to be pure if φ C = ν ( φ B ) for each p.p. L -formula φ ( x ).In the following, we assume for notational simplicity that our language L is one-sorted, and we denotethe structures A , B , C by A , B , C , respectively. Lemma 4.14. Let → A ι −−→ B ν −−→ C → be a short exact sequence of morphisms of L -structures,where ι is an embedding and ν is a projection. Then ι is pure iff ν is pure. ISTALITY IN VALUED FIELDS AND RELATED STRUCTURES 33 Proof. First assume that ι is pure. Consider a p.p. L -formula φ ( x ) = ∃ x ′ V ni =1 R i (cid:0) t i ( x, x ′ ) (cid:1) , whereeach t i is a tuple of L -terms and each R i is a relation symbol of L or an equation between componentsof t i , and let c ∈ C x with C | = φ ( c ). Take c ′ ∈ C x ′ such that C | = V i R i (cid:0) t i ( c, c ′ ) (cid:1) , and let b , b ′ bepreimages of c , c ′ , respectively, under ν . Since ν is a projection, we can take appropriate tuples a i in A such that B | = V i R i (cid:0) t i ( b, b ′ ) + ι ( a i ) (cid:1) . Since ι is pure, there are a ∈ A x , a ′ ∈ A x ′ such that A | = V i R i (cid:0) t i ( a, a ′ ) + a i (cid:1) . This implies B | = V i R i (cid:0) t i ( b − ι ( a ) , b ′ − ι ( a ′ )) (cid:1) . So b − ι ( a ) is a preimageof c under ν satisfying φ . This shows that ν is pure.For the converse assume that ν is pure, and let a ∈ A x where ι ( a ) satisfies a p.p.-formula φ ( x ) asabove. So there is b ′ ∈ B x ′ such that B | = V i R i (cid:0) t i ( ι ( a ) , b ′ ) (cid:1) . Therefore C | = V i R i (cid:0) t i (0 , ν ( b ′ ) (cid:1) and byassumption there is a ′ ∈ A x ′ such that B | = V i R i (cid:0) t i (0 , b ′ − ι ( a ′ ) (cid:1) . This implies B | = V i R i (cid:0) t i ( ι ( a, a ′ )) (cid:1) .So A | = V i R i (cid:0) t i ( a, a ′ ) (cid:1) since ι is an embedding, and a satisfies φ . (cid:3) A short exact sequence 0 → A ι −−→ B ν −−→ C → L -structures where ι is a pureembedding and ν is a pure projection is called pure . Remark. If B is the direct sum of the abelian L -structures A and C (defined in the obvious way), thenthe resulting sequence A ι −→ B ν −→ C is pure exact. All pure exact sequences where A is |L| + -saturatedare of this form. Lemma 4.15. Let ν : B → C be a pure projection, φ ( x, x ′ ) be a p.p. L -formula, b ∈ B x , and c ′ ∈ C x ′ .Then the following are equivalent: (1) There is b ′ ∈ B x ′ such that B | = φ ( b, b ′ ) and ν ( b ′ ) = c ′ ; (2) B | = ∃ x ′ φ ( b, x ′ ) and C | = φ (cid:0) ν ( b ) , c ′ (cid:1) .Proof. The direction (1) ⇒ (2) is clear; we only use that ν is morphism. For the converse assume (2).Take b ′ ∈ B x ′ such that B | = φ ( b, b ′ ). Since ν is a pure projection, there are b ∈ B x and b ′ ∈ B x ′ such hat ν ( b ) = ν ( b ), ν ( b ′ ) = c ′ and B | = φ ( b , b ′ ). So B | = φ ( b − b , b ′ − b ′ ). By the last lemma, A := ker ν is (the underlying set of) a pure substructure of B . Since b − b ∈ A , purity gives an a ′ ∈ A x ′ such that B | = φ ( b − b , a ′ ). So we have B | = φ ( b, b ′ ) for b ′ = b ′ + a ′ . We see now that ν ( b ′ ) = c ′ ,and (1) holds. (cid:3) We now consider a sequence(4.2) 0 → A ι −−→ B ν −−→ C → L -structures. We let L a , L b , L c be pairwise disjoint copies of L (for A , B , C ,respectively), introduce a three-sorted language L abc = L a ∪ L b ∪ L c ∪ { ι, ν } , and view ( A, B, C ) asan L abc -structure in the natural way. This L abc -structure ( A, B, C ) is also abelian, hence Proposi-tion 4.13 applies to ( A, B, C ). (As a consequence, ( A, B, C ) is stable [40, A.1.13].) Let the multivari-ables x a , x b , x c be of sort A , B and C , respectively, and similarly with y in place of x .4.5.1. Pure exact sequences. In this subsection we assume that the sequence (4.2) is pure exact.Furthermore we consider an arbitrary expansion ( A, C ) ∗ of the reduct ( A, C ) of ( A, B, C ) with lan-guage L ∗ ac , and we let L ∗ abc := L ∗ ac ∪ L b . Unless mentioned otherwise, in the following, “equivalent”means “equivalent in the L ∗ abc -structure ( A, B, C )”. By an ac-existential quantification of an L ∗ abc -formula ψ we mean a formula of the form ∃ x a ∃ x c ψ , for some multivariables x a , x c . Lemma 4.16. Every p.p. L ∗ abc -formula φ ∗ abc ( x a , x b , x c ) is equivalent to an ac-existential quantificationof a formula φ b (cid:0) ι ( x a ) , x b (cid:1) ∧ φ ∗ ac (cid:0) x a , ν ( x b ) , x c (cid:1) , where φ b is a p.p. L b -formula and φ ∗ ac is a p.p. L ∗ ac -formula. Proof. Recall that each p.p. formula is equivalent to an existential quantification of a basic formula,i.e., a conjunction of atomic formulas. Since ν is a morphism of L -structures and ν ◦ ι = 0, everyterm ν ( t ) can be replaced by a sum of terms ν ( x b ). So every basic formula is equivalent to a formula ψ b (cid:0) ι ( t ) , x b (cid:1) ∧ ψ ∗ ac (cid:0) x a , ν ( x b ) , x c (cid:1) , where ψ b is a basic L b -formula, ψ ∗ ac is a basic L ∗ ac -formula, and t is atuple of L ∗ ac -terms in x a , ν ( x b ), and x c . We can replace t by existentially quantified multivariables x ′ a of sort A and add the equations x ′ a = t . Thus we may assume that our p.p. formula has the form ∃ y b (cid:0) ψ b ( ι ( x a ) , x b , y b ) ∧ ψ ∗ ac ( x a , ν ( x b ) , ν ( y b ) , x c ) (cid:1) . This formula in turn is equivalent to ∃ y c (cid:16) θ ( x a , x b , y c ) ∧ ψ ∗ ac ( x a , ν ( x b ) , y c , x c ) (cid:17) where θ := ∃ y b (cid:0) ψ b ( ι ( x a ) , x b , y b ) ∧ ν ( y b ) = y c (cid:1) ,and by Lemma 4.15, θ is equivalent to ∃ y b ψ b (cid:0) ι ( x a ) , x b , y b (cid:1) ∧ ψ c (cid:0) , ν ( x b ) , y c (cid:1) , where ψ c is the L c -copy of ψ b . (cid:3) For a p.p. L -formula φ ( x ) let A φ be the quotient group A x /φ A and π φ : A x → A φ be the naturalsurjection. Define the map ρ φ : B x → A φ on ν − ( φ C ) as the composition of the maps ν − ( φ C ) = φ B + ι ( A x ) → (cid:0) φ B + ι ( A x ) (cid:1) /φ B ∼ −−→ ι ( A x ) / (cid:0) φ B ∩ ι ( A x ) (cid:1) ∼ −−→ A φ , and identically zero outside ν − ( φ C ). The following lemma is clear from the definitions. Lemma 4.17. Let a ∈ A x , b ∈ B x . Then ι ( a ) + b ∈ φ B iff π φ ( a ) + ρ φ ( b ) = 0 and ν ( b ) ∈ φ C . We now fix a family of p.p. L -formulas which is fundamental for B . We expand ( A, C ) ∗ by a newsort A φ together with the corresponding projection map π φ , for every fundamental L -formula φ . Let L ∗ acq := L ∗ ac ∪ { π φ : φ fundamental } be the language of this expansion. We call terms of the form ρ φ (cid:0) t ( x b ) (cid:1) or ν ( x b ) for a fundamental φ and a tuple t of L b -terms special . Lemma 4.18. Every p.p. L ∗ abc -formula φ ∗ abc ( x a , x b , x c ) is equivalent to a formula φ ∗ acq (cid:0) x a , σ ( x b ) , . . . , σ m ( x b ) , x c (cid:1) where the σ i are special terms and φ ∗ acq is a suitable p.p. L ∗ acq -formula.Proof. By Lemma 4.16 it suffices to prove this for formulas φ ∗ abc ( x a , x b ) = φ b (cid:0) t b ( ι ( x a ) , x b ) (cid:1) where φ b is fundamental and t b is a tuple of L b -terms. We may arrange that t b (cid:0) ι ( x a ) , x b (cid:1) = ι (cid:0) r a ( x a ) (cid:1) + s b ( x b ) for a tuple r a of L a -terms and a tuple s b of L b -terms. Let φ c and s c be the L c -copies of φ b and s b , respectively; then by Lemma 4.17, φ ∗ abc ( x a , x b ) is equivalent to π φ (cid:0) r a ( x a ) (cid:1) + ρ φ (cid:0) s b ( x b ) (cid:1) =0 ∧ φ c (cid:0) s c ( ν ( x b )) (cid:1) . (cid:3) We now obtain versions of Theorem 4.2 and Corollary 4.3 for our pure exact sequence (4.2): Theorem 4.19. Every L abc -formula φ ( x a , x b , x c ) is equivalent to a formula φ acq (cid:0) x a , σ ( x b ) , . . . , σ m ( x b ) , x c (cid:1) where the σ i are special terms and φ acq is a suitable L acq -formula.Proof. By Proposition 4.13, every L abc -formula is equivalent to a boolean combination of p.p. L abc -formulas. Now apply Lemma 4.18 to the trivial expansion of ( A, C ). (cid:3) ISTALITY IN VALUED FIELDS AND RELATED STRUCTURES 35 Corollary 4.20. Every L ∗ abc -formula φ ∗ ( x a , x b , x c ) is equivalent to a formula φ ∗ acq (cid:0) x a , σ ( x b ) , . . . , σ m ( x b ) , x c (cid:1) where the σ i are special terms and φ ∗ acq a suitable L ∗ acq -formula.Proof. This follows from the theorem like Corollary 4.3 follows from Theorem 4.2. (cid:3) Weakly pure exact sequences. In this subsection we assume that (4.2) is weakly pure exact ,i.e., ι a pure embedding, ν a pure projection, and im ι ⊆ ker ν . As in Section 4.4 let δ := ν ◦ ι . Thepair ( A, C ) is then an abelian L acd -structure, where L acd = L ac ∪ { δ } . Let ( A, C ) ∗ be an expansionof ( A, C ) with language L ∗ acd , let L ∗ abcd := L ∗ acd ∪ L b . “Equivalent” now means “equivalent in the L ∗ abcd -structure ( A, B, C )”, and we define ac-existential quantifications as in the previous subsection.We have then the following generalization of Lemma 4.16: Lemma 4.21. Every p.p. L ∗ abcd -formula φ ∗ abcd ( x a , x b , x c ) is equivalent to an ac-existential quantifi-cation of a formula φ b (cid:0) ι ( x a ) , x b (cid:1) ∧ φ ∗ acd (cid:0) x a , ν ( x b ) , x c (cid:1) , where φ b is a p.p. L b -formula and φ ∗ acd is a p.p. L ∗ acd -formula.Proof. The proof is the same as the proof of Lemma 4.16, except that terms ν ( ι ( t )) are not replacedby 0 but by δ ( t ). Note that we use here, in Lemma 4.15, that ν is a pure projection. (cid:3) Let C := coker δ = C/ im δ equipped with its induced structure under the natural surjection c c : C → C . This surjection c c is a pure projection; composition with ν yields a pure projec-tion ν : B → C as in Section 4.4. The natural inclusion ker δ → A is a pure embedding. Moreover,ker ν = A , and the short exact sequence0 → A ι −−→ B ν −−→ ν → L -structures associated to (4.2) is pure exact. We define for every p.p. L -formula φ ( x )the map ρ φ : B x → A φ = A x /φ A as in the last subsection but according to the pure exact sequenceassociated to (4.2) displayed above. Lemma 4.17 then becomes: Lemma 4.22. Let a ∈ A x , b ∈ B x ; then ι ( a ) + b ∈ φ B iff π φ ( a ) + ρ φ ( b ) = 0 and δ ( a ) + ν ( b ) ∈ φ C .Proof. The direction from left to right is clear since ι ( a ) + b ∈ φ B implies ν ( ι ( a ) + b ) ∈ φ C . Theconverse follows from Lemma 4.17 since δ ( a ) + ν ( b ) ∈ φ C implies ν ( b ) ∈ φ C . (cid:3) As in the last subsection we fix now a family of p.p. L -formulas which is fundamental for B andexpand ( A, C ) ∗ by the new sorts A φ for every fundamental φ together with the projection map π φ .Let L ∗ acdq be the language of the resulting expansion. Lemma 4.18 is now: Lemma 4.23. Every p.p. L ∗ abcd -formula φ ∗ abcd ( x a , x b , x c ) is equivalent to a formula φ ∗ acdq (cid:0) x a , σ ( x b ) , . . . , σ m ( x b ) , x c (cid:1) where the σ i are special terms and φ ∗ acdq is a suitable p.p. L ∗ acdq -formula.Proof. As the proof Lemma 4.18, except that φ b ( t b ( ι ( x a ) , x b )) is equivalent to π φ (cid:0) r a ( x a ) (cid:1) + ρ φ (cid:0) s b ( x b ) (cid:1) = 0 ∧ φ c (cid:0) δ ( r a ( x a )) + s c ( ν ( x b )) (cid:1) (cid:3) . As in the last subsection we can conclude: Corollary 4.24. Every L ∗ abcd -formula φ ∗ ( x a , x b , x c ) is equivalent to a formula φ ∗ acdq (cid:0) x a , σ ( x b ) , . . . , σ m ( x b ) , x c (cid:1) where the σ i are special terms and φ ∗ acdq a suitable L ∗ acdq -formula. Remarks. (1) There is always a fundamental family of p.p. L -formulas, namely the set of all p.p. L -formulas.So, by the previous corollary and following the proofs of Lemma 4.5 and Theorem 4.6, wesee that a weakly pure exact sequence ( A, B, C ) of abelian L -structures with an expansion( A, C ) ∗ of ( A, C, δ ) is NIP (or distal) if and only if ( A, C ) ∗ is NIP (or distal).(2) If ( A, C, δ ) comes from a weakly pure exact sequence, then δ : A → im δ is a pure projectionand the natural inclusion im δ → C a pure embedding. The converse is may be true, but weknow it only if ker δ is a direct summand of A or im δ is a direct summand of C .5. Eliminating Field Quantifiers in Henselian Valued Fields In this section we discuss two frameworks for elimination of field quantifiers in henselian valued fieldsof characteristic zero construed as multi-sorted structures. The first one is the familiar RV (leadingterm) setting, for which we use [32] as our reference. Here the additional sorts are quotients ofthe multiplicative group of the underlying field by groups of higher 1-units. (See Sections 5.1–5.3.)In our second context we instead use, besides the value group, certain imaginary sorts obtainedfrom quotient rings of the valuation ring, and employ the results of Section 4 to prove the relevantelimination theorems. In the equicharacteristic zero case, which we treat first, this setting simplifieseven more, to quotients of the multiplicative group of the residue field; see Section 5.4 below. Each ofthese various settings has advantages that make it more convenient for some tasks rather than others;in this spirit, the elimination theorems from the present section are applied in combination to proveour main theorem in the next section.5.1. Quantifier elimination in henselian valued fields. Throughout this section we fix a valuedfield K of characteristic zero. We let v : K × → Γ = v ( K × ) be the valuation of K , and O its valuationring. As in Section 4.3 we consider the abelian monoid Γ ∞ := Γ ∪{∞} with absorbing element ∞ / ∈ Γ,and extend the ordering of Γ to a total ordering on Γ ∞ with γ < ∞ for all γ ∈ Γ; as usual we denotethe extension of v to a monoid morphism K → Γ ∞ also by v . Let γ , δ range over Γ ≥ . Let m δ := { x ∈ K : vx > δ } , so m δ is an ideal of O with m γ ⊆ m δ if γ ≥ δ . The maximal ideal of O is m := m , and its residuefield is k := O / m . Let also RV δ := K/ (1 + m δ ) , RV × δ := RV δ \{ } , with residue morphism rv δ : K → RV δ . Thus for a ∈ K × we have rv δ ( a ) = a (1 + m δ ) ∈ RV × δ , and rv δ sends 0 ∈ K to the absorbing element 0 of RV δ . We writeRV := RV = K/ (1 + m ) , rv := rv . For a ∈ O \ m , the element a (1 + m ) of RV × only depends on the coset a + m , and we hence obtain agroup embedding k × → RV × which sends the element a + m of k × to a (1 + m ) ∈ RV × . Together withthe group morphism v rv : RV × → Γ induced by the valuation v : K × → Γ, this group embedding fitsinto a pure short exact sequence 1 → k × → RV × v rv −−−→ Γ → . We denote the extension of v rv to a morphism RV → Γ ∞ of monoids by the same symbol. Besides theinduced multiplication, RV δ also inherits a partially defined addition from K via the ternary relation(5.1) ⊕ δ ( r, s, t ) ⇐⇒ ∃ x, y, z ∈ K (cid:0) r = rv δ ( x ) ∧ s = rv δ ( y ) ∧ t = rv δ ( z ) ∧ x + y = z (cid:1) . For γ ≥ δ we also have a natural surjective monoid morphism rv γ → δ : RV γ → RV δ . ISTALITY IN VALUED FIELDS AND RELATED STRUCTURES 37 It turns out that for what follows, not all of RV δ will be needed. Therefore, from now on we let γ and δ (possibly with decorations) range over { } if char k = 0, and over the set v ( p N ) := (cid:8) v ( p n ) : n ≥ (cid:9) if char k = p > 0. We introduce a many-sorted structure K whose sorts are K and the sets RV δ ,equipped with the following primitives:(K1) the ring primitives on K ;(K2) on each sort RV δ , the monoid primitives and the partial addition relation ⊕ δ defined above;(K3) for each δ , the map rv δ : K → RV δ ; and(K4) for each γ ≥ δ , the maps rv γ → δ : RV γ → RV δ .We also denote by RV ∗ the structure with underlying sorts RV δ and primitives listed under (K2)and (K4) above, with associated language L RV ∗ . Remark . The relation v rv ( x ) ≤ v rv ( y ) on RV is definable in RV ∗ [32, Proposition 2.8(1)]. Namely, v rv ( x ) ≤ ⇐⇒ ¬ ⊕ ( x, , , v rv ( x ) = 0 ⇐⇒ v rv ( x ) ≤ ∧ ∃ y (cid:0) x · y = 1 ∧ v rv ( y ) ≤ (cid:1) and hence v rv ( x ) = v rv ( y ) ⇐⇒ ∃ z (cid:0) v rv ( z ) = 0 ∧ x = y · z (cid:1) , v rv ( x ) < v rv ( y ) ⇐⇒ x = 0 ∧ ⊕ ( x, y, x ) . Hence the multiplicative group ker v rv ∼ = k × is definable in RV ∗ . As a consequence the ordered abeliangroup Γ = v ( K × ) is interpretable in RV ∗ , and using ⊕ it follows that the field k is also interpretablein RV ∗ . Remark . Our valued field viewed as a structure ( K, O ) in the language of rings expanded by aunary predicate for the valuation ring O of K is bi-interpretable with K (regardless of the character-istic of k ). Hence ( K, O ) is distal, respectively has a distal expansion, iff K has the correspondingproperty, by Fact 1.14(1). Fact 5.3 (Flenner [32, Propositions 4.3 and 5.1]) . Suppose K is henselian. (1) Let S ⊆ K be A -definable in K , for some parameter set A in K . Then there are a , . . . , a m ∈ K ∩ acl( A ) and an acl( A ) -definable D ⊆ RV δ × · · · × RV δ m , for some δ , . . . , δ m , such that S = (cid:8) x ∈ K : (cid:0) rv δ ( x − a ) , . . . , rv δ m ( x − a m ) (cid:1) ∈ D (cid:9) ;(2) RV ∗ is fully stably embedded ( i.e., the structure on RV ∗ induced from K , with parameters, isprecisely the one described above ) . Fact 5.3 is uniform in K ; moreover, it continues to hold if we add arbitrary additional structureon RV ∗ ; see the discussion before [32, Proposition 4.3]. Remarks . (1) Among the primitives of RV ∗ we have the projections rv γ → δ ( γ ≥ δ ); thus in Fact 5.3 we mayassume that δ = · · · = δ m = δ , after possibly modifying D and taking δ := max { δ , . . . , δ m } .(2) Note that for any x ∈ K , y ∈ K × , we have rv δ ( x ) = rv δ ( y ) iff v ( x − y ) > vy + δ ; hence forany z ∈ K and x, y ∈ K \ { z } , rv δ ( x − z ) = rv δ ( y − z ) iff v ( x − y ) > v ( y − z ) + δ .5.2. The finitely ramified case. For later use we analyze the kernels of the group morphismsrv γ → δ : RV × γ → RV × δ ( γ ≥ δ ) . In the following well-known lemma and its corollary we assume that we have a generator π for themaximal ideal: π O = m . Lemma 5.5. Suppose n ≥ . Then the map ϕ : 1 + π n O → O /π O = k , ϕ (1 + π n a ) := a + π O for a ∈ O is a surjective group morphism from the multiplicative abelian group π n O to the additive abeliangroup k with kernel π n +1 O . Thus, as abelian groups, (1 + π n O ) / (1 + π n +1 O ) ∼ = k . We leave the proof of Lemma 5.5 to the reader; an easy induction on r based on this lemma yields: Corollary 5.6. Suppose k is finite. Then | (1 + π n O ) / (1 + π n + r O ) | = | k | r for each n ≥ and r ∈ N . We now obtain our desired result: Lemma 5.7. Suppose K is finitely ramified with finite residue field k = O / m of characteristic p .Then for each n , the kernel of the group morphism rv v ( p n +1 ) → v ( p n ) : RV × v ( p n +1 ) → RV × v ( p n ) is finite.Proof. Take π ∈ O with m = π O ; then p = π e u where e ∈ N , e ≥ u ∈ O × . By Corollary 5.6,(1 + p n m ) / (1 + p n +1 m ) = (1 + π en +1 O ) / (1 + π ( en +1)+ e O )is finite, as required. (cid:3) We also need additive versions of the results above. In the following lemma and its corollary, we againassume that π satisfies π O = m : Lemma 5.8. The map π n a a + π O : π n O → O /π O = k is a surjective group morphism from the additive abelian group π n O to the additive abelian group k with kernel π n +1 O . Thus π n O /π n +1 O ∼ = k . Corollary 5.9. Suppose k is finite. Then | π n O /π n + r O| = | k | r for each r ∈ N . Now given a prime p and some n we let R p n := O /p n m (so R p = k ). In the same way as Corollary 5.6gave rise to Lemma 5.7, from the previous corollary we obtain: Lemma 5.10. Suppose K is finitely ramified with finite residue field of characteristic p . Then foreach n , the kernel of the natural surjective group morphism R p n +1 → R p n is finite. ( Hence R p n isfinite for each n . )5.3. NIP for RV ∗ . In this subsection K is henselian, and the structure K and its reduct RV ∗ areas introduced in Section 5.1. We allow RV ∗ to be equipped with additional structure, and equip itsexpansion K with the corresponding additional structure. Recall that then, by part (2) of Fact 5.3and the remark following it, RV ∗ is fully stably embedded in K . As a warm-up to the proof ofProposition 6.1 below, we show a version of Fact 2.17: Proposition 5.11. Suppose k is finite or of characteristic zero. Then K is NIP if and only if K isfinitely ramified and RV ∗ is NIP. Here the forward direction is obvious by Remark 5.2, Fact 2.18, and the fact that NIP is preservedunder reducts. The proof of the converse relies on an analysis of indiscernible sequences in valuedfields, with the distinction of cases similar to [15] or [10, Section 7.2]. (A similar case distinction isat the heart of the proof of Proposition 6.1.) Given a linearly ordered set I we let I ∞ := I ∪ {∞} where ∞ is a new element, equipped with the extension of the ordering ≤ of I to the linear orderingon I ∞ , also denoted by ≤ , such that i < ∞ for all i ∈ I . Recall that I ∗ denotes the set I equippedwith the reversed ordering ≥ . In the two lemmas and their corollary below we let ( a i ) i ∈ I be anindiscernible sequence of singletons of the field sort in K where I does not have a largest or smallestelement. For the first lemma see [9]. (Also compare with Lemma 2.11 above.) Lemma 5.12. Exactly one of the following cases occurs: (1) v ( a i − a j ) < v ( a j − a k ) for all i < j < k in I ( we say that ( a i ) is pseudocauchy) ; ISTALITY IN VALUED FIELDS AND RELATED STRUCTURES 39 (2) v ( a i − a j ) > v ( a j − a k ) for all i < j < k in I ( so the sequence ( a i ) i ∈ I ∗ is pseudocauchy ) ; or (3) v ( a i − a j ) = v ( a j − a k ) for all i < j < k in I ( we refer to such a sequence ( a i ) as a fan) . Note that if ( a i ) i ∈ I is pseudocauchy and a ∞ ∈ K is such that ( a i ) i ∈ I ∞ is indiscernible, then ( a i ) i ∈ I ∞ is also pseudocauchy, and similarly with “fan” in place of “pseudocauchy”. Lemma 5.13. Suppose ( a i ) i ∈ I is pseudocauchy, and let a ∞ ∈ K such that ( a i ) i ∈ I ∞ is indiscernible.Then the sequence i v ( a ∞ − a i ) is strictly increasing.Proof. Since ( a i ) i ∈ I ∞ remains pseudocauchy, if i < j are in I , then v ( a j − a i ) < v ( a ∞ − a j ) and so v ( a ∞ − a i ) = v (cid:0) a ∞ − a j + ( a j − a i ) (cid:1) = v ( a j − a i ) < v ( a ∞ − a j ). (cid:3) Corollary 5.14. Suppose K is finitely ramified. Then with ( a i ) i ∈ I and a ∞ as in Lemma 5.13, (5.2) v ( a ∞ − a i ) > v ( a ∞ − a j ) + δ for all δ and i > j in I .Proof. Assume that we have some δ such that v ( a ∞ − a i ) ≤ v ( a ∞ − a j ) + δ for some i > j in I ;then by δ -indiscernibility (as δ ∈ dcl( ∅ )), v ( a ∞ − a i ) ≤ v ( a ∞ − a j ) + δ for all i > j in I ,so for each j the interval (cid:2) v ( a ∞ − a j ) , v ( a ∞ − a j ) + δ (cid:3) in Γ is infinite, contradicting finite ramification. (cid:3) Now suppose k is finite or of characteristic zero, K is finitely ramified, and RV ∗ is NIP. To show that K is NIP we may assume that K is a monster model of its theory. Suppose K is not NIP. Then there is anindiscernible sequence ( a i ) i ∈ Z of elements of the field sort of K and a definable S ⊆ K such that i ∈ Z is even iff a i ∈ S . By Fact 5.3 and the remark following it we may choose b = ( b , . . . , b m ) ∈ K m ,some δ , as well as a definable subset D of RV mδ , such that for a ∈ K : a ∈ S ⇐⇒ (cid:0) rv δ ( a − b ) , . . . , rv δ ( a − b m ) (cid:1) ∈ D. By Lemma 5.12, one of the following three cases occurs. Case 1: ( a i ) i ∈ Z is pseudocauchy. Using saturation take some a ∞ ∈ K such that ( a i ) i ∈ Z ∞ is indis-cernible. Let i , j range over Z , and let k ∈ { , . . . , m } . Suppose first that v ( b k − a ∞ ) > v ( a ∞ − a j ) forall j . Using (5.2) we then obtain v ( b k − a ∞ ) > v ( a ∞ − a j ) + δ and hence rv δ ( b k − a j ) = rv δ ( a ∞ − a j ),for all j . Now suppose v ( b k − a ∞ ) ≤ v ( a ∞ − a j ) for some j ; then v ( b k − a ∞ ) + δ < v ( a ∞ − a i ) forall i > j , and hence rv δ ( b k − a i ) = rv δ ( b k − a ∞ ) for i > j . Permuting the components of b , we canthus arrange that we have some l ∈ { , . . . , m + 1 } and some j such that for i > j and k = 1 , . . . , m we have rv δ ( b k − a i ) = ( rv δ ( a ∞ − a i ) if k < l rv δ ( b k − a ∞ ) otherwise.Put r i := rv δ ( a i − a ∞ ) for i > j and s k := rv δ ( a ∞ − b k ) for k = l, . . . , m . The sequence ( r i ) i>j isindiscernible, and for i > j we have( r i , . . . , r i , s l , . . . , s m ) ∈ D ⇐⇒ i is even,in contradiction with RV ∗ being NIP. Case 2: ( a i ) i ∈ Z ∗ is pseudocauchy. Then we apply Case 1 to the sequence ( a − i ) i ∈ Z in place of ( a i ) i ∈ Z . Case 3: ( a i ) i ∈ Z is a fan. Note that then k necessarily is infinite, hence char k = 0 by hypothesis,so δ = 0. Let i , j range over Z and k over { , . . . , m } , and let γ be the common value of v ( a i − a j )for all i = j . Let c ∈ K and j be given; if γ < v ( c − a j ), then γ = v ( c − a i ) for all i = j , whereasif γ > v ( c − a j ) then v ( c − a i ) = v ( c − a j ) < γ for each i = j . Hence we can choose an even j such that for each k we either have γ > v ( b k − a i ) for all i ≥ j or γ = v ( b k − a i ) for all i ≥ j .Now if γ > v ( b k − a j ), then rv( b k − a i ) = rv( b k − a j ) for i > j , whereas if γ = v ( b k − a j ), thenrv( b k − a i ) = rv( b k − a j ) ⊕ rv( a j − a i ) for i > j . Hence by reindexing the components of b wecan arrange that we have some l ∈ { , . . . , m + 1 } such that with r i := rv( a i − a j ) for i > s k := rv( a j − b k ) for k = 1 , . . . , m , for i > j and k = 1 , . . . , m :rv( a i − b k ) = ( r i ⊕ s k if k < ls k otherwise.The sequence ( r i ) i>j is indiscernible, and for i > j we have( r i ⊕ s , . . . , r i ⊕ s l − , s l , . . . , s m ) ∈ D ⇐⇒ i is even,in contradiction with RV ∗ being NIP. (cid:3) A quantifier elimination in equicharacteristic zero. We use the quantifier eliminationresult for pure short exact sequences from Section 4 to prove a variant of the QE result of Flenner,already used earlier, in the equicharacteristic zero case. As above we extend the valuation v : K × → Γto a monoid morphism K → Γ ∞ , also denoted by v , with v (0) = ∞ . Recall that Γ ∞ = Γ ∪ {∞} where γ < ∞ for all γ ∈ Γ and γ + ∞ = ∞ + γ = ∞ for all γ ∈ Γ ∞ . We also extend the residuemorphism a res( a ) := a + m : O → k = O / m to K by setting res( a ) := 0 for a ∈ K \ O . In the rest of this subsection k has characteristic zero. We consider K as a three-sorted structure with sorts k , K , Γ ∞ in the language L rkg = L r ∪ L k ∪ L g ∪ { v, res } where L r = { r , r , + r , − r , · r } , L k = { k , k , + k , − k , · k } , L g = { g , + g , <, ∞} . For our quantifier elimination result we expand ( k , Γ ∞ ) by a new sort k / ( k × ) n for every n ≥ π n : k → k / ( k × ) n . Let L rgq = L r ∪ L g ∪ { π , π , . . . } be the language of this expansion.Define, for every n , a map res n : K → k / ( k × ) n in the following way: If v ( a ) / ∈ n Γ, set res n ( a ) := 0.Otherwise, let b be any element of K with nv ( b ) = v ( a ) and set res n ( a ) := π n res( a · b − n ). This doesnot depend on the choice of b since nv ( c ) = v ( a ) implies that b · c − has value 0, so is a unit in O andres( a · c − n ) = res( a · b − n ) · res( b · c − ) n . One verifies easily that the restriction of res n to v − ( n Γ) is agroup morphism v − ( n Γ) → k × / ( k × ) n . We identify k with k / ( k × ) in the natural way, so res = res .We also extend the multiplicative inverse function a a − : K × → K × to a function K → K bysetting 0 − := 0, and let L rkgq := L rkg ∪ { − , π , π , . . . , res , res , . . . } . Let the multivariables x r , x k , x g be of sort k , K , and Γ ∞ , respectively. We call L rkgq -terms of theform v (cid:0) p ( x k ) (cid:1) , res (cid:0) p ( x k ) q ( x k ) − (cid:1) or res n (cid:0) p ( x k ) (cid:1) (where n ≥ p , q with integercoefficients, special . We have the following analogue of Theorem 4.2: Theorem 5.15. In the theory of henselian valued fields with residue field of characteristic zero, viewedas L rkgq -structures in the natural way, every L rkg -formula φ ( x r , x k , x g ) is equivalent to a formula φ rgq (cid:0) x r , σ ( x k ) , . . . , σ m ( x k ) , x g (cid:1) where the σ i are special terms and φ rgq is a suitable L rgq -formula. ISTALITY IN VALUED FIELDS AND RELATED STRUCTURES 41 In the proof we make use of Flenner’s quantifier elimination theorem, already stated in Section 5.1above. For convenience let us slightly paraphrase this result, in the case of equicharacteristic zero.Recall that in this case the structure RV ∗ has a single new (interpretable) sortRV = K/ (1 + m ) , which comes equipped with the binary operation · rv which gives RV the structure of an abelian monoidand makes the natural projection rv : K → RV a monoid morphism. Note that 0 RV := rv(0) is anabsorbing element of RV and RV × := RV \ { RV } = K × / (1 + m ) is a group. The projection rv andthe valuation v : K → Γ ∞ also induce morphisms ι : k → RV and ν : RV → Γ ∞ of abelian monoids,which give rise to a pure short exact sequence(5.3) 1 → k × → RV × → Γ → L rv = L r ∪ L g ∪ {· rv , ι, ν } be the language of the structure ( k , RV , Γ ∞ ), and let L rkg , rv := L rkg ∪ { rv , · rv , ι, ν } = L r ∪ L k ∪ L g ∪ { v, res , rv , · rv , ι, ν } . Now Flenner’s result [32, Proposition 4.3] is: Fact 5.16. In the theory of henselian valued fields with residue field of characteristic zero, formulatedin the language L rkg , rv , every L rkg -formula φ ( x r , x k , x g ) is equivalent to a formula φ rv (cid:0) x r , rv( q ( x k )) , . . . , rv( q k ( x k )) , x g (cid:1) where the q i are polynomials with integer coefficients and φ rv is a suitable L rv -formula. Actually, Flenner’s result is a bit stronger, allowing variables ranging over the RV-sort; moreover,Fact 5.16 also works for arbitrary expansions of the L rv -structure ( k , RV , Γ ∞ ). (See the discussionpreceding [32, Proposition 4.3].)We now apply the material of Section 4.3 to the short exact sequence (5.3). Let φ ( x r , x k , x g ) be an L rkg -formula and take q , . . . , q k and φ rv as in Fact 5.16. Corollary 4.8 and Remark 4.9 applied to φ rv show that φ ( x r , x k , x g ) is equivalent to a formula φ rgq (cid:0) x r , σ ( x k ) , . . . , σ m ( x k ) , x g (cid:1) where the σ j are terms of the form ρ (cid:0) rv( q ( x k )) e · · · rv( q k ( x k )) e k (cid:1) ( e , . . . , e k ∈ Z )or ρ n (cid:0) rv( q ( x k )) e · · · rv( q k ( x k )) e k (cid:1) ( e , . . . , e k ∈ N , n ≥ ν (cid:0) rv( q ( x k )) e · · · rv( q k ( x k )) e k (cid:1) ( e , . . . , e k ∈ N ) , and φ rgq is a suitable L rgq -formula. Here the maps ρ n : RV → k / ( k × ) n are as defined in Section 4.3.Since rv is a monoid morphism, for each appropriate tuple a of the field sort and e , . . . , e k ∈ Z wehave rv( q ( a )) e · · · rv( q k ( a )) e k = rv (cid:0) p ( a ) q ( a ) − (cid:1) where p = Y e j ≥ q e j j and q = Y e j < q − e j j . We have ν ◦ rv = v . Recall that ρ n is identically zero outside ν − ( n Γ), hence ρ n ◦ rv = res n . Thuseach term σ j is special. This finishes the proof of Theorem 5.15. (cid:3) Remarks. (1) Suppose K rgq is equipped with additional structure, and we equip its expansion to an L rkgq -structure with the corresponding additional structure. The theorem above then remains truein this setting; this is shown just as in Corollary 4.8. As a consequence, K rgq is fully stablyembedded in the L rkgq -structure K , and the induced structure on K rgq is the given one.(2) Suppose now that k and Γ ∞ come equipped with additional structure, and the L rkgq -struc-ture K is expanded by these structures on its sorts k and Γ ∞ ; then the sorts k , Γ ∞ are fullystably embedded in K , with the induced structure on these sorts just the given ones.We finish this subsection with observing that the structure RV ∗ introduced in Section 5.1 is onlyostensibly richer than the structure ( k , RV , Γ ∞ ) of RV viewed as pure short exact sequence: Lemma 5.17. The L rv -structure ( k , RV , Γ ∞ ) and the L RV ∗ -structure RV ∗ are bi-interpretable. To see this note that the relation ⊕ = ⊕ on RV introduced in (5.1) is definable in ( k , RV , Γ ∞ ):for a, b, c ∈ RV × we have ⊕ ( a, b, c ) ⇐⇒ h ν ( a ) = ν ( b ) & ∃ y ∈ k (cid:0) ι ( y ) · rv a = b & ι (1 + y ) · rv a = c (cid:1)i ∨ h ν ( a ) > ν ( b ) & b = c i ∨ h ν ( b ) > ν ( a ) & a = c i . Conversely, Remark 5.1 shows that k , Γ ∞ and the morphisms ι , ν are interpretable in RV ∗ . Note thatin this lemma we may allow k and Γ ∞ to be equipped with additional structure, and RV ∗ with thecorresponding structure, that is, by all relations S ⊆ RV m where S ⊆ (ker v rv ) m = ( k × ) m is definablein k or S = v − ( v rv ( S )) where v rv ( S ) ⊆ Γ m is definable in Γ ∞ . Corollary 5.18. Suppose that k and Γ ∞ are equipped with additional structure; then K is NIP iffboth k and Γ ∞ are NIP.Proof. By Proposition 5.11 and the remark preceding the corollary, K is NIP iff ( k , RV , Γ ∞ ) is NIP,and by Lemma 4.5, ( k , RV , Γ ∞ ) is NIP iff k and Γ ∞ are NIP. (cid:3) A generalization. In this subsection we put the QE result for weakly pure exact sequences fromSection 4.4 to work by proving a version of Theorem 5.15 for henselian valued fields of characteristiczero with arbitrary residue field. Only Corollary 5.23 from this subsection is used later. Throughoutthis subsection we assume that K is henselian, and we let M , N range over N ≥ .Let R N be the ring O /N m , and extend the residue morphism x res N ( x ) := x + N m : O → R N to a map K → R N , also denoted by res N , by setting res N ( x ) := 0 for x ∈ K \ O . The valuation v : K → Γ ∞ induces a map v N : R N → Γ ∞ with v N ( r ) = ( v ( x ) if r = res N ( x ) = 0, ∞ if r = 0.Note that 0 ≤ v N ( r ) ≤ v ( N ) for r ∈ R N , r = 0. We have R = k . If char k = 0, then R N = k and v N ( R N ) = { , ∞} for all N . If char k = p > 0, then R M = R N if M and N are divisible by the samepowers of p . If M is a multiple of N , let res MN : R M → R N be the natural surjection; its kernel isres M ( N m ) = (cid:8) r ∈ R M : v N ( r ) > v ( N ) (cid:9) , ISTALITY IN VALUED FIELDS AND RELATED STRUCTURES 43 and v N (cid:0) res MN ( r ) (cid:1) = v M ( r ) for r ∈ R M with res MN ( r ) = 0. Let L rng = (cid:8) + N , · N , v N , res MN : N divides M (cid:9) ∪ L g be the language of the multi-sorted structure K rng = ( R , R , . . . , Γ ∞ ). The pair consisting of thefamily of rings ( R N ) and the family of morphisms (res MN ) N | M forms an inverse system; let lim ←− R N denote its inverse limit. The morphisms res N : O → R N induce a ring morphism O → lim ←− R N whosekernel is ˙ m := \ N N m = (cid:8) x ∈ K : v ( x ) > v ( N ) for every N (cid:9) , and hence induces an embedding ϕ : O / ˙ m → lim ←− R N . Clearly we have: Lemma 5.19. Suppose K rng is ℵ -saturated; then ϕ is an isomorphism. We now consider K as a many-sorted structure ( K, R , R , . . . , Γ ∞ ) in the language L rkng = L k ∪ L rng ∪ { v, res , res , . . . } . Lemma 5.20. Suppose n divides N , and for i = 1 , let x i ∈ K with v ( x i ) + 2 v ( n ) ≤ v ( N ) . Thenwith r i = res N ( x i ) , the following are equivalent: (1) x · x − ∈ ( K × ) n ; (2) r r n = r or r = r r n for some r ∈ R N .Proof. Suppose x · x − ∈ ( K × ) n , and say v ( x ) ≤ v ( x ); then x z n = x for some z ∈ O , so r r n = r for r = res N ( z ). Conversely, suppose r r n = r where r ∈ R N . Take y ∈ O with v ( x y n − x ) > v ( N );then v ( x x − y n − > v ( N ) − v ( x ) ≥ v ( n ) . Hensel’s Lemma (in the Newton formulation) applied to the polynomial x x − y n − X n ∈ O [ X ] yieldsan x ∈ K such that x x − y n − x n = 0, so x x − ∈ ( K × ) n . (cid:3) From Lemma 5.20 we see that for N , n as in the lemma, r ∼ nN r : ⇐⇒ ∃ s ( r s n = r ∨ r = r s n )defines an equivalence relation on the subset R nN = (cid:8) r ∈ R N : v N ( r ) + 2 v ( n ) ≤ v ( N ) (cid:9) of R N . For such N , n we introduce a new sort S nN := ( R nN / ∼ nN ) ∪ { } together with the map π nN : R N → S nN which agrees with the quotient map R nN → R nN / ∼ nN on R nN and is 0 on R N \ R nN . Let L rngq = L rng ∪ { π nN : n divides N } be the language of the expansion ( K rng , S nN ) of K rng . Note that ( K rng , S nN ) is interpretable in K rng .Finally, we define, for every n such that n divides N , the following map res nN : K → S nN : If there issome γ ∈ Γ such that 0 ≤ v ( x ) − nγ ≤ v ( N ) − v ( n ), choose y ∈ K with v ( y ) = γ and setres nN ( x ) = π nN (cid:0) res N ( x · y − n ) (cid:1) ;one verifies easily that this does not depend on the choice of γ and y . If there is no such γ ,set res nN ( x ) := 0. We view each henselian valued field of characteristic zero in the natural wayas an L krngq -structure where L rkngq := L rkng ∪ { res nN : n divides N } . Let the multivariables x r , x k , x g be of sort R , R , . . . , K , and Γ ∞ , respectively. We call L rkngq -termsof the form v (cid:0) p ( x k ) (cid:1) , res N (cid:0) p ( x k ) q ( x k ) − (cid:1) or res nN (cid:0) p ( x k ) (cid:1) (where n ≥ p , q withinteger coefficients, special . We then have the following theorem. Theorem 5.21. In the L rkngq -theory of characteristic zero henselian valued fields, every L rkng -for-mula φ ( x r , x k , x g ) is equivalent to a formula φ rngq (cid:0) x r , σ ( x k ) , . . . , σ m ( x k ) , x g (cid:1) where the σ i are special terms and φ rngq is a suitable L rngq -formula. For the proof of this theorem, suppose our valued field K (as always, of characteristic zero) is henselian.Let r , a , γ be finite tuples in K of the same sort as x r , x k , x g , respectively. Let σ , σ , . . . list all specialterms, and let σ ( a ) denote the tuple σ ( a ) , σ ( a ) , . . . . We have to show that the type of (cid:0) r, σ ( a ) , γ (cid:1) in K rng determines the type of ( r, a, γ ) in the L rkngq -structure K . For this we may assume that K isspecial of some suitable cardinality κ , e.g., κ = i ω ( ω ) (see [40, Theorem 10.4.2(c)]). The followingclaim is then clear (see [40, Theorems 10.4.4 and 10.4.5 (a)]: Claim 1. The type of (cid:0) r, σ ( a ) , γ (cid:1) in K rng determines the isomorphism type of (cid:0) K rng , r, σ ( a ) , γ (cid:1) . In the following we use the notation and terminology of [2, Section 3.4]. Let ∆ be the smallest convexsubgroup of Γ containing all v ( N ). Let ˙ v : K × → ˙Γ := Γ / ∆ be the coarsening of v by ∆, with residuefield ˙ K of characteristic zero, and let v : ˙ K × → ∆ be the corresponding specialization of v . Thevaluation ring of the valuation v on ˙ K is O ˙ K := O / ˙ m , where˙ m := (cid:8) x ∈ K : v ( x ) > v ( N ) for all N (cid:9) is the maximal ideal of the valuation ring˙ O := (cid:8) x ∈ K : v ( x ) > − v ( N ) for some N (cid:9) of ˙ v , and the maximal ideal of O ˙ K is m ˙ K := m / ˙ m . The valued field ˙ K is henselian [2, Lemma 3.4.2].(In fact, even better: ˙ K is complete with archimedean value group; cf. the proof of Claim 2 below.)We view ˙ K as the two-sorted structure (cid:0) ˙ K, Γ ∞ , v (cid:1) , with the ring structure on ˙ K and the orderedgroup structure on Γ, and the valuation v : ˙ K × → ∆ ⊆ Γ extended to a map ˙ K → ˙Γ ∞ as usual. Thenatural surjection O → O ˙ K induces an isomorphism R N = O /N m → O ˙ K /N m ˙ K = ( O / ˙ m ) / ( N m / ˙ m ) , and we identify R N with its image; note that then R N is interpretable in ˙ K , and we may view r as atuple of elements in ˙ K eq . The maps ˙res n : K → ˙ K/ ( ˙ K × ) n are defined as before Theorem 5.15, for thevaluation ˙ v in place of v . Now let θ ( a ) be a sequence enumerating all terms of the form ˙res n (cid:0) q ( a ) (cid:1) or v (cid:0) q ( a ) (cid:1) for polynomials q with integer coefficients. Claim 2. The isomorphism type of (cid:0) K rng , r, σ ( a ) , γ (cid:1) determines that of (cid:0) ˙ K, r, θ ( a ) , γ (cid:1) .Proof. By Lemma 5.19, since K rng is ℵ -saturated, we have an isomorphism O ˙ K = O / ˙ m ∼ = −→ lim ←− R N ,and ˙ K is the fraction field of O ˙ K . It remains to show that σ ( a ) determines each value ˙res n ( b ) where b = q ( a ) for some polynomial q with integer coefficients. For this we may assume ˙ v ( b ) ∈ n ˙Γ. Take c ∈ K with n ˙ v ( c ) = ˙ v ( b ), so bc − n ∈ ˙ O ; then with y := ˙res( bc − n ) ∈ ˙ K × we have˙res n ( b ) = y · ( ˙ K × ) n ∈ ( ˙ K × ) / ( ˙ K × ) n , where ˙res: ˙ O → ˙ K is the natural surjection. If necessary replacing b , c , y by their respective inverses,we can arrange that 0 ≤ v ( b ) − nv ( c ) ≤ v ( M ) for some M . Set N := n M ; then res nN ( b ) ∈ S nN is the equivalence class of ˙res N ( y ) ∈ R N ; here ˙res N : O ˙ K → R N is the natural surjection. Now ISTALITY IN VALUED FIELDS AND RELATED STRUCTURES 45 suppose σ ( a ) = σ ( a ′ ) where a ′ is a tuple in K of the same sort as a , and let b ′ := q ( a ′ ). Then v ( b ) = v ( b ′ ), so n ˙ v ( c ) = ˙ v ( b ′ ) and 0 ≤ v ( b ′ ) − nv ( c ) ≤ v ( M ). Thus setting y ′ := ˙res( b ′ c − n ), we have˙res n ( b ′ ) = y ′ · ( ˙ K × ) n ∈ ( ˙ K × ) / ( ˙ K × ) n . By hypothesis we have res nN ( b ) = res nN ( b ′ ) and hence ˙res N ( y ) ∼ nN ˙res N ( y ′ ). By Lemma 5.20 appliedto ˙ K in place of K we therefore obtain y/y ′ ∈ ( ˙ K × ) n and thus ˙res n ( b ) = ˙res n ( b ′ ) as required. (cid:3) Let ˙RV := K/ (1 + ˙ m ) be the abelian monoid introduced in Section 5.4, with ˙ v in place of v , andlet ˙rv : K → ˙RV be the natural surjection. Note that since ˙ m ⊆ m , we have a natural surjectivemonoid morphism ˙RV → RV = K/ (1 + m ), and we hence obtain a sequence(5.4) 1 → ˙ K × ι −−→ ˙RV × ν −−→ Γ → ι is injective, ν is surjective, and ker ν ⊆ im ι ; since ∆ = im( ν ◦ ι )and Γ / ∆ are both torsion-free, this sequence is weakly pure exact, by Lemma 4.11. We consider nowthe structure ( ˙ K, ˙RV , Γ ∞ ) in the three-sorted language L rv (see Section 5.4), which comprises of thefield ˙ K , the abelian monoid structures on Γ ∞ and ˙RV, and the maps ι , ν . Let τ ( a ) be an enumerationof all terms ˙rv (cid:0) q ( a ) (cid:1) , where q ranges over polynomials with integer coefficients. Claim 3. The type of (cid:0) r, θ ( a ) , γ (cid:1) in ˙ K determines the type of (cid:0) r, τ ( a ) , γ (cid:1) in ( ˙ K, ˙RV , Γ ∞ ) .Proof. This follows from Theorem 4.12 applied to the weakly pure exact sequence (5.4) as in the proofof Theorem 5.15. (cid:3) Claim 4. The type of (cid:0) r, τ ( a ) , γ (cid:1) in ( ˙ K, ˙RV , Γ ∞ ) determines the type of ( r, a, γ ) in the L rkngq -struc-ture K .Proof. This follows from Flenner’s QE (Fact 5.16). To see this, let ( ˙ K, ˙RV , ˙Γ ∞ ) be the L rv -structureassociated to the ∆-coarsening of the valued field K , as in Section 5.4: that is, ( ˙ K, ˙RV , ˙Γ ∞ ) consistsof the field ˙ K , the abelian monoids ˙Γ ∞ , ˙RV, the map ι : ˙ K → ˙RV from above, and the compo-sition ˙ ν : ˙RV → ˙Γ ∞ of ν with the natural surjection π : Γ ∞ → ˙Γ ∞ . Expand this structure by asort for Γ ∞ as well as the primitives ν , π . Note that ˙Γ = Γ /ν ( ι ( ˙ K × )) and ˙ ν = π ◦ ν . Hencethe type of (cid:0) r, τ ( a ) , γ (cid:1) in ( ˙ K, ˙RV , Γ ∞ ) determines the type of (cid:0) r, τ ( a ) , γ (cid:1) in this expanded struc-ture ( ˙ K, ˙RV , ˙Γ ∞ ). Now by Fact 5.16 and the remark following it, the type of (cid:0) r, τ ( a ) , γ (cid:1) in ( ˙ K, ˙RV , ˙Γ ∞ )implies the type of ( r, a, γ ) in the ∆-coarsening of K , viewed as L rkg , rv -structure in the natural way,and expanded by a sort for Γ ∞ and the primitives ν , π . This L rkg , rv -structure defines the valuation v on K (as v = ν ◦ ˙rv), and hence interprets K viewed as L rkngq -structure. This yields the claim. (cid:3) The combinations of the four claims above completes the proof of Theorem 5.21. Remark . Theorem 5.21 implies a quantifier elimination result for arbitrary expansions of L rngq just as in Corollary 4.8.In the following corollary we assume that Γ ∞ comes equipped with additional structure, and the L krng -structure K is expanded by this structure on its sort Γ ∞ ; by Remark 5.22, Γ ∞ is then stablyembedded in K , and the structure induced on Γ ∞ is the given one. Corollary 5.23. Suppose k is finite. Then K is NIP iff K is finitely ramified and Γ ∞ is NIP.Proof. The forward direction is clear by earlier results. For the converse, suppose K is finitely ramifiedbut not NIP. We may assume that K is a monster model of its theory. Then there is an indiscerniblesequence ( a i ) i ∈ N of elements of the field sort and a definable subset S ⊆ K such that for all i , we have a i ∈ S iff i is even. By Theorem 5.21 there are special terms σ ( x k ) , . . . , σ m ( x k ) and a suitable L rngq -formula ψ (possibly involving parameters) such that for a ∈ K : a ∈ S ⇐⇒ K | = ψ (cid:0) σ ( a ) , . . . , σ m ( a ) (cid:1) . In particular, K | = ψ (cid:0) σ ( a i ) , . . . , σ m ( a i ) (cid:1) ⇐⇒ i is even.Since k is finite, so are R N and hence all S nN , by Lemma 5.10. Hence after modifying ψ and the σ j suitably, we can assume that each σ j has the form σ j ( x k ) = v (cid:0) q j ( x k ) (cid:1) for some polynomial q j ( x k )with integer coefficients. From [63, Lemma A.18] we obtain γ , . . . , γ m ∈ Γ, r , . . . , r m ∈ N , and anindiscernible sequence ( α i ) of elements of Γ such that v (cid:0) q j ( a i ) (cid:1) = γ j + r j α i for sufficiently large i .With x g a variable of sort Γ ∞ and ψ g ( x g ) := ψ (cid:0) γ + r x g , . . . , γ m + r m x g (cid:1) , for sufficiently large i we then have K | = ψ g ( α i ) ⇐⇒ i is even,showing that Γ ∞ has IP. (cid:3) Distality in Henselian Valued Fields The main aim of this section is to prove the theorem stated in the introduction. In Section 6.3 weconsider when naming a henselian valuation on a distal field preserves distality. After some valuation-theoretic preliminaries in Section 6.4, we investigate the structure of fields with a distal expansionsin Section 6.5. Using work of Johnson [46], we obtain some consequences in the dp-minimal casein Section 6.6.6.1. Reduction to RV ∗ . In this subsection K is a henselian valued field of characteristic zero, andthe structure K and its reduct RV ∗ are as introduced in Section 5.1, where RV ∗ may carry additionalstructure. The aim of the present subsection is to prove the following: Proposition 6.1. K is distal if and only if K is finitely ramified and RV ∗ is distal. The “only if” part is straightforward by Lemma 1.15, full stable embeddedness of RV ∗ in K (seeFact 5.3(2)), and Corollary 2.19. In the rest of this subsection we assume that K is finitely ramifiedand RV ∗ is distal, and show that then K is also distal. We may assume that K is a monster modelof Th( K ). Note that K is automatically NIP by Fact 2.1 and Proposition 5.11. Suppose towardsa contradiction that K is not distal. By Corollary 1.11 there are an indiscernible sequence ( a i ) i ∈ Q with a i ∈ K and finite tuples b = ( b , . . . , b n ) in K and c in RV ∗ , as well as a formula φ ( x, b, c ), suchthat ( a i ) i ∈ Q = is bc -indiscernible, where Q = := Q \ { } , but | = φ ( a i , b, c ) iff i = 0. By Fact 5.3 andRemark 5.4(1), φ ( x, b, c ) is equivalent to a formula of the form(6.1) ψ (cid:0) rv δ ( x − b ′ ) , . . . , rv δ ( x − b ′ m ) , c ′ (cid:1) for some δ , some m and b ′ = ( b ′ , . . . , b ′ m ) ∈ K m , some tuple c ′ from RV ∗ , and an L RV ∗ -formula ψ ,where in addition b ′ , . . . , b ′ m , c ′ ∈ acl( bc ). In particular, ( a i ) i ∈ Q = is b ′ c ′ -indiscernible, hence afterreplacing our original formula with this new one, we can assume that φ ( x, b, c ) itself is of the form (6.1)with b = b ′ . So for i ∈ Q :(6.2) | = ψ (cid:0) rv δ ( a i − b ) , . . . , rv δ ( a i − b n ) , c (cid:1) ⇐⇒ i = 0 . ISTALITY IN VALUED FIELDS AND RELATED STRUCTURES 47 As the structure induced on RV ∗ is distal by Fact 5.3 and KKK is NIP, Proposition 1.17 impliesthat ( a i ) i ∈ Q is c -indiscernible. By Lemma 5.12, the following three cases exhaust all the possibil-ities. Case 1: ( a i ) i ∈ Q is pseudocauchy. Take a ∞ ∈ K such that ( a i ) i ∈ Q = ∞ is bc -indiscernible and ( a i ) i ∈ Q ∞ is c -indiscernible. (Such an a ∞ exists by assumption and saturation.) Then the sequence (cid:0) v ( a ∞ − a i ) (cid:1) i ∈ Q is strictly increasing. Now for each k ∈ { , . . . , n } , one of the following must occur.(a) v ( b k − a ∞ ) > v ( a ∞ − a i ) for all i ∈ Q . As the sequence ( a i ) i ∈ Q is endless, in view of (5.2) wethen have v ( b k − a ∞ ) > v ( a ∞ − a i ) + δ and hence rv δ ( b k − a i ) = rv δ ( a ∞ − a i ), for all i ∈ Q .(b) v ( b k − a ∞ ) < v ( a ∞ − a i ) for all i ∈ Q . As in (a), this implies that v ( b k − a ∞ ) + δ < v ( a ∞ − a i )and hence rv δ ( b k − a i ) = rv δ ( b k − a ∞ ), for all i ∈ Q .(c) There are i > j in Q such that v ( a ∞ − a i ) ≥ v ( b k − a ∞ ) ≥ v ( a ∞ − a j ) . After increasing i or decreasing j if necessary we can assume that i, j = 0. As the relation v ( x ) ≤ v ( y ) is ∅ -definable, we obtain a contradiction with b r -indiscernibility of ( a i ) i ∈ Q = ∞ .Permuting the components of b , we can thus assume that we have some l ∈ { , . . . , n + 1 } such thatfor each i ∈ Q and k = 1 , . . . , n we haverv δ ( a i − b k ) = ( rv δ ( a i − a ∞ ) if k < l rv δ ( a ∞ − b k ) otherwise.Set r i := rv δ ( a i − a ∞ ) for i ∈ Q as well as s k := rv δ ( a ∞ − b k ) for k = l, . . . , n , and r := ( r l , . . . , r n ). Nowthe sequence ( r i ) i ∈ Q is indiscernible, and ( r i ) i ∈ Q = is sc -indiscernible (as ( a i ) i ∈ Q = ∞ is bc -indiscernible).As RV ∗ is distal, by Proposition 1.10 this implies that ( r i ) i ∈ Q is also sc -indiscernible. But then | = ψ (cid:0) rv δ ( a − b ) , . . . , rv δ ( a − b n ) , c (cid:1) ⇐⇒ | = ψ ( r , . . . , r , s, c ) ⇐⇒ | = ψ ( r , . . . , r , s, c ) ⇐⇒ | = ψ (cid:0) rv δ ( a − b ) , . . . , rv δ ( a − b n ) , c (cid:1) , contradicting (6.2). Case 2: ( a i ) i ∈ Q ∗ is pseudocauchy. Then we apply Case 1 to the sequence ( a − i ) i ∈ Q in place of ( a i ) i ∈ Q . Case 3: ( a i ) i ∈ Q is a fan. Note again that then k is infinite, hence char k = 0 by Fact 2.1, andthus δ = 0. Take some a ∞ as in Case 1, and let γ be the common value of v ( a i − a j ) for all i = j in Q ∞ . Let k ∈ { , . . . , n } ; then one of the following must occur.(a) v ( b k − a ∞ ) < γ . Then rv( b k − a i ) = rv( b k − a ∞ ) for all i ∈ Q .(b) v ( b k − a i ) > γ for some i ∈ Q = . Then v ( b k − a i ) > v ( a ∞ − a i ) and v ( b k − a j ) ≤ v ( a ∞ − a j )for each j ∈ Q \ { , i } , contradicting b k -indiscernibility of ( a i ) i ∈ Q = ∞ .(c) v ( b k − a ) > γ . Then rv( a − a ∞ ) = rv( b k − a ∞ ). Note that the sequence (cid:0) rv( a i − a ∞ ) (cid:1) i ∈ Q is indiscernible, and hence not totally indiscernible, by distality and stable embeddedness ofRV ∗ . So (cid:0) rv( a i − a ∞ ) (cid:1) i ∈ Q = is not indiscernible over rv( a − a ∞ ) = rv( b k − a ∞ ) by Corollary 1.6.But this again contradicts the b k -indiscernibility of ( a i ) i ∈ Q = ∞ .(d) v ( b k − a i ) = γ for all i ∈ Q . Then rv( b k − a ∞ ) = γ and thus rv( b k − a i ) = rv( b k − a ∞ ) ⊕ rv( a ∞ − a i ) for all i ∈ Q .Reindexing the components of b , we can thus assume that we have some l ∈ { , . . . , n + 1 } such thatfor i ∈ Q and k = 1 , . . . , n , with r i := rv( a i − a ∞ ) and s k := rv( a ∞ − b k ):rv( a i − b k ) = ( r i ⊕ s k if k < ls k otherwise. Let s := ( s , . . . , s n ). Then ( r i ) i ∈ Q is indiscernible and ( r i ) i ∈ Q = is sc -indiscernible, since ( a i ) i ∈ Q ∞ isindiscernible and ( a i ) i ∈ Q = ∞ is bc -indiscernible. Hence ( r i ) i ∈ Q is sc -indiscernible by Proposition 1.10,as RV ∗ is distal. But then | = ψ (cid:0) rv( a − b ) , . . . , rv( a − b n ) , c (cid:1) ⇐⇒ | = ψ ( r ⊕ s , . . . , r ⊕ s l − , s l , . . . , s n , c ) ⇐⇒ | = ψ ( r ⊕ s , . . . , r ⊕ s l − , s l , . . . , s n , c ) ⇐⇒ | = ψ (cid:0) rv( a − b ) , . . . , rv( a − b n ) , c (cid:1) , contradicting (6.2). This finishes the proof of Proposition 6.1. (cid:3) Reduction of distality from RV ∗ to k and Γ . In this subsection we assume that the structureon RV ∗ is obtained from structures on k , Γ ∞ by expanding RV ∗ by all relations S ⊆ RV m where S ⊆ (ker v rv ) m = ( k × ) m is definable in k or S = v − ( v rv ( S )) and v rv ( S ) ⊆ Γ m is definable in Γ . Wethen have: Proposition 6.2. Suppose K is finitely ramified. Then RV ∗ is distal if and only if both k and Γ are. For the proof, it is natural to distinguish two cases.6.2.1. char k > . Here we may assume that k is finite, by Fact 2.1. The structure induced on Γ isthe given one; see the remarks preceding Corollaries 5.23. The forward direction now follows fromLemma 1.15. For the converse, suppose Γ is distal; then Γ is NIP and hence so is the structure RV ∗ interpretable in K , by Corollary 5.23. By Lemma 5.7, the group morphisms rv γ → : RV × γ → RV × =RV × have finite fibers; moreover, since v rv : RV × → Γ has kernel k × , this group morphism also hasfinite fibers. Hence each element of RV ∗ is algebraic over Γ. As Γ is distal, applying Corollary 1.26we conclude that RV ∗ is distal.6.2.2. char k = 0 . In this case, we note that RV ∗ is bi-interpretable with the pure short exact sequence1 → k × → RV × → Γ → 0, in the sense of Section 4.1, where k , Γ carry the given additional structure.But then the conclusion holds by Theorem 4.6. (cid:3) Combining Propositions 6.1 and 6.2 with Remark 5.2 finishes the proof of the main theorem.6.3. When naming a henselian valuation preserves distality. Let ( K, O ) be a henselian valuedfield with residue field k and value group Γ. The following is [42, Theorem A]: Fact 6.3. If k is not separably closed, then O is definable in the Shelah expansion K Sh of the field K . Together with Lemma 1.30 this immediately implies: Corollary 6.4. If the field K has a distal expansion and k is not separably closed, then the valuedfield ( K, O ) has a distal expansion. Our main theorem allows us to treat the case of separably closed residue field: Corollary 6.5. Suppose k is separably closed. Then the valued field ( K, O ) has a distal expansion ifand only if Γ has a distal expansion and k has characteristic zero.Proof. Note that k is necessarily infinite, and if k has characteristic zero, then k is algebraicallyclosed, hence has distal expansion: just add a predicate for a maximal proper subfield of k . Now theclaim follows from our main theorem. (cid:3) In view of Conjecture 3.16 we expect that in order for ( K, O ) to have a distal expansion, we only needto require that k has a distal expansion. Before we turn to discussing our conjectural classification offields with distal expansion, we recall some definitions and basic facts about canonical valuations. ISTALITY IN VALUED FIELDS AND RELATED STRUCTURES 49 Canonical valuations. In this subsection we let K be a field. We collect some notions andbasic facts used in the next subsection. Let O , O be valuation rings K . One says that O is coarser than O , and that O is finer than O , if O ⊆ O , that is, if O is the valuation ring of a coarseningof ( K, O ).Let now H be the set of henselian valuation rings of K , and let H c be the subset of H consisting ofthose valuation rings with separably closed residue field. Then H \ H c is linearly ordered by inclusion.If H c = ∅ , then H c contains a coarsest valuation ring O c of K ; this valuation ring is (strictly) finerthan every valuation ring in H \ H c . If H c = ∅ , then there is a finest henselian valuation ring of K ,which we also denote by O c . We refer to [28, Theorem 4.4.2] for these facts. The valuation ring O c is called the canonical henselian valuation ring of the field K .Let now p be a prime. We denote by K ( p ) the compositum of all finite normal field extensions L | K of p -power degree. If K ( p ) = K , then K is called p -closed. Lemma 6.6. Suppose K is separably closed and p = char K ; then K is p -closed.Proof. If char K = 0, then K is algebraically closed, and if char K = q > K is a power of q . (cid:3) Following [47, Section 9.5] we say that K is p -corrupted if no finite extension of K is p -closed; as aconsequence of a theorem of Becker [4], one has (see [47, Lemma 9.5.2]): Lemma 6.7 (Johnson) . Every perfect field which is neither algebraically closed nor real closed has afinite p -corrupted extension. A valuation ring O of K is said to be p -henselian if only one valuation ring of K ( p ) lies over O .Let H p be the set of p -henselian valuation rings of K , and let H p c be the subset of H p consisting ofthose valuation rings with p -closed residue field. As before, H p \ H p c is linearly ordered by inclusion.If H p c = ∅ , then H p c contains a coarsest valuation ring O p c of K , which is then finer than every valuationring in H p \ H p c . If H p c = ∅ , then there is a finest p -henselian valuation ring of K , also denoted by O p c .One calls O p c the canonical p -henselian valuation ring of K . See [44], which also contains a proof ofthe following fact: Proposition 6.8 (Jahnke-Koenigsmann) . If K is not orderable and contains all p th roots of unity,then O p c is ∅ -definable in K . Here we recall that K is said to be orderable if there is an ordering on K making K an ordered field.6.5. Distal fields. In this subsection K is a field. The following is commonly attributed to Shelah: Conjecture 6.9. If K is NIP, then K is finite, separably closed, real closed, or admits a non-trivialhenselian valuation. This conjecture has numerous consequences; for example, by [38, Proposition 6.3], it implies that everyNIP valued field is henselian. In [42, Theorem B] it is shown that if K is NIP and O is a henselianvaluation ring of K , then the valued field ( K, O ) is also NIP. Hence if Conjecture 6.9 holds, then everyvaluation ring on a NIP field is henselian, and its residue field is NIP. Moreover, under Conjecture 6.9,any two (externally) definable valuation rings on a NIP field are comparable [38, Corollary 5.4]. InTheorem 6.12 below we show that Conjecture 6.9 also gives rise to a classification of all fields admittinga distal expansion. We first note that the non-trivial henselian valuation stipulated in Conjecture 6.9may be taken to be ∅ -definable, by results in [42, 43] (see also [38, Corollary 7.6]): Lemma 6.10. Suppose Conjecture 6.9 holds, and suppose K is infinite and NIP; then K is separablyclosed or real closed, or K has an ∅ -definable non-trivial henselian valuation ring. Proof. Suppose K is neither separably closed nor real closed; so according to Conjecture 6.9, K has anon-trivial henselian valuation. If K has such a valuation with residue field which is separably closedor real closed, then by [43, Theorem 3.10 and Corollary 3.11, respectively], there is an ∅ -definable non-trivial henselian valuation ring of K . Hence we may assume that the residue field of each henselianvaluation on K is not separably closed and not real closed. In particular, the residue field k of O := O c is neither separably closed nor real closed. Hence O is the finest henselian valuation ring of K ; inparticular, k does not have a non-trivial henselian valuation. Now k is NIP, and so by Conjecture 6.9applied to k , this field is finite. Its absolute Galois group is non-universal, so O is ∅ -definable by [43,Theorem 3.15 and Observation 3.16]. (cid:3) We also recall that every infinite field with a distal expansion has characteristic zero. Corollary 6.11. Suppose Conjecture 6.9 holds, and K is infinite and has a distal expansion. Then K is algebraically closed or real closed, or K has an ∅ -definable non-trivial henselian valuation ring O whose residue field (1) is finite, or (2) is a field of characteristic zero with a distal expansion.Proof. Suppose K is neither algebraically closed nor real closed; then by Lemma 6.10 we can takean ∅ -definable non-trivial henselian valuation ring O of K . Let k be the residue field of O ; then k also has a distal expansion by the forward direction in our main theorem; in particular, if char k > k is finite. (cid:3) In connection with option (1) in Corollary 6.11 recall that if ( K, O ) is an infinite NIP henselianvalued field of characteristic zero with finite residue field, then ( K, O ) has a specialization whichis p -adically closed of finite p -rank, for some prime p . (Remark 2.20.) We do not know whetherwe can upgrade (2) in this corollary to “is algebraically closed of characteristic zero, or real closed”(even while simultaneously weakening the condition that O be ∅ -definable in K to O being externallydefinable, say). Instead we show: Theorem 6.12. Suppose Conjecture 6.9 holds, and K is NIP and does not define a valuation ringwhose residue field is infinite of positive characteristic; then K has a henselian valuation ring, type-definable over the empty set, whose residue field is algebraically closed of characteristic zero, realclosed, or finite. Before we give the proof of Theorem 6.12, we establish analogues of two results from [47] (9.5.4and 9.5.7, respectively): Lemma 6.13. Suppose Conjecture 6.9 holds and K is NIP, non-orderable, and contains all p -th rootsof unity, where p is a prime. Let O = O p c be the canonical p -henselian valuation ring of K ; then O is ∅ -definable, and its residue field is finite, has characteristic p , or is p -closed.Proof. Proposition 6.8 yields the ∅ -definability of O . Suppose the residue field k of O is infinite,has characteristic = p , and is not p -closed. Then by Lemma 6.6, k cannot be separably closed;since K is non-orderable, k is also not real closed. Hence by Conjecture 6.9 we may equip k witha non-trivial henselian valuation ring; let k → k be the corresponding place. Composition of theplaces K → k → k then gives rise to a henselian valuation ring O of K with residue field k suchthat k is a specialization of ( K, O ), and then O is a strictly finer p -henselian valuation ring than O ,a contradiction. (cid:3) Lemma 6.14. Suppose Conjecture 6.9 holds, and suppose K is infinite NIP, and the residue fieldof each ∅ -definable valuation ring of K has characteristic zero. Let O ∞ be the intersection of all ∅ -definable valuation rings of K . Then O ∞ is a valuation ring of K whose residue field is algebraicallyclosed of characteristic zero or real closed. ISTALITY IN VALUED FIELDS AND RELATED STRUCTURES 51 Proof. The hypothesis and the remarks following Conjecture 6.9 yield that the set of all valuationrings of K is linearly ordered by inclusion; in particular, O ∞ is a valuation ring of K . As in the proofof [47, Theorem 9.5.7(2)] one also sees that O ∞ equals the intersection of all definable valuation ringsof K . Let k ∞ be the residue field of O ∞ . We have char k ∞ = 0, since otherwise some ∅ -definablevaluation ring O ⊇ O ∞ of K would have residue field k with char k = char k ∞ > k ∞ is neither algebraically closed nor real closed. By Lemma 6.7we then obtain a prime p and a finite p -corrupted extension l of k ∞ . Let v ∞ : K × → Γ ∞ denote thevaluation associated to O ∞ . Choose a finite field extension L of K which contains all 4 p -th rootsof unity and such that the residue field of the unique valuation w ∞ on L extending v ∞ contains l ,and hence is not p -closed. Lemma 6.13 yields a valuation w on L which is ∅ -definable (that is, itsvaluation ring is ∅ -definable in the field L ) and not a coarsening of w ∞ . Let v be the restriction of w to a valuation on K ; then v is definable, hence a coarsening of v ∞ , say v = ( v ∞ ) ∆ where ∆ is aconvex subgroup of Γ ∞ . Let ∆ L be the convex hull of ∆ in the value group of w ∞ . The restrictionof the ∆ L -coarsening ( w ∞ ) ∆ L of w ∞ to K is v . But v is henselian, so w = ( w ∞ ) ∆ L is a coarseningof w ∞ , a contradiction. (cid:3) Now Theorem 6.12 follows easily: If K has an ∅ -definable valuation ring with residue field of positivecharacteristic, then this residue field is finite by hypothesis, and we are done. Thus we may assumethat the residue field of every ∅ -definable valuation ring of K has characteristic zero. Then Lemma 6.14yields a henselian valuation ring O ∞ , type-definable over ∅ , whose residue field is algebraically closedof characteristic zero, or real closed. (cid:3) Corollary 6.15. Suppose Conjectures 3.16 and 6.9 hold, and K is NIP; then the following areequivalent: (1) K has a distal expansion; (2) K does not interpret an infinite field of positive characteristic; (3) K does not define a valuation ring whose residue field is infinite of positive characteristic; (4) K has a henselian valuation ring whose residue field is algebraically closed of characteristiczero, real closed, or finite.Proof. The implications (1) ⇒ (2) ⇒ (3) are clear (using Fact 2.1), and (3) ⇒ (4) follows fromTheorem 6.12. To show (4) ⇒ (1), suppose K has characteristic zero. If O is a henselian valuationring of K whose residue field k is algebraically closed of characteristic zero, real closed, or finite,then k has a distal expansion, and after choosing a distal expansion of the value group of O , our maintheorem yields that ( K, O ) has a distal expansion, which is also a distal expansion of K . (cid:3) We also note a consequence of Theorem 6.12 for ordered fields. In [23], a field is defined to be almostreal closed if it has a henselian valuation ring with real closed residue field. Corollary 6.16. Suppose Conjecture 6.9 holds, and K is orderable and has a distal expansion; then K is almost real closed.Proof. Equip K with an ordering making it an ordered field; it is well-known that then every henselianvaluation ring of K is convex, and hence its residue field is orderable. Now use Theorem 6.12. (cid:3) Based on Theorem 6.12 we conjecture: Conjecture 6.17. Suppose K has a distal expansion; then K has a henselian valuation ring whoseresidue field is algebraically closed of characteristic zero, real closed, or finite. Distality in the dp -minimal case. In this subsection we show that for dp-minimal K , theconclusion of Corollary 6.15 holds even without assuming Conjectures 3.16 and 6.9; this relies againon work of Johnson [47]. We first recall a few facts about dp-minimal fields and related structures. Fact 6.18. (1) Every dp -minimal expansion of an ordered abelian group is distal ( by Fact 1.5 ) . (2) Every dp -minimal valued field is henselian [46, 45] . Combining Fact 6.18 and the main theorem of this paper, we get: Corollary 6.19. A dp -minimal valued field is distal ( has a distal expansion ) if and only if its residuefield is distal ( respectively, has a distal expansion ) . A dp-minimal (pure) field can fail to admit a distal expansion only in the most obvious way: Corollary 6.20. Let K be an infinite dp -minimal field; then the following are equivalent: (1) K has a distal expansion; (2) K does not interpret an infinite field of positive characteristic; (3) K does not define a valuation ring whose residue field is infinite of positive characteristic; (4) K has a henselian valuation ring whose residue field is algebraically closed of characteristiczero, real closed, or finite.Proof. As in the proof of Corollary 6.15, the implications (1) ⇒ (2) ⇒ (3) are clear. For (3) ⇒ (4),we argue as in the proof of the corresponding implication in Theorem 6.12: If K has an ∅ -definablevaluation ring with residue field of positive characteristic, then (4) holds. Otherwise, let O ∞ be theintersection of all ∅ -definable valuation rings of K ; by [47, Theorem 9.1.4], O ∞ is a henselian valuationring of K (with O ∞ = K if K admits no ∅ -definable non-trivial valuations) whose residue field k ∞ is algebraically closed, real closed, or finite. Moreover, char k ∞ = 0 by [47, Theorem 9.4.18(3),Remark 9.5.6]. Finally, (4) ⇒ (1) is shown as in the proof of (4) ⇒ (1) in Corollary 6.15, usingFacts 3.1 and 6.18 in place of Conjecture 3.16. (cid:3) Note that there are indeed dp-minimal fields of characteristic zero without distal expansions. Example. Let Q unr p be the maximal unramified extension of the valued field Q p . Its value groupis Z and its residue field is the algebraic closure F a p of F p . Let O K be the unique valuation ring of K = Q unr p (cid:0) p /p , p /p , . . . (cid:1) lying over that of Q unr p . Its value group S n p n Z is archimedean (henceregular) but non-divisible, and ( K, O K ) is henselian; so it follows from [41, Theorem 5] that O K is ∅ -definable in the field K . Hence K is a field of characteristic zero which is dp-minimal by [47,Theorem 9.1.5, 1(c)] but has no distal expansion since it interprets an infinite field of characteristic p .7. Distality in Expansions of Fields by Operators In this section we use a “forgetful functor” approach to show that various expansions of distal fieldsby operators remain distal. Most of the results of this section were obtained and circulated in 2014.We have learned that recently some of them were observed independently in [21].7.1. An abstract distality criterion. We fix a language L and a complete L -theory T = Th( M ).As usual all variables here are assumed to be (finite) multivariables. Recall that by Fact 1.8, T isdistal if and only if every partitioned L -formula ϕ ( x ; y ) has a strong honest definition in T , i.e., thereis a formula ψ ( x ; y , . . . , y N ), where y , . . . , y N are disjoint multivariables (for some N ∈ N ), each ofthe same sort as y , such that for all a ∈ M x and finite subsets B of M y with | B | ≥ 2, there are b , . . . , b N ∈ B such that ψ ( x ; b , . . . , b n ) isolates tp ϕ ( a | B ):(1) a ∈ ψ ( M x ; b , . . . , b N ); and(2) for all b ∈ B , either ψ ( M x ; b , . . . , b N ) ⊆ ϕ ( M x ; b ) or ψ ( M x ; b , . . . , b N ) ∩ ϕ ( M x ; b ) = ∅ . ISTALITY IN VALUED FIELDS AND RELATED STRUCTURES 53 We also consider an extension L ( F ) of the language L by a set F of new function symbols. We assumethat L ( F ) has the same sorts as L , and we consider F itself as a language by declaring the sorts of F to be those of L . Finally, we let T ( F ) be a complete L ( F )-theory extending T . Proposition 7.1. Suppose T is distal and the following conditions hold: (1) T ( F ) has quantifier elimination; (2) all function symbols in F are unary; and (3) for every L ( F ) -term t ( x ) there are an L -term s in n variables of the appropriate sorts and F -terms t ( x ) , . . . , t n ( x ) such that T ⊢ t ( x ) = s (cid:0) t ( x ) , . . . , t n ( x ) (cid:1) . Then T ( F ) is distal.Proof. Fix a model M of T ( F ), and let ϕ ( x ; y ) be a partitioned L ( F )-formula; we show that ϕ ( x ; y ) hasa strong honest definition in T ( F ). By assumption (1), we may assume that ϕ ( x ; y ) is quantifier-free.Then by assumptions (2) and (3) there are an L -formula ϕ ′ ( x ′ ; y ′ ) as well as F -terms s ( x ) , . . . , s m ( x )and t ( y ) , . . . , t n ( y ), such that for all a ∈ M x , b ∈ M y we have M | = ϕ ( a, b ) ⇐⇒ M | = ϕ ′ (cid:0) s ( a ) , t ( b ) (cid:1) , where s ( a ) := (cid:0) s ( a ) , . . . , s m ( a ) (cid:1) and t ( b ) := (cid:0) t ( b ) , . . . , t n ( b ) (cid:1) . Suppose y = ( y , . . . , y k ) where k = | y | ; we can assume that the terms t , . . . , t n contain theterms y , . . . , y k ; thus b t ( b ) : M y → M y ′ is injectve. By distality of T , take a strong honestdefinition ψ ′ ( x ′ ; y ′ , . . . , y ′ N ) for ϕ ′ ( x ′ ; y ′ ) in T , where y ′ , . . . , y ′ N are disjoint new multivariables ofthe same sort as y ′ ; thus for all a ′ ∈ M x ′ and any finite subset B ′ of M y ′ with | B ′ | ≥ 2, there are b ′ , . . . , b ′ N ∈ B ′ such that(1) M | = ψ ′ ( a ′ ; b ′ , . . . , b ′ N ); and(2) for all b ′ ∈ B ′ , either ψ ′ ( M x ′ ; b ′ , . . . , b ′ N ) ⊆ ϕ ′ ( M x ′ ; b ′ ) or ψ ′ ( M x ′ ; b ′ , . . . , b ′ N ) ∩ ϕ ′ ( M x ′ ; b ′ ) = ∅ . We claim that ψ ( x ; y , . . . , y N ) := ψ ′ (cid:0) s ( x ); t ( y ) , . . . , t ( y N ) (cid:1) where y , . . . , y N are disjoint new multivariables of the same sort as y , is a strong honest definitionfor ϕ ( x ; y ) in T ( F ). Let a ∈ M x , and let B ⊆ M y be finite with | B | ≥ 2. Set a ′ := s ( a ), B ′ := t ( B ) ⊆ M y ′ (so | B ′ | = | B | ≥ b , . . . , b N ∈ B such that (1) and (2) above hold with b ′ i := t ( b i ),for i = 1 , . . . , N . Then M | = ψ ( a ; b , . . . , b N ), and ψ ( x ; b , . . . , b N ) isolates tp ϕ ( a | B ), as required. (cid:3) In a similar way as the preceding proposition, one shows: Lemma 7.2. Suppose T is distal and for every partitioned L ( F ) -formula ϕ ( x ; y ) , where | x | = 1 , thereis a partitioned L -formula ϕ ′ ( x ; z ) and a tuple of L ( F ) -terms t ( y ) of length | z | such that T ⊢ ϕ ( x ; y ) ↔ ϕ ′ (cid:0) x ; t ( y ) (cid:1) . Then T ( F ) is distal.Proof. Let ϕ ( x ; y ) be a partitioned L ( F )-formula, where | x | = 1; by Proposition 1.9 it is enough toshow that ϕ ( x ; y ) has a strong honest definition in T ( F ). By our hypothesis we can assume ϕ ( x ; y ) = ϕ ′ (cid:0) x ; t ( y ) (cid:1) where ϕ ′ ( x ; y ′ ) is an L -formula and t ( y ) = (cid:0) t ( y ) , . . . , t n ( y ) (cid:1) is an appropriate tuple of L ( F )-terms whose components contain the terms y , . . . , y k for y = ( y , . . . , y k ). Distality of T yields a stronghonest definition ψ ′ ( x ; y ′ , . . . , y ′ N ) for ϕ ′ ( x ; y ′ ) in T , where y ′ , . . . , y ′ N are disjoint new multivariablesof the same sort as y ′ . Then ψ ( x ; y , . . . , y N ) := ψ ′ (cid:0) x ; t ( y ) , . . . , t ( y N ) (cid:1) is a strong honest definitionfor ϕ ( x ; y ) in T ( F ). (cid:3) In practice, condition (3) in Proposition 7.1 is easily verified whenever T is a relational expansion ofthe theory of fields, and the functions symbols in F are interpreted as derivations in models of T ( F ).We now give several applications of these criteria.7.2. Transseries. In this subsection we assume that the reader is familiar with [2, Chapter 16].Consider the language L ΛΩ = { , , + , − , · , ∂ , ι, ≤ , , Λ , Ω } introduced there. The L ΛΩ -theory T nl of ω -free newtonian Liouville closed H -fields eliminates quan-tifiers [2, Theorem 16.0.1] and has two completions: T nlsmall , of which the differential field T oflogarithmic-exponential transseries is a model, and T nllarge . Both completions are distal: Corollary 7.3. The L ΛΩ -theories T nlsmall and T nllarge are distal.Proof. Let L := L ΛΩ \ { ∂ } (so L ( ∂ ) = L ΛΩ ), let T ( ∂ ) = T nlsmall , and let T be the L -theory of T . Eachmodel of T is a real closed ordered field K , viewed as a structure in the language { , , + , − , · , ι, ≤} in the natural way, equipped with a convex dominance relation and interpretations of the unaryrelation symbols Λ and Ω as certain convex subsets of K . By Baisalov-Poizat [3], the theory of eachexpansion of an o-minimal structure by convex subsets of its domain is weakly o-minimal, hence distal;in particular, T is distal. (Alternatively, we could use Fact 1.29.) Proposition 7.1 (and the quotientrule for derivations) implies that T nlsmall = T ( ∂ ) is distal. The argument for T nllarge is similar. (cid:3) Combining Fact 2.1 with the preceding corollary shows that no infinite field of positive characteristicis interpretable in T . We venture the following: Conjecture 7.4. The only infinite fields interpretable in T are T , R , and their respective algebraicclosures T [ i ] , C = R [ i ] . Other distal differential fields. Proposition 7.1 can be used to show that many other theoriesof interest are distal as well. In general, whenever T is the theory of an expansion of a differentialfield (perhaps with several derivations) by relations and constants, and we know that(1) T has QE, and(2) the reduct of T to the language without derivations is distal,then Proposition 7.1 implies that T itself is distal. In the literature, one finds many theories whichsatisfy these conditions. For instance: Corollary 7.5. The following theories are distal: (1) CODF , the model completion of the theory of ordered differential fields from [65] ; (2) CODF m , the model completion of the theory of ordered differential fields with m commutingderivations from [58, 67] ; (3) pCDF d,m , the model completion of the theory of p -valued fields of p -rank d with m commutingderivations from [67] . The fact that CODF is NIP was first shown (also using the “forgetful functor”) in [53], and generalizedto CODF m in [36]. The paper [33] considers a generalization of CODF m : Given a complete, modelcomplete o-minimal theory T expanding the theory of real closed ordered fields, the theory whosemodels are models of T equipped with m commuting derivations which satisfy the Chain Rule withrespect to the continuously differentiable definable functions in T has a model completion T m , andif T has quantifier elimination and a universal axiomatization, then T m has quantifier elimination [33,Theorem 6.8]. (Note that the latter hypothesis on T can always be achieved by expanding the languageby function symbols for all ∅ -definable functions and expanding T accordingly.) Our criterion impliesthat then T m is distal; this has also been observed in [33, Proposition 6.10]. ISTALITY IN VALUED FIELDS AND RELATED STRUCTURES 55 The topological fields with generic valuations considered in [36] are also distal. For example, let L = { , , + , − , · , ≤ , } and let OVF be the L -theory of ordered fields equipped with a non-trivialconvex dominance relation; its model completion is RCVF, the theory of real closed valued fields(see [2, Section 3.6]). By [36, Corollary 6.4], the L ( ∂ )-theory whose models are the expansions ofmodels of OVF by a derivation ∂ , has a model completion; this model completion is distal becauseRCVF is weakly o-minimal. In [56] it is shown that the L -theory of pre- H -fields with gap 0 hasa model completion. Here, a pre- H -field is a model of the universal part of the theory T nl fromSection 7.2, and such a pre- H -field has gap 0 if it satisfies the L -sentence ∀ y ( y ′ y → y C in the distal field of reals in the usual way [66, 72].Corollary 7.5 implies a qualitative analog for the stable theories DCF ,m of differentially closed fieldsof characteristic 0 with m commuting derivations. For this we need the following facts [64, 68]: Fact 7.6. If K | = CODF , then the differential field extension K [ i ] of K ( where i = − is adifferentially closed field of characteristic , i.e., K [ i ] | = DCF . More generally, if K | = CODF m ,then K [ i ] | = DCF ,m . This immediately yields (see Lemma 1.28): Corollary 7.7. The theory DCF ,m has a distal expansion. Problem 7.8. By [8, Lemma 4.5.9], the theory CODF is not strongly dependent. Does DCF admita strongly dependent distal expansion?7.4. Henselian valued fields with analytic structure. We finish by showing that the forgetfulfunctor argument (in the form of Lemma 7.2) also allows us to extend the main theorem from theintroduction to the analytic expansions of henselian valued fields introduced in [20]; for this we relyon some arguments from [57, Section 5]. We need to recall the relevant definitions from [20].We fix a noetherian commutative ring A and an ideal I = A of A such that A is separated andcomplete for its I -adic topology. Let A h X i = A h X , . . . , X m i be the ring of power series in the distinctindeterminates X , . . . , X m with coefficients in A whose coefficients I -adically converge to 0, and set A m,n := A h X i [[ Y ]] where X = ( X , . . . , X m ) and Y = ( Y , . . . , Y n ) are disjoint tuples of distinctindeterminates over A . We expand the (one-sorted) language of valued fields to a language L A byintroducing a unary function symbol ι as well as an ( m + n )-ary function symbol for each elementof A m,n (which we denote by the same symbol). We let T A be the L A -theory whose models are the L A -structures expanding a valued field ( K, O ) of characteristic zero, such that with m = maximalideal of O :(A1) ι is interpreted by the map K → K with a /a if a = 0 and 0 f ∈ A m,n is interpreted by a function f K : K m × K n → K which isidentically zero outside of O m × m n and such that f K ( O m × m n ) ⊆ O ;(A3) the map f f K is a ring morphism from A m + n to the ring of functions K m × K n → K ;(A4) each f ∈ A m,n , viewed as an element of A m,n +1 under the natural inclusion A m,n ⊆ A m,n +1 ,is interpreted as a function K m × K n +1 → K which does not depend on the last coordinate,and similarly for the inclusion A m,n ⊆ A m +1 ,n ;(A5) each a ∈ I ⊆ A = A , is interpreted by a constant function with value in m ; (A6) for a = ( a , . . . , a m ) ∈ O m and b = ( b , . . . , b n ) ∈ m n we have X Ki ( a, b ) = a i ( i = 1 , . . . , m )and Y Kj ( a, b ) = b j ( j = 1 , . . . , n ).The valued field underlying each model of T A is automatically henselian; see [57, Proposition 3.5].Let now K | = T A , and as in Section 5.1 expand K to a multi-sorted structure K whose sorts are K (called the field sort below) and the sets RV δ (called the RV-sorts below), with the primitives specifiedin (K1)–(K4). Let K ∗ be an expansion of K obtained by imposing additional structure on thereduct RV ∗ of K , including,(A7) for each u ∈ A m + n , the function u Kδ : RV m + nδ → RV δ satisfying u Kδ (cid:0) rv δ ( a ) (cid:1) = rv δ (cid:0) u K ( a ) (cid:1) for a ∈ K m + n .(See [57, Corollary 3.9].) Let also L be the reduct of the language L ∗ of K ∗ obtained by removingall symbols listed under (A1)–(A7) above. The following is a consequence of [57, Corollary 5.5] (ageneralization of a theorem in [26]): Proposition 7.9. Let ϕ ( x, y, r ) be an L ∗ -formula where the multivariables x , y , are of the field sortwith | x | = 1 , and r is of the RV -sort. Then there exists an L -formula ϕ ′ ( x, z, r ) and an appropriatetuple of L A -terms t ( y ) such that K ∗ | = ϕ ( x, y, r ) ↔ ϕ ′ (cid:0) x, t ( y ) , r (cid:1) . We now use this result to show a variant of our main theorem: Corollary 7.10. Let K | = T A ; if the valued field underlying K is distal ( has a distal expansion ) ,then the L A -structure K is distal ( has a distal expansion, respectively ) .Proof. Suppose first that the valued field underlying K has a distal expansion; by the forward directionof our main theorem, this valued field is finitely ramified, and its value group Γ and residue field k have a distal expansion. Consider now the structure K ∗ introduced before Proposition 7.9, where weequip RV ∗ with the functions (A7) as well as the structure coming from the distal expansions of Γand k as explained at the beginning of Section 6.2. By Propositions 6.1 and 6.2, the L -reduct K of K ∗ is distal. Now Lemma 7.2 and Proposition 7.9 yield that the expansion K ∗ of K is distal. Thisshows that if the valued field underlying K has a distal expansion, then so does K . 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