Interpretable fields in real closed valued fields and some expansions
aa r X i v : . [ m a t h . L O ] F e b FIELDS INTERPRETABLE IN REAL CLOSED VALUED FIELDS AND SOMEEXPANSIONS
ASSAF HASSON AND YA’ACOV PETERZILA
BSTRACT . Let M = h K ; O i be a real closed valued field and let k be its residue field.We prove that every interpretable field in M is definably isomorphic to either K , K ( √− , k , or k ( √− . The same result holds when K is a model of T , for T an o-minimal expan-sion of a real closed field, and O is a T -convex subring.The proof is direct and does not make use of known results about elimination of imagi-naries in valued fields.
1. I
NTRODUCTION
The goal of this work is to classify fields interpretable (i.e. fields which are given asdefinable quotients) in real closed valued fields or, more generally, in expansions of o-minimal structures by a proper T -convex subring. This type of classification of definablequotients originates in Poizat’s influential model theoretic consideration of the Borel-Titstheorem, [22] A similar study of interepretable groups and fields, in the setting of (pure)algebraically closed valued fields, was carried out By Hrushovski and Rideau-Kikuchi in[9] using different techniques.We prove (see Section 3 for the definition of T conv ): Theorem.
Let K be a real closed valued field, or more generally a model of T conv , where T is an o-minimal expansion of a real closed field. Let k be its residue field. If F is a fieldinterpretable in K then F is definably isomorphic to either K , K ( √− , k , or k ( √− . This result extends a similar theorem of Bays and the second author, [1], proved for definable fields in real closed valued fields. The analogous problem of definable fields in Q p was addressed in [21], but its generalisation to interpretable fields remains open.The study of imaginaries in valued fields was suggested by Holly in [8] and first studiedin depth and in full generality, in the setting of algebraically closed valued fields, by Haskell,Hrushovski and Macpherson in [4] and [5]. Elimination of imaginaries in real closed valuedfields analogous to that of [4] was proved by Mellor in [17]. Some results analogous to [5]for real closed valued fields were obtained by Ealy, Haskell and Marikova in [3].The work in [9] uses the theorems on elimination of imaginaries, stable domination andother structural results on algebraically closed valued fields, and accomplishes considerablymore than the classification of interpretable fields. We adopt a different approach circum-venting elimination of imaginaries, and avoiding almost completely the so called geometricsorts. In fact, our main result covers expansions of real closed valued fields by analyticfunctions where no elimination of imaginaries results are currently available (see [6]). Ourproof is based on the analysis of one dimensional (equivalently dp-minimal) subsets of the Date : February 2, 2021.The second author was partially supported by Israel Science Foundation grant number 290/19 . interpreted field F , and as such it borrows ideas from Johnson’s work on fields of finitedp-rank (see for example [11] and [10]), as well as [20].The outline of the proof is as follows: We identify F with X/E for a definable X ⊆ K n and a definable equivalence relation E . We then find a one-dimensional definable set J ⊆ X intersecting infinitely many E -classes. After possibly shrinking J we endow it with thestructure of a weakly o-minimal structure I . Its universe I = J/E is our basic buildingblock (see [11] for the similar notion of a quasi-minimal sets or [20] for the notion of G -minimal set). Next, we show that, after possibly shrinking J further, I can be definablyembedded in one of four weakly o-minimal structures: h K ; < i , h k ; < i , h K/O ; < i or h Γ; < i (the value group). By analysing definable functions in K/O and showing that they arelocally “affine” with respect to the additive structure of
K/O , we eliminate the possibilitythat I could be embedded into K/O . As for Γ , we consider two cases: If T is power-bounded then the o-minimal K -induced structure on Γ cannot interpret a field (see [26]),and in fact we show that I cannot be embedded in Γ . In the exponential case we showthat h Γ , + i is definably isomorphic to h K/O, + i and therefore, by the above, I cannot beembedded into Γ . Thus, regardless of whether T is power bounded or exponential, we areleft with the first two possibilities, of K and k . Using the notion of infinitesimals we provein these two cases that the field is definably isomorphic to a definable field in K conv or in k .The final result follows from the work on definable fields in o-minimal structures ([19]).We remark that given the o-minimal expansion of K in a signature L , as in the state-ment of Theorem 1, it suffices to prove the result for some h K ; O i ≺ h K ′ ; O i . Therefore,throughout, we tacitly assume that h K ; O i is ( |L| + ℵ ) + -saturated as will be all struc-tures considered below.As a corollary to our main theorem, and using the work of Hempel and Palacin [7], weobtain a theorem about definable division rings: Corollary 1.1.
Let K be a real closed valued field, or more generally a model of T conv ,where T is an o-minimal expansion of a real closed field. If D is a division ring interpretablein K then D is definably isomorphic to either K , K ( √− , or the quaternions over K , orto k , k ( √− , or the quaternions over k .Proof. By [7, Theorem 2.9], D is a finite extension of its center, the field F . By the theoremabove, F is definably isomorphic to a field which is definable in h K ; + , ·i , or in h k ; + , ·i .Thus D , as a finite extension, is definably isomorphic to a division ring definable in these.The result follows from [19, Theorem 1.1] on division rings in o-minimal structures. (cid:3) Remark. (1) Weakly o-minimal structures are dp-minimal (namely, have dp-rank ),thus if F is an interpretable field in a real closed valued field (or in a model of T conv ) then dp-rk ( F ) < ω . It is therefore possible that the algebraic classificationof F , into a real or an algebraically closed field of characteristic , follows from theseminal work of Johnson on fields of finite dp-rank (see for example [12]). Notehowever that our theorem gives the additional connection between F and the fields K and k .(2) A weakly o-minimal theory is distal [24, Chapter 9], as is T eq (see Exercise 9.12there). Thus an interpretable field in T conv is definable in a distal structure. As waspointed out by Chernikov, such fields cannot have characteristic p > due to thecombinatorial regularity in [2]. So our field F must have characteristic zero.However, our arguments do not make use of either of the above algebraic facts. IELDS INTERPRETABLE IN REAL CLOSED VALUED FIELDS AND SOME EXPANSIONS 3 (3) As the two comments above suggest, there are various model theoretic frameworkswhich fit our setting. The field F has finite dp-rank. It is also definable in a distalstructure. Finally, the structure h K ; O i , as well as the induced structures on K/O , k and Γ are al weakly o-minimal expansions of a groups thus they are all uniformdp-minimal structures in the sense of Simon and Walsberg, [25]. While no directapplication of their paper appears in this final version we found the results there tobe very helpful during our work on this project.(4) We expect that a similar direct proof could be given for the theorem of Hrushovskiand Rideau-Kikuchi, on interpretable fields in algebraically closed valued fields.2. D IMENSION IN WEAKLY O - MINIMAL THEORIES
Throughout this section M denotes a model of a weakly o-minimal theory. . We first collect some useful facts concerning dimension in weakly o-minimal structures.The set M is equipped with the order topology and M n with the product topology. By anopen box in M n we mean a cartesian product of convex subsets of M (since M is weaklyo-minimal the end points might not be in M ).We recall that if M is a weakly o-minimal structure and S ⊆ M r is a definable set then dim( S ) = l if l is the maximal natural number such that the projection of S on l of thecoordinates has non empty interior.We remind that definable functions in weakly o-minimal structures may be locally con-stant without being piecewise continuous, so that dcl need not, in general, satisfy the ex-change principle. Though dcl satisfies exchange in the main sorts of T conv (because itcoincides with dcl L ), it is not the case for K/O .The following can be inferred readily from [23, Theorem 3.8] and the subsequent com-ment.
Fact 2.1. If Y ⊆ M n is definable then dp-rk ( Y ) = dim( Y ) . For a a tuple in M n and A ⊆ M we let dim( a/A ) be the minimal dimension of an A -definable subset of M n containing a . Given X ⊆ M n definable over A , we say that a is generic in X over A if a ∈ X and dim( a/A ) = dim( X ) . Our standing saturationassumption assures that generics over small parameter sets always exist. Corollary 2.2. If a ∈ M n and A ⊆ M then dp-rk ( a/A ) = dim( a/A ) .Proof. By [23, Corollary 3.5] in dp-minimal structures, dp-rk () is local, namely dp-rk ( a/A ) is the minimal dp-rank of an A -definable set containing a . By definition, the same is truefor dim( a/A ) , so we can use Fact 2.1. (cid:3) By the sub-additivity of the dp-rank, [13], we have:
Fact 2.3. dim( ab/A ) ≤ dim( a/A ) + dim( b/Aa ) . We shall use several times the fact that a definable subset of M n in weakly o-minimaltheories has a finite decomposition into definable cells, each homeomorphic, via an appro-priate projection to an open set in some M k (see [15, Theorem 4.11]). Also, we shall usethe fact, [15, Theorem 4.7], that dimension of definable sets is preserved under definablebijection. ASSAF HASSON AND YA’ACOV PETERZIL
From now on we only use the weakly o-minimal dimension (and not dp-rank), providing– for the sake of completeness – self contained proofs of the properties we need.
Lemma 2.4.
Assume that X ⊆ M n is definable over A and that there is an externallydefined set W (i.e. W is definable in an elementary extension N ) such that X ( M ) ⊆ W .If dim( W ) ≤ s then dim( X ) ≤ s .Proof. By weak o-minimality, there is a cell decomposition C , . . . , C l for W and pro-jections π i : C i → M t i such that π i ( C i ) is open and π i | C i is a homeomorphism. So dim( C i ) = t i . It follows that we can find N -definable sets W , . . . , W r ⊆ W such thatevery W i has a finite-to-one projection onto s of the coordinates. Without loss of generality, W is one of these W i and hence there is a finite-to-one projection π : W → N s . Since X ( M ) ⊆ W the restriction of π to X ( M ) is also finite-to-one. Hence dim( X ) ≤ s . (cid:3) We also need the following:
Lemma 2.5.
Let ( J, ) be a definable infinite linearly ordered set without a maximumelement. Let { X b : b ∈ J } be a definable family of subsets of M n such that for b b , X b ⊆ X b . Assume that for every b ∈ J we have dim( X b ) ≤ m . Let U = S b ∈ I X b . Then dim( U ) ≤ m .Proof. Let b ∗ ∈ J ( N ) be an element in an elementary extension N , such that b ∗ is greaterthan all elements of J ( M ) (since J has no maximum such a b ∗ exist). Since dimensionis definable in parameters, dim( X b ∗ ) ≤ m . For every b ∈ M , X b ( M ) ⊆ X b ∗ therefore U ( M ) ⊆ X b ∗ . It follows from Lemma 2.4 that dim( U ) ≤ m . (cid:3) Lemma 2.6.
Let c , . . . , c m be each a tuple of elements from M , of possibly differentlengths, and let C ⊆ M be an A -definable convex set.Given any initial segment I ⊆ C (possibly defined over additional parameters), thereexists b ∈ I such that for every i = 1 , . . . , m , dim( bc i /A ) = 1 + dim( c i /A ) .Proof. For each i = 1 , . . . , m we fix W i , definable over A , such that dim( c i /A ) = dim W i .Consider the type p ( x ) : { x ∈ I } ∪ m [ i =1 {¬ X ( x, c i ) : dim( X ) ≤ dim( c i /A ) and X ⊆ M × W i definable over A } . Since C is definable over A , we may take above only those sets X such that for every w ∈ W , X ( M, w ) is a nonempty subset of C .If p is consistent then its realization will be the desired b . So, we assume towards contra-diction that p is inconsistent.It follows that there are X , . . . , X r as above, all A -definable, and for each j = 1 , . . . , r ,there is i ( j ) ∈ { , . . . , m } , such that ∀ x x ∈ I → r _ j =1 X i ( x, c i ( j ) ) . In addition, for each j = 1 , . . . , r , dim( X j ) ≤ dim( W i ( j ) ) . By weak o-minimality, each set X j ( M, c i ( j ) ) is a finite union of convex subsets of C ,and since I is a non-empty initial segment of C , the above implies that for some fixed j , theset X j ( M, c i ( j ) ) contains an initial segment of C . For simplicity, we write X for X j and IELDS INTERPRETABLE IN REAL CLOSED VALUED FIELDS AND SOME EXPANSIONS 5 assume that i ( j ) = 1 . Namely, we assume that X ( M, c ) contains an initial segment of C .This is an A -definable property of c , therefore we may assume, after possibly shrinking W , that for every w ∈ W , the set X ( M, w ) contains an initial segment of C . Moreover,by choosing in each X ( M, w ) the first convex component, we may assume that for every w ∈ W , the set X ( M, w ) contains a single convex set, which is an initial segment of C .For b ∈ C , let X b = { w ∈ W : h b, w i ∈ X } . Notice that S b ∈ C X b = W . Ourassumptions imply that for b ≤ b ∈ C , we have X b ⊆ X b . It now follows from lemma2.5 (with the ordering < reversed) that for some b ∈ C we have dim( X b ) = dim( W ) .Also, our assumptions imply that for every b < b in C and w ∈ X b we have ( h b, w i ∈ X , namely C dim W . But by ourassumption, dim( X ) ≤ dim( c /A ) = dim( W ) . This is a contradiction and the lemma isproved. (cid:3) As a corollary we obtain:
Corollary 2.7.
Assume that M has definable Skolem funcions, and let X ⊆ M n be an A -definable set. Let c , . . . , c m be tuples from M (of possibly different lengths). Then thereexists b ∈ X such that for each i = 1 , . . . , m , dim( bc i /A ) = dim( X ) + dim( c i /A ) . Proof.
By cell decomposition we may assume that X is an open subset of M n . We useinduction on n .The case n = 1 follows from Lemma 2.6. For a general n , note first that X containsan A -definable open box I × · · · × I n . Indeed, since X is open it contains an open boxand by the existence of Skolem functions we can find such a box defined over A . Applyinginduction, we can find a ′ = h a , . . . , a n i ∈ I × · · · × I n such that for each i = 1 , . . . , m , dim( a ′ c i /A ) = ( n −
1) + dim( c i /A ) .We now let c ′ i = h a ′ , c i i , for i = 1 , . . . , m , and using again the case n = 1 , find a ∈ I such that , for all i dim( a c ′ i /A ) = n + dim( c i /A ) . The tuple h a a ′ i is the desired b . (cid:3) In the above existence of Skolem functions somewhat simplifies the proof, but is not, infact, needed.2.1.
Dimension and domination in quotients by convex equivalence relations.
We es-tablish here several properties of quotients of one-dimensional sets by convex equivalencerelations. Recall:
Definition 2.8.
An equivalence relation E on a linearly ordered set ( J, < ) is convex if allits equivalence classes are.If M is weakly o-minimal, E a definable convex equivalence relation with infinitelymany classes then M/E is linearly ordered and itself weakly o-minimal. If in addition
T h ( M ) is weakly o-minimal then so is T h ( M /E ) . It follows, in particular (see nextsection), that if h K ; O i | = T conv then the theories of K , K/O , Γ and k are weakly o-minimal.Let M be a weakly o-minimal structure, E a definable convex equivalence relation on M , and let π : M → M/E be the quotient map. For simplicity we also denote by E theequivalence relation on M n given by E n and let π : M n → ( M/E ) n be the associatedquotient map. We prove here a domination result for such quotients. ASSAF HASSON AND YA’ACOV PETERZIL
Lemma 2.9.
Let I , . . . , I n ⊆ M be open convex sets such that each I j /E is infinite, and X ⊆ B = I × · · · × I n . If π ( X ) = π ( B ) then dim( π ( B \ X )) < n .Proof. Let us say that a definable set X crosses an E -class [ a ] if both X and X c intersect [ a ] .We proceed by induction on n . The result for n = 1 follows from weak o-minimality of K/E : Indeed, if X ⊆ I is definable then, since X consists of at most r -convex sets andeach E -class is convex, then X crosses at most r -many classes. Thus, since X intersectsall E -classes, π ( I \ X )) is finite.Assume now that X ⊆ M n and π ( X ) = π ( B ) . let Y = B \ X , and assume towardsa contradiction that dim( π ( Y )) = n . This implies that π ( Y ) has nonempty interior, andtherefore contains a box. The pre-image of such a box is itself a box (since E is convex,and we may assume that its domain is an open box). Thus there exists an open box B ⊆ Y such that π ( B ) is an open subset of π ( X ) . By our assumption, we also have π ( B ∩ X ) = π ( B ) = π ( B ∩ Y ) .Thus we may assume, by replacing B with B , that π ( X ) = π ( Y ) = π ( B ) . To simplifynotation let B ′ = I × · · · × I n − . We define X ∗ = {h a, b i ∈ B ′ × I n : X a crosses [ b ] } . By the case n = 1 , for each a ∈ B ′ , X a crosses at most finitely many E -classes in I n . Itfollows that π ( X ∗ ) ⊆ π ( B ) projects finite-to-one into π ( B ′ ) , and thus dim( π ( X ∗ )) < n .Hence, there exists an open box R ⊆ π ( B ) which is disjoint from π ( X ∗ ) . By replacing B with π − ( R ) we may assume that X ∗ = ∅ . Namely, for every h a, b i ∈ B ′ × I n , either X a contains [ b ] or it is disjoint from [ b ] .For every b ∈ I n , we let X b = { a ∈ B ′ : h a, b i ∈ X } . We claim that for each b ∈ I n , π ( X b ) = π ( B ′ ) . Indeed, if a ∈ B ′ then there is h a ′ , b ′ i ∈ X such that h a ′ , b ′ i E h a, b i (since π ( X ) = π ( B ) ). By our assumption, X a ′ does not cross [ b ′ ] , therefore [ b ′ ] is contained in X a ′ . We have bEb ′ , hence h a ′ , b i ∈ X . It follows that π ( a ) = π ( a ′ ) ∈ π ( X ) . Our inductionassumption thus implies that dim( B ′ \ X b ) < n − .We also claim that for b Eb , we have X b = X b (and hence also B ′ \ X b = B ′ \ X b ).Indeed, if a ∈ X b then X a ∩ [ b ] = ∅ and therefore [ b ] ⊆ X a , so b ∈ X a , namely a ∈ X b . The opposite inclusion follows.Thus, for every b ∈ I n , we have π ( B \ X ) π ( b ) = π ( B ′ \ X b ) , which has dimension < n − . It follows that for every c ∈ π ( I n ) , dim( π ( B \ X ) c ) < n − . By the sub-additivity of dimension, dim( π ( B \ X )) < n , contradicting our assumption. (cid:3) The following could be viewed as domination of types in M n by generic types in M n /E . Proposition 2.10.
In the above setting, assume that p ⊢ ( M/E ) n is a complete generictype over A ⊆ M . Then q = π − ( p ) is a complete type over A .Proof. If q is not complete then there is an A -definable set X ⊆ M n such that q ⊢ π ( X ) and q ⊢ π ( X c ) . But then dim( π ( X )) = dim( π ( X c )) = n , contradicting Lemma 2.9. (cid:3)
3. T
HE THEORY T conv Let T be an o-minimal expansion of a real closed field in a signature L . The theory T conv was introduced by Lowenberg and v.d. Dries in [29]: Given K | = T , a T -convexsubring of K is a subring O ⊆ K with the property that f ( O ) ⊆ O for any ∅ -definablecontinuous function f : K → K . Since O is convex it is a valuation ring, its maximalideal is denoted µ , the value group is Γ and the residue field is k . The theory T conv , in the IELDS INTERPRETABLE IN REAL CLOSED VALUED FIELDS AND SOME EXPANSIONS 7 language L conv := L ∪ { O } , is the extension of T by the axioms saying that O is a propervaluation ring which is T -convex. Here are some results from [29] and [26]: Fact 3.1. (1)
The theory T conv has quantifier elimination relative to T , it is completeand weakly o-minimal. ( [29, 3.10, 3.13,3.14] .Let T conv,c be the extension of T conv by the formula c / ∈ O , for a new constant c .Then: (2) The theory T conv,c has definable Skolem functions ( [26, Remark 2.4] ). (3) In models of T conv,c the definable closure in L and in L conv are the same (see [26, Lemma 2.6] ). Moreover, every L conv -definable function coincides with an L -definable function around generic points of the domain.From now on, we add a constant symbol c interpreted as a positive element outside O ,and still denote the new language L conv . Throughout the text we will be using the aboveresults without further reference.It will be convenient to define in L conv the relation v ( x ) > v ( y ) if x/y ∈ O ∧ y/x / ∈ O .Since O = { x : v ( x ) ≥ v (1) } , quantifier elimination for T conv in L conv is equivalent toquantifier elimination for L expanded by the the above binary predicate, and we will use thetwo languages interchangeably.The following could probably be read off [26], but we need the explicit formulationbelow. It is a consequence of quantifier elimination relative to T : Lemma 3.2.
Let S ⊆ K n be L conv -definable, a ∈ S generic over the parameters defining S (in the weakly o-minimal sense). Then there exists an open box B ∋ a (defined possiblyover new parameters) such that B ∩ S is L -definable.Proof. We first note that the associated valuation induces the same topology on K as theorder topology. Quantifier elimination implies that any definable set is of the form [ i ¯ T i (¯ x ) ∩ \ j ( t i,j (¯ x ) ∈ O ) ± where ¯ T i are L -definable sets, t i,j (¯ x ) are L -terms and ( x ∈ O ) − means x / ∈ O . It sufficesto verify the lemma for sets of the form ¯ T (¯ x ) ∩ \ j ( t j (¯ x ) ∈ O ) ± The set O is clopen in K and since a is generic with respect to the o-minimal structure,each t i,j defines a continuous function at a . It follows that a is an interior point of boththe preimage of O and its complement under each t j . Thus there is an open box B ⊆ K n containing a such that B is contained in the set T j ( t j (¯ x ) ∈ O ) ± . We have T ∩ B = S ∩ B ,so it is L -definable. (cid:3)
4. D
EFINABLE QUOTIENTS OF K In this section we classify, locally, definable quotients of K itself. The first lemma willeventually allow us to reduce much of the work from quotients of the form K n /E to quo-tients of the form ( K/E ) n . ASSAF HASSON AND YA’ACOV PETERZIL
Lemma 4.1.
Let M be a weakly o-minimal structure with definable Skolem functions. Let S = X/E , for some definable X ⊆ K n and E a definable equivalence relation. If S isinfinite then there exists an infinite S ′ ⊆ S (definable over additional parameters) suchthat S ′ is in definable bijection with J/E ′ for some interval J ⊆ M and E ′ a definableequivalence relation on M .Proof. We use induction on n , where the case n = 1 is trivial. Let W := π ( X ) be theprojection of X onto the last n − coordinates (assuming, without loss of generality, thisprojection has infinite image). If for some w ∈ W the fibre π − ( w ) meets infinitely many E -classes then take S ′ = X w = { x ∈ M : h x, w i ∈ X } and E ′ the relation on S ′ obtainedfrom π − ( w ) .So we may assume that | π − ( w ) /E | < ∞ for all w ∈ π ( X ) . Since M has definableSkolem function we can find, uniformly for each w ∈ W a finite set of representatives Y ( w ) of π − ( w ) /E . More precisely, there exists a definable set Z ⊆ K n such that π ( Z ) = π ( X ) and for each w ∈ π ( X ) , Z ∩ π − ( w ) has exactly one representative for each E -classintersecting π − ( w ) .By weak o-minimality, there exists a cell decomposition Z , . . . , Z l of Z such that each Y i is definably homeomorphic to an open cell C i ⊆ M dim( Z i ) . Assume, without loss, that | Z /E | = ∞ . If π : Z → C is the definable homeomorphism let E := π ( E | Z ) and weconclude by induction. (cid:3) We now fix an infinite field hF ; + , ·i interpretable in K . We fix an interval J ⊆ K and adefinable equivalence relation E on J as provided by the above lemma. As we show next,after possibly shrinking J , the set J/E can be definably embedded into one of four weaklyo-minimal structures.
Theorem 4.2.
Let K | = T conv , E a definable equivalence relation on J ⊆ K with infinitelymany classes. Then there exists a definable ordering on J/E such that h J/E, < i , with theinduced structure, is weakly o-minimal, and there exists an < -interval I ⊆ J/E which is indefinable bijection with an interval in K , in Γ , in k or in K/O .Proof.
By weak o-minimality each E -equivalence class is a finite union of convex sets. Soby replacing E with the equivalence relation E ′ , choosing the first component in each E -class, we get a convex equivalence relation with D/E ′ = J/E for some definable D ⊆ J .So we assume that E = E ′ . Thus J/E is linearly ordered and weakly o-minimal.If E is the identity on an infinite set we can find a convex set I ⊆ J where this is true,and then I/E ֒ → K , so we are done. So we assume that this is not the case. It followsthat for all but finitely many classes, sup([ x ]) is not an element of K (for otherwise theset ∃ y ( x = sup[ y ]) is an infinite discrete set in K ). For the same reason inf([ x ]) is not anelement of K for all but finitely many of the E -classes. By shrinking J we may assume thatfor all x ∈ J , the class [ x ] is a bounded convex set without supremum or infimum.For a valuational ball B (either open or closed) write x < B if x < y for all y ∈ B . Wewrite x ≤ B if x < B ∨ x ∈ B . Recall (see, e.g., [17]) that a disc cut is a set of the form x (cid:3) B where (cid:3) ∈ { <, ≤} and B is a definable ball (either open or closed) or a point in K .By quantifier elimination for T conv , any non-empty K -definable subset of K is a booleancombination of a finite set of disc cuts. Since the E -classes are convex it follows from ourassumptions that they are intersections of two disc cuts where B is actually a ball (and nota point).Let us call a set of the form B (cid:3) x (cid:3) B , where (cid:3) i ∈ { <, ≤} , a ball-interval . The form of a ball interval is the data specifying which of the balls B i is open, and which of the IELDS INTERPRETABLE IN REAL CLOSED VALUED FIELDS AND SOME EXPANSIONS 9 inequalities is weak.Below we use valuational radius for balls, namely for γ ∈ Γ , an open ball of radius γ is a set of the form { x ∈ K : v ( x − x ) > γ } and a closed ball is of the form { x ∈ K : v ( x − x ) ≥ γ } . Claim . If C is a ball interval B (cid:3) x (cid:3) B then B and B are uniquely determined by C .More explicitly, B and B are definable over a code for C . Proof.
We deal first with the case where (cid:3) is ≤ . If B is a ball of radius γ (either closedor open) then γ = inf { δ : ( ∀ x ∈ C ∃ y ∈ C )( y < x ∧ B δ ( y ) ⊆ C ) } . And B = B γ ( y ) for any small enough y ∈ C . We can recognise whether B is open orclosed by verifying whether γ is attained as a minimum in the above formula or not. B isdefined similarly when (cid:3) is ≤ .The case of strong inequalities follows from the previous case by replacing I with itscomplement. (cid:3) claim.The claim allows us to definably assign to each class [ x ] the left and righ “end-balls” ofthe ball-interval, B [ x ] and B [ x ] , respectively. There are finitely many possible forms ofball-intervals, hence we may further assume that all equivalence classes [ x ] ⊆ J have thesame form. Note that if [ x ] = [ y ] then B [ x ] = B [ y ] , for otherwise [ x ] ∩ [ y ] = ∅ . So thefunction [ x ] B [ x ] is a definable injection into the family of open (closed) balls. If forsome γ ∈ Γ the fibre of the function sending [ x ] to the valuational radius of B [ x ] is infinitethen, depending on whether B [ x ] is open or closed, we get an injection of some interval in J/E into
K/O or k .If all fibres of this function are finite we get a finite-to-one map from an interval in J/E into Γ . The linear order on J/E implies the existence of an injection from an interval in
J/E into Γ , finishing the proof of Theorem 4.2. (cid:3)
5. D
EFINABLE FUNCTIONS IN
K/O
Combining Lemma 4.1 with Theorem 4.2 we conclude that there exists an infinite defin-able I in our interpretable field F such that I is in definable bijection with an interval inone of the four weakly o-minimal structures K , Γ , k or K/O . While the definable sets in K , Γ and k are well understood, the situation is less clear in K/O . Our goal in this sectionis to show that definable functions in
K/O are locally affine with respect to the additivestructure of
K/O .We start with the following general lemma.
Lemma 5.1.
Assume that p ⊢ K n is a partial L -type and that H : K n → K is a partial L -definable function such that p ⊢ dom( H ) .Assume also that C ⊆ K is an L conv -definable convex set which is bounded above (orbelow) and for every α | = p , H ( α ) ∈ C . Then there exists c ∈ C such that for every α | = p we have H ( α ) < c (or H ( α ) > c ).Proof. We are using the saturation of M conv , and assume that C is bounded above.For every L -definable set X ⊆ K n , for which p ⊢ X , let s ( X ) be the supremum of H ( W ) , which exists by o-minimality and the boundedness of C . We claim that for somesuch X we have s ( X ) ∈ C (and then we may take c = s ( X ) ). Indeed, if not then for every X in p there exists α ∈ X such that s ( α ) > C . But then, by saturation we may find α | = p such that H ( α ) > C , contradicting our assumption. (cid:3) Definition 5.2.
A subset of K is called long if it contains infinitely many cosets of O . Asubset of K n is long if it contains a cartesian n -product of long subsets of K . A type p ⊢ K n is long if every set in p is long. Definition 5.3.
We say that a (partial) function F : K n → K descends to K/O if whenever a − b ∈ O n also F ( a ) − F ( b ) ∈ O . Example 5.4. If a ∈ O then the linear function x a · x descends to an endomorphismof h K/O, + i . In the case that a ∈ µ , the map descends to an endomorphism of K/O withinfinite kernel. Thus we obtain a definable locally constant, surjective endomorphism onf
K/O .For a K -differentiable F : K n → K , and i = 1 , . . . , n , we let F x i denote the partialderivative with respect to x i , and let ∇ F = ( F x , . . . , F x n ) .We are going to use several times the following (see Exercise 3.2.19(2)): If F : K n → K is an L -definable partial function and p ⊢ dom( F ) is a generic type then the restriction of F to p ( M ) is monotone in each coordinate separately. Lemma 5.5.
Assume that F : K n → K is an L ( A ) -definable (partial) function whichdescends to K/O . Assume that p ⊢ dom( F ) is a partial long L conv ( A ) -type which impliesa complete generic L ( A ) -type in K n ,Then for every a | = p , and i = 1 , . . . , n , we have F x i ( a ) ∈ O .Proof. Because p implies a generic type in K n the function F is smooth at every a | = p .Assume towards a contradiction that the conclusion of the lemma does not hold. Withoutloss of generality, assume that F x ( a ) > O for some a = ( a , . . . , a n ) | = p .Using monotonicity of F x in the first coordinate, we can find some α > O and an in-terval J ⊆ K of length greater than such that every element in J × {h a , . . . , a n i} stillrealizes p , and for every x ∈ J , F x ( x, a , . . . , a n ) > α . By Lagrange’s Mean Value Theo-rem for o-minimal structures, for x, x +1 ∈ J , F ( x +1 , a , . . . , a n ) − F ( x, a , . . . , a n ) > α ,contradicting the assumption that F descends to K/O . (cid:3) Lemma 5.6.
Assume that G : K n → K is an L ( A ) -definable partial function, and p ⊢ dom( G ) is a long partial L conv ( A ) -type which implies a complete generic L ( A ) -type in K n . Assume also that for every a | = p , G ( a ) ∈ O . Then for all a | = p and all i ∈{ , . . . , n } , we have G x i ( a ) ∈ µ .Proof. We prove the result for i = 1 . By Lemma 5.1, there is c ∈ O such that for all a | = p we have | G ( a ) | ≤ c . Assume towards contradiction that for some a = ( a , . . . , a n ) | = p we have | G x ( a ) | > µ . Then we can find a positive α ∈ O \ µ , such that | G x ( a ) | > α .Using the montonicity of G x in the first coordinate, we may assume that for every positive b ∈ O , we have G x ( a + b, a , . . . , a n ) > α . Pick b > c/α ∈ O . Then, by Lagrange’smean value theorem in o-minimal structures, | G ( a + b, a , . . . , a n ) − G ( a , . . . , a n ) | > | G x ( ξ, a , . . . , a n )) | (2 c/α ) for ξ ∈ ( a , a + b ) . By our assumptions, | G x ( ξ, a , . . . , a n ) | > α . It follows that G ( a + b, a , . . . , a n ) − G ( a , . . . , a n ) | > c , in contradiction to the fact that G | ( x ) | ≤ c for all x | = p . (cid:3) We conclude:
IELDS INTERPRETABLE IN REAL CLOSED VALUED FIELDS AND SOME EXPANSIONS 11
Lemma 5.7.
Assume that F : K n → K is a partial L ( A ) -definable function which de-scends to K/O . Let p ⊢ dom( F ) be a long L conv ( A ) -type which implies a completegeneric L ( A ) -type. Then there are a , . . . , a n ∈ O , b ∈ K and a long box B ⊆ p ( K ) suchthat for all x = h x , . . . , x n i ∈ B , F ( x ) − n X i =1 a i ( x i − α i ) − b ∈ µ, .Proof. By Lemma 5.5, each F x i takes values in O , on the type p . By Lemma 5.6, appliedto the each of the functions F x i , we have F x i ,x j ( a ) ∈ µ for all a | = p and i, j = 1 , . . . , n .By Lemma 5.1, there is some fixed positive β ∈ µ such that | F x i ,x j ( a ) | < β for all a | = p .Fix α | = p and let ( a , . . . , a n ) = ∇ F ( α ) .By [28, Lemma 7.2.9 ], for all x | = p ,(*) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F ( x ) − F ( α ) − n X i =1 a i ( x i − α i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | x − α | · max y ∈ [ α,x ] |∇ F ( x ) − ∇ F ( α ) | , where [ α, x ] is the closed line segment in K n connecting α and x and |∇ F ( x ) | is theoperator norm.Fix an x | = p and for i = 1 , . . . , n , and t ∈ [0 , , consider the function g i ( t ) = F x i ((1 − t ) α + tx ) . The derivative of dg i dt is, by the chain rule, ( F x ,x i ( y ) , . . . , F x n ,x i ( y )) · ( x − α ) . Applying Lagrange’s Theorem to g i and substituting, we get ξ ∈ (0 , such that | F x i ( x ) − F x i ( α ) | = | g i (1) − g i (0) | = | g ′ i ( ξ ) | = | ( F x ,x i ( y ) , . . . , F x n ,x i ( y )) · ( x − α ) | , where y = (1 − ξ ) α + ξx .Since each | F x j ,x i ( y ) | < β we have | F x i ( x ) − F x i ( α ) | ≤ β | x − α | for all i = 1 , . . . , n .Thus |∇ F ( x ) − ∇ F ( α ) | ≤ nβ | x − α | , so by (*), | F ( x ) − F ( α ) − n X i =1 a i ( x i − α i ) | ≤ nβ | x − α | . Since p is long and β ∈ µ , we can find a long box B ⊆ p ( M ) , centered at α such thatfor every x ∈ B , we have | x − α | ≤ (1 /β ) / . Thus, for x ∈ B we have | F ( x ) − F ( α ) − n X i =1 a i ( x i − α i ) | ≤ n p β ∈ µ. If we now let b = F ( α ) then we have the desired result. (cid:3) In fact, for the next corollary it would have been sufficient to show above that F ( x ) − F ( α ) − P ni =1 a i ( x i − α i ) + b ∈ O . Corollary 5.8.
Assume that f : ( K/O ) n → K/O is a (partial) ∅ -definable function whosedomain is open. Then for every generic c ∈ dom( f ) there exists an open box R ⊆ dom( f ) centered at c , definable endomorphisms of h K/O ; + i denoted by ℓ , . . . ℓ n , and d ∈ K/O ,such that for every y = h y , . . . , y n i ∈ R we have f ( y ) = n X i =1 ℓ i ( y ) + d. Proof.
Let p = π − ( tp K/O ( c/ ∅ )) . It is clearly a long type and by Proposition 2.10, it isin fact a complete L conv -type which implies a generic L -type. Using definable Skolemfunctions we may lift f to an L -definable partial F : K n → K , which descends to K/O .Namely, for every x ∈ dom( F ) , we have F ( x ) ∈ f ( x + O n ) . We have p ⊢ dom( F ) .Applying Lemma 5.7, we can find a long box B ⊆ K n centered at α | = p , and a , . . . , a n ∈ O , b ∈ K , such that for every x ∈ B we have F ( x ) − b − n X i =1 a i ( x i − α i ) ∈ µ. Each function x a i · x descends to an endomorphism ℓ i of h K/O ; + i , so if we let R be the image of B in K/O and e = b + O , then we have for all y ∈ R , f ( y ) = n X i =1 ℓ i ( y i − α i ) + e = n X i =1 ℓ i ( y i ) + d, for d = e − P ni =1 ℓ i ( α i ) . (cid:3) We end this section by commenting that in the o-minimal setting the local affiness ofdefinable functions provided by Corollary 5.8, would imply that the structure is linear (inthe sense of [14]), and thus does not interpret a field. This is not true for
K/O : Proposition 5.9.
The field k is definably isomorphic to a field interpretable (in inducedstructure on) K/O .Proof.
Fix some t such that v ( t ) < . Consider the balls tO ⊇ tµ . Since O ⊆ tµ , thesubgroups tO/O and tµ/O are definable subgroups of K/O . Thus, we get that the quotient h tO/tµ ; + i is interpretable in K/O . The former definably isomorphic to h k ; + i , hence wecan definably endow it with multiplication which makes it isomorphic to h k ; + , ·i . (cid:3) As we shall show in Proposition 6.7, no field is definable in K/O .6. S
UBSETS OF F THAT ARE STRONGLY INTERNAL TO I As we already saw in Theorem 4.2, there exists an infinite definable I ⊆ F such that I isin definable bijection with an interval in one of the structures K , Γ , k or K/O . Either way,the induced structure on I is weakly o-minimal. Let us be more precise about this: Remark 6.1.
Recall that given an ℵ -saturated structure M a definable set S ⊆ M n is stably embedded if for any definable X ⊆ M nk the set X ∩ S k is definable using onlyparameters from S . Thus the M -induced structure on S does not depend on the model oron the choice of parameters.However, intervals in a weakly o-minimal structure need not be stably embedded so thatthe notion of “the structure induced on I ” is not well defined. For A ⊆ K we let I A denotethe structure obtained by taking the traces on I n , n ∈ N , of all A -definable sets. Thestructure I K has a weakly o-minimal theory. So the same is true of any ordered reduct ofthe full K -induced structure on I and in particular of reducts of the form I A . We assumethat F , I , and an embedding of I into K, k, Γ or K/O are all definable over ∅ , and initiallywork in I ∅ . We will, throughout the course of the proof, add small sets of parameters (not IELDS INTERPRETABLE IN REAL CLOSED VALUED FIELDS AND SOME EXPANSIONS 13 necessarily coming from I ), tacitly expanding the structure I ∅ in into one of the form I A .We denote the resulted structure by I . Definition 6.2.
A definable Y ⊆ F is strongly internal to I (or strongly I -internal ) if thereexists a definable injection, possibly over additional parameters, from Y into I r for some r . We now fix Y ⊆ F which is strongly internal to I and has maximal dimension, denotedby n , in the sense of I (equivalently, maximal dp-rank). We identify Y with its image in I r and view it with the induced I -topology. We assumefor simplicity that Y is definable over ∅ . Unless otherwise stated we use + , − , · , () − forthe field operations of F . Proposition 6.3.
Let h b, c, d i ∈ I × Y be such that dim( b, c, d/ ∅ ) = 2 n + 1 . Then thereexists an initial segment J ⊆ I >b and a definable S ⊆ Y , all defined over an additionalparameter set B , such that dim( c, d/B ) = 2 n , h c, d i ∈ S , (in particular, dim( S ) = 2 n ),and such that for every h x, y, z i ∈ J × S ⊆ J × Y , we have ( x − b ) y + z ∈ Y .Proof. For h x, y, z i ∈ I × Y × Y consider the function f y ( x, z ) = xy − z . Claim
The point h b, d i ⊆ I × Y ⊆ I r is not an isolated point of the set f − c ( f c ( b, d )) . Proof.
Assume h b, d i were isolated in f − c ( f c ( b, d )) . By the weak o-minimality of T h ( I ) ,there is a number r such that for every d ∈ Im( f c ) , the set f − c ( d ) has at most r -manyisolated points. Because I is linearly ordered, each of these points is in dcl( c, d ) . Thus,there is a c -definable (partial) function σ : F → I × Y , such that for every w ∈ dom( σ ) , σ ( w ) is an isolated point of f − c ( w ) , and if w = f c ( b, d ) then σ ( w ) = h b, d i . In particular, f c ◦ σ ( w ) = w , so σ is injective.The image of σ is a c -definable set T containing h b, d i on which f c is injective. It followsthat dim( T ) = n + 1 . Because I and Y were strongly I -internal then so is f c ( T ) , and wehave dim( f c ( T )) = n + 1 . This contradicts the maximality of dimension of Y . (cid:3) Thus h b, d i is a cluster point of f − c ( bc − d ) . It now follows that there exists an openinterval J = ( b, b ′ ) ⊆ I (or ( b ′ , b ) ) such that for every x ∈ J there is z ∈ Y with xc − z = bc − d . In particular, the map x xc − ( bc − d ) sends J injectively into Y . We now applyLemma 2.6 and obtain b , b < b < b ′ , such that dim( b cd/b ) = 1 + 2 n . In particular, dim( cd/b b ) = 2 n Our assumption is that for every x in the interval J = ( b, b ) , we have ( x − b ) c + d ∈ Y .This is a first order property of c, d over the parameters bb . Thus there is a bb -definable set S ⊆ Y , with h c, d i ∈ S , such that for every h y, z i ∈ S and x ∈ J , we have ( x − b ) y + z ∈ Y . It follows that dim( S ) = 2 n and h c, d i is a generic point in S over bb . We now replace J with J . (cid:3) Before we proceed we recall some definitions.
Definition 6.4.
Let M be an o-minimal expansion of real closed field R . A power functionon R is a definable endomorphism of the ordered mutiplicative group R > . M is called power bounded (generalizing “polynomially bounded”) if every definablefunction of one variable is eventually bounded by some power function. An o-minimaltheory T is called power bounded if every model of T is power bounded.By Miller’s [18], M is power bounded if and only if it is not exponential, namely onecannot define a (necessarily monotone) isomorphism of ( R, +) and ( R > , · ) . While the structure of models of T conv is not well understood in case T is exponentialwe can still show the following : Lemma 6.5.
Let T be an exponential o-minimal expansion of a real closed field, T conv its expansion by a T -convex valuation ring. If K | = T then h K/O ; <, + i is in definablyisomorphic with h Γ; <, + i .Proof. We first note that exp( O ) = O > \ µ . Indeed, the right-to-left inclusion follows fromthe fact that O is T -convex, and exp ( x ) > x . For the converse, assume for contradictionthat a = exp( b ) ∈ µ for some b ∈ O . Then a − = exp ( − b ) / ∈ O , contradicting the factthat O is T -convex.Since O \ µ = O × it follows that exp induces a definable isomorphism between h K/O ; <, + i and h K > /O × > ; <, + i , which is isomorphic to h Γ; <, + i . (cid:3) The structure of the residue field and the value group of T conv is described by the workof v.d. Dries: Fact 6.6 (Theorem A, Theorem B) . [26] ] Let T be an o-minimal expansion of a real closedfield R and O a T -convex subring of R . Then: (1) The residue field with its induced structure can be given a structure of a T -model,and it is stably embedded as such. In particular it is o-minimal, so by [19] , any fieldinterpretable in the residue field is definably isomorphic to the residue field itself orto its algebraic closure. (2) If, in addition, T is power bounded, then the value group is, up to a change ofsignature, an ordered vector space over the field of exponents of T and is stablyembedded as such. To sum up the last fact in few words, if we denote k ind the induced structure on k (inthe signature L ind ) then k ind is o-minimal, and up to a change of signature elementarilyequivalent to K . In case T is power bounded, the value group, too, is an o-minimal vectorspace with no additional structure. From now on, we denote the induced structure on k by k ind and its language by L ind . Proposition 6.7.
The set I is neither a subset of K/O nor a subset of Γ .Proof. Assume for a contradiction that I is either a subset of K/O or of Γ , and denotethese by V . In both cases V has an underlying ordered group structures (which is in fact anordered Q -vector space). In order to distinguish it from the additive structure of the field wedenote here the additive structure on V by h V ; ⊕i . We still use + , · to denote the operationsof F .Let h b, c, d i ∈ I × Y be such that dim( b, c, d/ ∅ ) = 2 n + 1 We fix J ⊆ I and S ⊆ Y asprovided by Proposition 6.3. For simplicity we assume that all these sets are definable over ∅ . For h x, y, z i ∈ J × S , we let F ( x, y, z ) = ( x − b ) y + z . Our choice of S and J assuresthat F takes values in Y .Using cell decomposition in either K/O or Γ we may assume that S is an open subset of I s . Claim.
There exists an additive (with respect to ⊕ ) function L : V n → V s and e ∈ V ,such that F ( x, y, z ) = L ( x, y, z ) + e on a definable subset of J × S of the same dimension. Notice that there is an error in [26, §6] in the discussion of the exponential case. See [27] for more details.
IELDS INTERPRETABLE IN REAL CLOSED VALUED FIELDS AND SOME EXPANSIONS 15
Proof.
The claim follows from Corollary 5.8 if V is K/O . If T is exponential then byLemma 6.5, Γ is definably isomorphic to K/O , so the claim holds for Γ as well. If T ispower bounded then by Fact 6.6, Γ with its induced structure is an o-minimal vector spaceover the field of definable exponents. In that case the result follows from quantifier elimi-nation for ordered vector spaces. (cid:3) Claim
For simplicity of notation we still denote the set provided by the last claim as J × S .By the definition of the function F in the field F , for every h y , z i 6 = h y , z i , there isat most one x such that F ( x, y , z ) = F ( x, y , z ) .Fix x ∈ J ⊆ V such that x = 0 V . The map h y, z i 7→ F ( x , y, z ) sends Z × Z into Y . Since dim( Z × Z ) = 2 n > n , it follows that the map cannot be injective and hencethere are h y , z i 6 = h y , z i for which F ( x , y , z ) = F ( x , y , z ) .It follows that L ( x , y , z ) = L ( x , y , z ) , hence L ( x , y , z ) ⊖ L ( x , y , z ) = 0 .The function T : x L ( x, y , z ) ⊖ L ( x, y , z ) : V → V s is an additive function from V into V s , and x ∈ ker( T ) . It follows that Z x ⊆ ker( T ) , so by weak o-minimality, ker( T ) is a convex subgroup of ( V, ⊕ ) . Thus T is locally constant. This implies that F ( x, y , z ) = F ( x, y , z ) for infinitely many x , a contradiction. (cid:3) From now on we assume that I ⊆ K or I ⊆ k .7. F IELDS INTERPRETABLE IN T conv We fix Y a definable subset of F strongly internal to I and of maximal I -dimension. Asin the previous section, we identify Y with a subset of I s , and equip it with the topologyinduced from the weakly o-minimal topology of I s . For simplicity we assume that I and Y are ∅ -definable.7.1. Infinitesimal neighborhoods.
We introduce the notion of infinitesimals with respectto the ambient weakly o-minimal structures ( K , or k ):Given a ∈ I , the infinitesimal neighborhood of a (with respect to the ambient weaklyo-minimal structure I ) is the partial type over M consisting of all M -definable I -open setscontaining a . Equivalently, it is the collection { c < x < d : c < a < d, c, d ∈ I } . Wedenote it by ν ( a ) . For a = ( a , . . . , a n ) ∈ I n , we let ν ( a ) = ν ( a ) × · · · × ν ( a n ) . Notethat ν ( a ) is a partial type in the language < (in either K or k ).For a definable Y ⊆ I s , and a ∈ Y , we let ν Y ( a ) = Y ∩ ν ( a ) . In the case where I ⊆ k , Y is definable in the induced o-minimal language on k and therefore the type ν Y ( a ) is givenby L ind -formulas. As we now note, this is also the case when I ⊆ K : Remark 7.1.
Assume that I ⊆ K , Y ⊆ I s and a is generic in Y . By Lemma 3.2, there isan L -definable set T and an open box B ∋ a such that B ∩ Y is L -definable. Thus, the type ν Y ( a ) is given by L -formulas.Finally, note that if f : I r → I s is an I -definable function continuous at a ∈ I r , then f ( ν ( a )) ⊆ ν ( f ( a )) .7.2. The infinitesimal subgroup of hF , + i associated to Y . Throughout + , − , · and ( ) − denote the operations in F . Lemma 7.2. If d is generic in Y then ν Y ( d ) − d is a type definable subgroup of ( F , +) .Moreover, for every d , d both generic in Y over ∅ , we have ν Y ( d ) − d = ν Y ( d ) − d . Proof.
We fix any b ∈ I and c ∈ Y such that dim( b, c, d ) = 2 n + 1 and apply Proposition6.3. It follows that the b -definable set {h x, y, z i ∈ I × Y × Y : ( x − b ) y + z ∈ Y } has dimension n + 1 and contains a set of the form J × S with J an interval ( b, b ′ ) , dim( S ) = 2 dim Y and h c, d i generic in S over bb ′ .Apply Lemma 2.6 to obtain b ∈ J such that dim( b , c, d/bb ′ ) = 2 n + 1 . Let Y ′ =( b − b ) Y and c ′ = ( b − b ) c . Note that Y ′ is strongly internal to I and that c ′ is inter-definable with c over b , b . Therefore dim( c ′ , d/b , b ) = 2 n , i.e., h c ′ , d i is generic in Y ′ × Y over b b . Since I is weakly o-minimal it follows (see [15, Theorem 4.8, Theorem 4.11])that the function ( y, z ) y + z , from Y ′ × Y ⊆ I s into Y ⊆ I s , is I -continuous at h c ′ , d i .In particular, it sends the types ν Y ′ ( c ′ ) × ν Y ( d ) into ν Y ( c ′ + d ) . We now work in an elementary extension of M and realize the various infinitesimal typesthere .The function y + z is injective in each coordinate and therefore for every y ∈ ν Y ′ ( c ′ ) the map z y + z induces a bijection between ν Y ( d ) and ν Y ( c ′ + d ) . Similarly, for every z ∈ ν Y ( d ) , the function x x + z induces a bijection of ν Y ( c ′ ) and ν Y ( c ′ + d ) . Finally,for every w ∈ ν Y ( c ′ + d ) the function y w − y induces a bijection between ν Y ′ ( c ′ ) and ν Y ( d ) .This shows that for every y ∈ ν Y ′ ( c ′ ) and w ∈ ν Y ( c ′ + d ) there exists z ∈ ν Y ( d ) suchthat w = y + z . Similarly we obtain that ν Y ′ ( c ′ ) − c ′ = ν Y ( d ) − d = ν Y ( c ′ + d ) − ( c ′ + d ) andthey are all subgroups of ( F , +) . E.g., if x, y ∈ ν ( d ) − d then x + c ′ ∈ ν Y ( c ′ ) , y + d ∈ ν Y ( d ) and therefore ( x + y + c ′ + d ) ∈ ν Y ( c ′ + d ) , so that x + y ∈ ν Y ( c ′ + d ) = ν Y ( d ) . Closureof ν Y ( d ) under inverses is proved similarly.In particular, the above shows that whenever h c, d i is generic in Y × Y then ν Y ( c ) − c = ν Y ( d ) − d . Since either Y ⊆ K s or Y ⊆ k s we know that Y is definable in a weakly o-minimal structure with definable Skolem functions. Therefore, given any d , d ∈ Y , eachgeneric in Y , we can find, using Corollary 2.7 c ∈ Y such that both h c, d i and h c, d i aregeneric in Y × Y . It follows that ν Y ( d ) − d = ν Y ( c ) − c = ν Y ( d ) − d . (cid:3) We may thus associate to every strongly I -internal Y of maximal dimension a type-definable subgroup ν Y of ( F , +) , which is any of ν Y ( c ) − c for c generic in Y . We nowconclude: Lemma 7.3.
For every Y , Y strongly internal to I of maximal dimension, we have ν Y = ν Y , as partial types in F .Proof. Let Y be the disjoint union of Y and Y (both embedded in some I r ). Then Y isstrongly internal to I and thus for every generic of Y or of Y can be viewed as a genericof Y . The result follows from Lemma 7.2. (cid:3) We let ν I denote the type-definable subgroup ν Y of ( F , +) associated with (any) strongly I -internal set Y ⊆ F of maximal dimension. We still work with a fixed such Y , but nowreplace it with Y − a for a generic in Y , so that ν I = ν Y (0) . Because ν Y (0) is an additivesubgroup of F , there is a definable Y ′ ⊆ Y , with ν Y (0) ⊢ Y ′ , such that − Y ′ ⊆ Y and Y ′ + Y ′ ⊆ Y . We replace Y with Y ′ ∩ − Y ′ , so now − Y = Y and Y + Y is still I -internal.Our next goal is to show that the field F equals to the set Y · ( Y \ { } ) − (thus in-terpretable in I ) and that in fact, it can be made definable in the o-minimal language ofeither K or k (without the valuation). For that purpose we apply several times ideas fromMarikova’s work, [16]. We start with a direct analogue of Lemma 2.10 in her article. IELDS INTERPRETABLE IN REAL CLOSED VALUED FIELDS AND SOME EXPANSIONS 17
Lemma 7.4.
Assume that a ∈ Y is generic over the parameters defining Y . Then thefunctions ( x, y, z ) x − y + z and ( x, y, z )
7→ − x + y − z are I -continuous at ( a, a, a ) .Furthermore, if I ⊆ K then the functions are L -definable in a neighborhood of ( a, a, a ) .Proof. Fix c ∈ Y such that h c, a i is generic in Y . By our assumptions on Y , c + a, c − a, − a and − c are all in I s . We view the map x − y + z via a composition of maps, and userepeatedly the weakly o-minimal fact that definable functions are continuous at genericpoints of their domains and, in the case that I ⊆ K , use Lemma 3.1(4) which says that L conv - definable functions coincide with L -definable functions on neighborhoods of genericpoints of their domain. ϕ : ( x, y, z ) ( c + x, − y, z ) : Y → Y ,ϕ : ( x, y, z ) ( x + y, z ) : Y → Y ,ϕ : ( x, y ) ( − c, x + y ) : Y → Y × Y,ϕ : ( x, y ) x + y : Y × Y → Y. Since a is generic in Y , the function x
7→ − x is continuous near a and since dim( c, a ) =2 n , the function x c + x is continuous near a . Thus, ϕ is continuous near ( a, a, a ) and sends ( a, a, a ) to ( c + a, − a, a ) . We now have ( c + a, − a ) generic in Y thus ϕ iscontinuous near ( c + a, − a, a ) and sends it to ( c, a ) .We proceed in the same way, and at each step use the fact that we work near a genericpoint, to conclude that ϕ i is continuous locally. Finally, x − y + z = ϕ ϕ ϕ ϕ ( x, y, z ) iscontinuous near ( a, a, a ) .In the case that I is a subset of K , the same argument shows that x − y + z and − x + y − z is L -definable near ( a, a, a ) . Applying this result to the point ( − a, − a, − a ) we get the sameresult for − x + y − z . (cid:3) We now proceed with our proof:
Proposition 7.5. (1)
For every nonzero c ∈ F ( M ) , the partial types c · ν I and ν I areequal. (2) The set of realizations of ν I in an elementary extension is a subring of F .Proof. To see (1), notice that for every nonzero c ∈ F , the set cY is also strongly internal,of the same dimension, hence by Lemma 7.3, cν Y = ν cY = ν Y , (as partial types).(2). We first argue in M and fix a generic in Y . Since aν Y = ν Y , there is, by com-pactness, a definable subset Y ⊆ Y such aY ⊆ Y . We fix b ∈ Y such that h a, b i isgeneric in Y . Since ν Y ( a ) − a does not depend on the choice of a it suffices to show that ( ν Y ( a ) − a )( ν Y ( b ) − b ) = ν Y ( ab ) − ab .We now work in an elementary extension of M , with realizations of the various partialtypes. For any x ∈ ν Y ( a ) and y ∈ ν Y ( b ) we have ( x − a )( y − b ) = xy − xb − ay + ab .Since h a, b i is generic in Y , F -multiplication is continuous near h a, b i , hence xb, ay, xy ∈ ν Y ( ab ) . Since a, b are independent generic also ab is generic in Y . Thus by Lemma 7.4, xy − xb + ab ∈ ν Y ( − ab ) . Thus, ( x − a )( y − b ) ∈ ν Y ( − ab ) + ab = ν ( I ) . (cid:3) We can now conclude the proof of our main theorem:
Theorem 7.6.
The field F is definably isomorphic to a definable field in k ind or to an L -definable field in K .Proof. Let ν I = ν Y (0) be as above. By Proposition 7.5, the partial type ν I is invariantunder mutiplication by scalars from F . For every c ∈ F , the set cY ∩ Y is infinite (as both sets contain ν I ). Take a nonzero s ∈ cY ∩ Y and then s = cr for some nonzero r ∈ Y andso c = s ( r − ) . It follows that(1) F = { xy − : x, y ∈ Y y = 0 } = Y ( Y \ { } ) − . In fact, since ν Y ( a ) − a = ν I for every generic a ∈ Y , the above shows that for twogeneric elements a, b ∈ Y , we have F = ( Y − a ) · (( Y \ { b } ) − b ) − . Moreover, we mayreplace Y by any relatively open neighborhoods U, V ⊆ Y of a and b respectively, and then F = ( U − a )( V ∗ − b ) − , where V ∗ = V \ { b } .Because Y is strongly internal to I , the above already implies that F is interpretable in I . Our standing assumption is: I ⊆ K or I ⊆ k . When I ⊆ k we can use elimination ofimaginaries in the o-minimal structure k ind to deduce that F is definable in k ind . So weare left with the case I ⊆ K , and we want to prove that F is definably isomorphic to an L -definable field in K .We fix h a, b i generic in Y such that a · b ∈ Y . By (1) and the comment right after it, forevery K -neighborhoods U ∋ a and V ∋ b , we have ( U − a )( V ∗ − b ) − = F . We define on U × V ∗ the following equivalence relation: h x , y i ∼ h x , y i if ( x − a )( y − b ) − = ( x − a )( y − b ) − ⇔ ( x − a )( y − b ) = ( x − a )( y − b ) . Note that there is a definable bijection between U × V ∗ / ∼ and F , given by [ h x, y i ] ( x − a )( y − b ) − . Claim 7.7.
There are L -definable U ∋ a and V ∋ b such that ∼ is L -definable on U × V ∗ ,and U × V ∗ / ∼ has an L -definable set of representatives.Proof. Unravelling the definition of ∼ , we obtain: h x , y i ∼ h x , y i ⇔ x y − ay − x b = x y − ay − x b. The function h x, y i 7→ xy is L -definable near h a, b i because of genericity. Thus, y ay is L -definable near b and x xb is L -definable near a . Finally, by Lemma 7.4, the map h x, y, z i 7→ − x + y − z is L -definable near the point ( ab, ab, ab ) . It follows that the function h x, y i 7→ xy − ay − xb is L -definable near h a, b i , thus ∼ is an L -definable relation near h a, b i .We fix L -definable relatively open neighbourhoods U, V ⊆ Y of a and b , respectively,such that the restriction of ∼ to U × V ∗ is L -definable. We require further, using Lemma7.4, that the functions ( x, y, z ) x − y + z and ( x, y, z )
7→ − x + y − z are L -definableon U · V . Using definable choice in o-minimal structures we can thus find an L -definableset of representatives, call it S , for ∼ . (cid:3) We have so far a definable bijection between F and S , an L -definable set in K n . Namely, F is definably isomorphic to an L conv - definable field (as the field operations might still bedefinable in L conv ). Our goal is to show that F is definably isomorphic to an L -definable field. One ap-proach for doing that is by noticing that the proof of the analogous result, [1, Theorem 4.2],in the case of real closed valued fields, works word-for-word for T conv . However, for thesake of completeness we give a different, self-contained proof that we can endow S with L -definable field operations making F into a L -definable field. We first need: IELDS INTERPRETABLE IN REAL CLOSED VALUED FIELDS AND SOME EXPANSIONS 19
Claim 7.8.
Given h x, y i ∈ ( U \ { a } ) × ( V \ { b } ) , for all Y -neighborhoods U ∋ a and V ∋ b sufficiently small, and for all a ∈ U \ { a } and b ∈ V \ { b } there are unique x ∈ U and unique y ∈ V such that h x, y i ∼ h a , y i ∼ h x , b i . Moreover, the two families of functions F x,y ( b ) = x and G x,y ( a ) = y are L -definable, as h x, y i varies.Proof. The uniqueness is clear from the definition. Let c = ( x − a ) − ( y − b ) . The sets c ( Y − a ) and ( Y − b ) both contain ν I . Thus, for a ∈ ν Y ( a ) there exists y ∈ ν Y ( b ) suchthat c ( a − a ) = y − b . It now follows that ( x − a )( y − b ) − = ( a − a )( y − b ) − , so h x, y i ∼ h a , y i .Similarly, there exists x ∈ ν Y ( a ) such that h x, y i ∼ h x , b i . The existence of U , V follows by compactness. Because S is L -definable, the map h x, y, b i 7→ x and the map h x, y, a i 7→ y are L -definable on their appropriate domains. (cid:3) We can now show that the field operations, as induced from F on the L -definable set S ,are L -definable. Given U ∋ a and V ∋ b , we let U ∗ = U \ { a } and V ∗ = V \ { b } . The definability of addition.
For h x , y i , h x , y i , h x , y i ∈ S , let M + (( x , y ) , ( x , y )) = h x , y i if ( x − a )( y − b ) − +( x − a )( y − b ) − = ( x − a )( y − b ) − . If x = a then M + (( x , y ) , ( x , y )) = h x , y i . Similarly, M + (( x , y ) , ( a, y )) = h x , y i . We thus assume that x ∈ U ∗ and y ∈ V ∗ .By Claim 7.8, we can replace h x , y i with h F x ,y ( y ) , y i and , h x , y i with h F x ,y ( y ) , y i ,uniformly and L -definably in the parameters. So we may assume that y = y = y . Setting x ′ i := F x i ,y i ( y ) we get M + (( x , y ) , ( x , y )) = h x , y i ⇔ ( x ′ − a ) + ( x ′ − a ) = ( x − a ) ⇔ x ′ − x ′ + x = a. By the choice of U , we may apply Lemma 7.4, to get that this is an L -definable relation. The definability of multiplication.
For h x , y i , h x , y i , h x , y i ∈ S , let M • (( x , y ) , ( x , y )) = h x , y i if ( x − a )( y − b ) − · ( x − a )( y − b ) − = ( x − a )( y − b ) − . To see that M • is L -definable, we first fix a ∈ U ∗ and b ∈ V ∗ such that a − a = b − b .Using F , as above, we can find x ′ such that h x ′ , y i ∼ h x , y i . Thus M • (( x , y ) , ( x , y )) = h x , y i ⇔ ( x − a )( y − b ) − · ( x ′ − a ) = x − a. We can now change, L -definably, h x , y i in a similar way to an equivalent h x , y ′ i sothat M • (( x , y ) , ( x , y )) = h x , y i ⇔ ( y ′ − b ) − · ( x ′ − a ) = 1 . We now use the fact that ( a − a )( b − b ) − = 1 to conclude: M • (( x , y ) , ( x , y )) = h x , y i ⇔ h x ′ , y ′ i ∼ h a , b i . Thus M • is L -definable, concluding the proof of the theorem. (cid:3) For the sake of clarity we now sum up everything done up until this point to get a com-plete proof of Theorem 1:
Corollary 7.9.
Let T be an o-minimal expansion of a field, T -conv a T -convex expansionof T . Any field F which is interpretable in K | = T conv is definably isomorphic to one of K , K ( √− , k , or k ( √− Proof.
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