aa r X i v : . [ m a t h . L O ] J u l DEFINABLE GROUPS IN TOPOLOGICAL FIELDS WITH AGENERIC DERIVATION
FRANC¸ OISE POINT ( † ) Abstract.
We continue the study of a class of topological L -fields endowed with ageneric derivation δ , focussing on describing definable groups. We show that one canassociate to an L ∪ { δ } definable group a type L -definable topological group. We usethe group configuration tool in o-minimal structures as developed by K. Peterzil. Introduction
Let K be an L -structure expanding a field of characteristic 0, endowed with a non-discrete definable field topology, as introduced by A. Pillay in [12]. We assume that K is a model of an L -open theory T of topological fields (see section 2.1 for a precisedefinition). The language L is a (multisorted) language which is, on the field sort,a relational expansion of the ring language possibly with additional constants (withfurther assumptions on relation symbols, as defined in [7], [2]). In particular the theory T is a complete L -theory admitting quantifier elimination on the field sort.Examples of such theories T are the theories of algebraically closed valued fields,real-closed fields, real-closed valued fields, p-adically closed fields or henselian valuedfields of characteristic 0. Note that the first four theories are not only dp-minimal butrespectively C -minimal, o-minimal, weakly o-minimal, p -minimal.W. Johnson established a link between topological fields and dp-minimal fields. Heshowed that if K is an expansion of a field, K infinite, and dp -minimal but not stronglyminimal, then K can be endowed with a non-discrete definable field topology, namely K has a uniformly definable basis of neighbourhoods of zero compatible with the fieldoperations [9, Theorem 9.1.3]. Moreover, in models of the theory of K , the topologicaldimension coincides with the dp-rank. This definable field topology is a V-topologyand so it is induced either by a non-trivial valuation or an absolute value.Here given an L -open theory T of topological fields, we consider the generic expan-sion of a model of T with a derivation δ . Namely, letting L δ := L ∪ { − } ∪ { δ } and T δ the L δ -theory consisting of T together with the axioms expressing that δ is a deriva-tion, we consider the class of existentially closed models of T δ . When T is a theory ofhenselian fields of characteristic 0, we identify the class of existentially closed models of T δ (see [2, Corollary 3.3.4] for a precise statement): we give an explicit axiomatisation T ∗ δ and we show various transfer of model-theoretic properties from the theory T to Date : July 24, 2020.1991
Mathematics Subject Classification.
Key words and phrases. open core, differential field, definable groups, group configuration.( † ) Research Director at the ”Fonds de la Recherche Scientifique (F.R.S.-F.N.R.S.)”. ( † ) the theory T ∗ δ . An easy but quite useful result is that T ∗ δ admits quantifier eliminationas well (on the field sort).The main aim of the present paper is to show that given an L δ -definable group (onthe field sort) in a model of T ∗ δ , one can associate a type L -definable group. There aretwo steps in the proof. First we give a more direct and slightly more informative andmore general proof that T ∗ δ has the L -open core property [2, Theorem 6.0.8]. Second weperform two constructions. Both were first done for o-minimal expansions of a group.The first construction is due to A. Pillay [13] who, on a definable group, put a definabletopology for which the group operations become continuous and the other one is dueto K. Peterzil who, starting from a group configuration, constructed a type definablegroup. In order to perform those constructions, besides the L -open core property, oneuses that the topological dimension function is a well-behaved dimension function inmodels of T [2, Proposition 2.4.1, Corollary 2.4.5] (the topological dimension coincideswith the algebraic dimension, has the exchange property and is a fibered dimensionfunction [4]). Finally one uses the transfer of the elimination of imaginaries from T to T ∗ δ [2, Theorem 4.0.5] under the L -open core property of T ∗ δ . In [2], in the case thetopology on the models of T is given by a valuation, we need an intermediate resulton continuity almost everywhere of definable functions to the value group and weonly show it holds under certain conditions on the value group [2, Proposition 2.6.11].Here we proceed more directly constructing, what we call an L -definable envelop ofan L δ -definable subset. It will help us to associate with an L δ -definable group a largedefinable L -definable subset of the so-called envelop on which we can recover the groupoperations on the differential points.The contents of the paper are as follows. In section 2, we review the propertiesthat we need on the topological dimension in models of an L -open theory; we recallthe definition of the theory T ∗ δ and various transfer properties between T and T ∗ δ . Insection 3, we give a direct proof of the L -open core property of T ∗ δ (Corollary 3.9):given a L δ -definable set X , one constructs a L -definable set that we call an L -openenvelop (Proposition 3.8). In section 4, we prove our main result, namely we show howto associate with an L δ -definable group a type L -definable topological group (Theorem4.15). In section 5 (the annex), we revisit the question of when a topological differentialfield embeds in a model of the scheme (DL). Acknowledgements
The first part of this paper owes a lot to a recent work with PabloCubid`es on differential topological fiels; among other things, it helped the author torevisit a former work on definable groups. For the second part, the author is indebtedto the work of Kobi Peterzil (and its nice exposition) on group configuration in ano-minimal setting. 2.
Preliminaries
Model theory and topological fields.
Let L ring := {· , + , − , , } and L field := L ring ∪ { − } denote the respective language of rings and of fields. Any field is an L field -structure by extending the multiplicative inverse to 0 by 0 − = 0.We will follow standard model theoretic notation and terminology. Lower-case let-ters like a, b, c and x, y, z will usually denote finite tuples and we let | x | denote the EFINABLE GROUPS IN TOPOLOGICAL FIELDS WITH A GENERIC DERIVATION 3 length of x . We will sometimes use ¯ x to denote a tuple of the form ( x , . . . , x n ) whereeach x i is itself a tuple. Given an L -structure K and an L -formula ϕ ( x ) with | x | = n ,we let ϕ ( K ) denote the set { a ∈ K n | K | = ϕ ( a ) } . Given a subset X ⊂ K n and a ∈ K ℓ ,0 < ℓ < n , the fiber of X over a is denoted by X a := { b ∈ K n − ℓ : ( a, b ) ∈ X } . By an L -definable set, we mean definable with parameters. If we want to restrict the subsetwhere the parameters vary we will use L ( A )-definable, A ⊂ K . If we wish to specifythat it is definable without parameters, we use L ( ∅ )-definable.Let L r be a relational extension of L field and assume that the language L (possiblymulti-sorted) extends L r in which every sort is an imaginary L r -sort. In particular, thefield-sort will always be the home sort of an L -structure. Let us recall the definition of L -open theories T . Let K be a field of characteristic 0 in the language L endowed witha definable field topology, namely there is an L -formula χ ( x, z ) be providing a basis ofneighbourhoods of 0 [12]. Throughout the text, we will assume that the topology on K is given by such formula χ . Let T be the L -theory of K .Then such theories T with the following further assumptions are called L -open [2]:( A ) (i) T has relative quantifier elimination with respect to the field sort,(ii) every quantifier-free L -definable subset of the field sort is a finite union ofsets of the form: an intersection of an L -definable open set with a Zariskiclosed set.One can choose the language L in such a way that the following theories are examplesof L -open theories: theories of real closed fields, p -adically closed valued fields, realclosed valued fields, algebraically closed valued field of characteristic 0 and the theoriesof any Hahn power series field k (( t Γ )), where k is a field of characteristic 0 and Γ is anordered abelian group [2, Examples 2.2.1], [6, Proposition 4.3].2.2. Definable sets in models of L -open theories. Let us recall a simple lemmain [2] on the form of L -definable subsets in models of L -open theories T . First we needto some notations.Let A be a finite subset of K [ x, y ]. We let A y := { P ∈ A | deg y ( P ) > } , We letthe L ring ( K )-formula Z A ( x, y ) be Z A ( x, y ) := V P ∈A P ( x, y ) = 0. Thus the algebraicsubset of K n +1 (Zariski closed set) defined by A corresponds to Z A ( K ). For an element R ∈ K [ x, y ] we let(2.2.1) Z R A ( x, y ) := Z A ( x, y ) ∧ R ( x, y ) = 0 . Lemma 2.1. [2, Corollary ]
Let K be a model of an L -open theory T . Then every L -definable set X ⊆ K n +1 (in the field sort) is defined by an L -formula ϕ ( x, y ) with x = ( x , . . . , x n − ) and y a single variable such that ϕ ( x, y ) ↔ _ j ∈ J Z S j A j ( x, y ) ∧ θ j ( x, y ) with J a finite set such that for each j ∈ J , θ j is an L -formula that defines an opensubset of K n , Z S j A j ( x, y ) is as defined in 2.2.1 with either(1) A j ⊆ K [ x ] and S j ∈ K [ x, y ] or FRANC¸ OISE POINT ( † ) (2) A j ⊆ K [ x, y ] , A y = { P } and S j = ( ∂∂y P ) R j for some R j ∈ K [ x ] . In a topological field K , one has a natural notion of topological dimension dim.For X a definable subset of K n , define dim( X ) := max { ℓ : there is a projection π nℓ : K n → K ℓ such that π ( X ) has non-empty interior } . Denote by X the closure of X in the topological sense, let Fr(X) := X \ X. For X ⊂ Y , let Int Y (X) be the subset ofelements of X which have a neighbourhood contained in Y and cl Y (X) be the subsetof elements of Y for which any neighbourhood has a non-empty intersection with X .2.3. Dimension functions.
In this paragraph we fix a complete L -theory T (withoutthe previous assumptions on L ), a sufficiently saturated model U | = T endowed witha fibered dimension function d (on definable sets) and we recall a few well-knownproperties of such fibered dimension. The notion of fibered dimension was introducedby L. van den Dries as follows. Definition 2.2. [4]
Let
Def( U ) be the set of L -definable subsets U . A fibered dimensionfunction d : Def( U ) → {−∞} ∪ On satisfies the following axioms. Let S, S , S , T ∈ Def( U ) .(1) d ( S ) = −∞ iff S = ∅ , d ( { a } ) = 0 , for each a ∈ U .(2) d ( S ∪ S ) = max { d ( S ) , d ( S ) } .(3) Let S ∈ U m , then for any permutation σ of { , · · · , m } , we have d ( S σ ) = d ( S ) .(4) Let S ⊂ U n + m , S ¯ x := { ¯ y ∈ A m : (¯ x, ¯ y ) ∈ S } and S ( γ ) := { ¯ x ∈ U n : d ( S ¯ x ) = γ } ,γ ∈ On . Then, S ( γ ) ∈ Def ( U ) and d ( { (¯ x, ¯ y ) ∈ S : ¯ x ∈ S ( γ ) } ) = d ( S ( γ )) + γ. If one adds that d ( U ) = 1 , one obtains a dimension function taking its values in {−∞} ∪ N and one can relax the condition (4) by asking it only for m = 1 (see [4] Proposition 1.4).
Given a (fibered) dimension function d on U , one may extend it to the space of types S Tn ( C ) over a subset C ⊂ U by d ( p ) := inf { d ( ϕ ( U , c ) : ϕ ( x, y ) an L ( ∅ ) − formula and ϕ ( x, c ) ∈ p } . Then we extend d on a tuple a of elements of U by defining d ( a/C ) as the dimensionof the type tp ( a/C ) of a over C . One then can show [1, Lemma 1.6]: Let a, b be twotuples of elements of U , then:(2.3.1) d ( ab/C ) = d ( a/C ∪ b ) + d ( b/C ) . Let B ⊂ U and let p ( x ) be a partial n -type. Let X ⊂ U n with X = p ( U ). Then X is finitely satisfiable in B if for any formula ϕ ( x ) ∈ p ( x ), we have ϕ ( M ) = ∅ . A tuple a in a definable set X is called generic over C (w.r. to the dimension d) if d ( X ) = d ( a/C ). Definition 2.3.
Let B ⊂ A ⊂ U n be two definable subsets of U , then B is almostequal to A (or large in A ) if d ( A \ B ) < d ( A ) [15, section 2].Let us recall the following result which can be found in [13], proven in the settingof o-minimal theories but it only uses the notion of a fibered dimension. EFINABLE GROUPS IN TOPOLOGICAL FIELDS WITH A GENERIC DERIVATION 5
Fact 2.1. [13, Proposition 1.13, Remark 1.14]
Let A be an L B -definable subset of U , with B ⊂ U and let ϕ ( x ; y ) be an L -formula.Then { c : ϕ ( U ; c ) ∩ A is almost equal to A } is B -definable.Moreover { c : ϕ ( U ; c ) ∩ A is almost equal to A } = { c : for every L -generic point u of A over c , U | = ϕ ( u ; c ) } . A cell decomposition theorem.
Throughout this section, let
K | = T , where T is L -open. Let us first observe a few properties of the topological dimension dim ondefinable subsets of K . First dim is a fibered dimension function. The proof followsthe same strategy as in [4] by noting that the dimension acl-dim is induced by thefield algebraic closure and has the exchange property (on the field sort) (namely T is ageometric theory on the field sort) and that it coincides with the topological dimension(see [2, Proposition 2.4.1]).Note that we may also define the dimension of an n -tuple ¯ a of elements of K as thecardinality of a maximal subtuple of acl-independent elements and it coincides withthe previous definition given above (see for instance [1, section 4]).Moreover, the dimension of the frontier of a definable set has the following property.(2.4.1) dim(Fr(X)) < dim(X) = dim(X) . As noted in [5, Chapter 4, Corollary 1.9], it implies for any definable set X ⊂ Y that(2.4.2) dim( X \ Int Y (X)) < dim(Y) . Indeed, note that X \ Int Y (X) = X ∩ cl Y (Y \ X) = cl Y (Y \ X) \ (Y \ X) ⊂ Fr(Y \ X)and so dim( X \ Int Y (X)) < dim(Y). Lemma 2.4. [13, Lemma 2.4]
Let ˜ K be an elementary saturated extension of K and X be definable subset of ˜ K with parameters in K . Let a ∈ X be generic over K and c ∈ ˜ K . Assume that tp ( a/Kc ) is finitely satisfiable in K , then a is generic over Kc .Proof: It amounts to show that dim( a/K ) = dim( a/Kc ). Since dim( a c/K ) =dim( a/Kc )+dim( c/K ) (see equation 2.3.1), we will show that dim( a c/K ) = dim( a/K )+dim( c/K ). Since dim = acl-dim, let a , respectively c , maximal acl-independent sub-tuples of a , respectively c over K . By the way of contradiction, suppose that a c is notacl-independent over K , then w.l.o.g. we may assume that c = c c with | c | = 1such that c ∈ acl( c a K ). Let ϕ ( x, z, y ; u ) with u ⊂ K such that ϕ ( c , c , a ; u )holds and ∃ ≤ n x ϕ ( x, c , a ; u ) for some natural number n . Since tp ( a/K c ) is finitelysatisfiable in K , there is b ⊂ K such that ϕ ( c , c , b ; u ) ∧ ∃ ≤ n x ϕ ( x, c , b ; u ) holds.This contradicts that c was chosen to be acl-independent over K . ✷ Finally, one has the following description of definable subsets of topological fieldsmodels of an L -open theory [2]; it is the analogue of the cell decomposition provenfor dp-minimal fields (see [15, Proposition 4.1]). Before stating the result, we need torecall the notion of correspondences. Definition 2.5. [15, section 3.1]
A correspondence f : E ⇒ F consists of two definablesubsets E, F together with a definable subset graph ( f ) of E × F such that < |{ y ∈ F : ( x, y ) ∈ graph ( f ) }| < ∞ , forall x ∈ E. FRANC¸ OISE POINT ( † ) The correspondence f is continuous at x ∈ E if for every V ∈ B there is U ∈ B suchthat ( f ( x ) , f ( x ′ )) ∈ V whenever ( x, x ′ ) ∈ U . A correspondence f is an m -correspondence if for all x ∈ E , |{ y ∈ F : ( x, y ) ∈ graph ( f ) }| = m . We denote by f ( x ) the set { y ∈ F : ( x, y ) ∈ graph ( f ) } . Note thata 1-correspondence is a function (together with its domain and image). Moreover acontinuous m -correspondence is locally given by m continuous functions [15, Lemma3.1] (see also [2, Lemma 2.6.2]). A m -correspondence f on an open definable set U iscontinuous on an open subset of U almost equal to U [2, Proposition 2.6.10].We follow the following convention: if f : K ⇒ K n , then graph ( f ) is identified witha finite set and if U is an open subset of K n and f : U ⇒ K , graph ( f ) is identifiedwith U . Proposition 2.6. [2, Theorem 2.7.1]
Let T be an L -open theory of topological fieldsand K be a model of T . Let X be a definable subset of K n . There are finitely manydefinable subsets X i with X = S i X i such that X i is, up to permutation of coordinates,the graph of a definable continuous m -correspondence f : U i ⇒ K n − d , where U i is adefinable open subset of K d , for some ≤ d ≤ n . Generic differential expansions of topological fields.
Let T be an L -opentheory of topological fields and K be a model of T . Let L δ be the language L extendedby a unary function symbol δ . Denote by K δ the expansion of K to an L δ -structure.Let T δ be the L δ -theory T together with the usual axioms of a derivation, namely, ( ∀ x ∀ y ( δ ( x + y ) = δ ( x ) + δ ( y )) ∀ x ∀ y ( δ ( xy ) = δ ( x ) y + xδ ( y )) . Notation 2.7.
Let K δ | = T δ . For m > a ∈ K , we define δ m ( a ) := δ ◦ · · · ◦ δ | {z } m times ( a ) , with δ ( a ) := a ,and ¯ δ m ( a ) as the finite sequence ( δ ( a ) , δ ( a ) , . . . , δ m ( a )) ∈ K m +1 .Similarly, given an element a = ( a , . . . , a n ) ∈ K n , we will write ¯ δ m ( a ) to de-note the element (¯ δ m ( a ) , . . . , ¯ δ m ( a n )) ∈ K ( m +1) n . Let ¯ m := ( m , . . . , m n ) ∈ N n and | ¯ m | := P ni =1 m i . We will write ¯ δ ¯ m ( a ) to denote the element (¯ δ m ( a ) , . . . , ¯ δ m n ( a n )) ∈ K | ¯ m | + n . For notational clarity, we will sometimes use ∇ m instead of ¯ δ m , especiallyconcerning the image of subsets of K n . For example, when A ⊆ K , we will usethe notation ∇ m ( A ) for { ¯ δ m ( a ) : a ∈ A } instead of ¯ δ m ( A ). Likewise for A ⊆ K n , ∇ ¯ m ( A ) := { (¯ δ m ( a ) , . . . , ¯ δ m n ( a n )) : a ∈ A } ⊆ K | ¯ m | + n .We will call ¯ δ ¯ m ( a ) a differential tuple and we will sometimes use the notation a ∇ ,suppressing the index ¯ m .Given x = ( x , . . . , x n ), we let K { x } be the ring of differential polynomials in n + 1differential indeterminates x , . . . , x n over K , namely it is the ordinary polynomial ringin formal indeterminates δ j ( x i ), 0 ≤ i ≤ n , j ∈ ω , with the convention δ ( x i ) := x i .We extend the derivation δ to K { x } by setting δ ( δ i ( x j )) = δ i +1 ( x j ). By a rationaldifferential function we simply mean a quotient of differential polynomials.For P ( x ) ∈ K { x } and 0 i n , we let ord x i ( P ) denote the order of P with respectto the variable x i , that is, the maximal integer k such that δ k ( x i ) occurs in a non-trivial monomial of P and − k exists. We let ord( P ), the order of P , be EFINABLE GROUPS IN TOPOLOGICAL FIELDS WITH A GENERIC DERIVATION 7 max i ord x i ( P ). Suppose ord( P ) = m . For ¯ x = (¯ x , . . . , ¯ x n ) a tuple of variables with | ¯ x i | = m + 1, we let P ∗ ∈ K [¯ x ] denote the corresponding ordinary polynomial suchthat P ( x ) = P ∗ (¯ δ m ( x )).Suppose ord x n ( P ) = m >
0. Then, there are (unique) differential polynomials c i ∈ K { x } such that ord x n ( c i ) < m and(2.5.1) P ( x ) = d X i =0 c i ( x )( δ m ( x n )) i . The separant s P of P is defined as s P := ∂∂δ m ( x n ) P ∈ K { x } . We extend the notion of separant to arbitrary polynomials with an ordering on theirvariables in the natural way, namely, if P ∈ K [ x ], the separant of P corresponds to s P := ∂∂x n P ∈ K [ x ]. By convention, we induce a total order on the variables δ j ( x i ) bydeclaring that δ k ( x i ) < δ k ′ ( x j ) ⇔ ( i < ji = j and k < k ′ . This order makes the notion of separant for differential polynomials compatible withthe extended version for ordinary polynomials, i.e. , s P ∗ = s ∗ P .We define an operation on K { x } sending P P δ as follows: for P written as in(2.5.1) P ( x ) P δ ( x ) = d X i =0 δ ( c i ( x ))( δ m ( x n )) i . A simple calculation shows that(2.5.2) δ ( P ( x )) = P δ ( x ) + s P ( x ) δ m +1 ( x n ) . Now we will describe a scheme of L δ -axioms generalizing the axiomatization ofclosed ordered differential fields (CODF) given by M. Singer in [ ? ]. Let χ ( x, z ) bean L -formula providing a basis of neighbourhoods of 0. For a = ( a , . . . , a n ) with | a i | = | z | , we let W a := χ ( K, a ) × · · · × χ ( K, a n ) . Definition 2.8.
Set T ∗ δ := T δ ∪ (DL), where (DL) is the following list of axioms: forevery differential polynomial P ( x ) ∈ K { x } with | x | = 1 and ord x ( P ) = m , for variables u = ( u , . . . , u m ) with | u i | = | z | and y = ( y , . . . , y m ) ∀ u (cid:16) ∃ y ( P ∗ ( y ) = 0 ∧ s ∗ P ( y ) = 0) → ∃ x (cid:0) P ( x ) = 0 ∧ s P ( x ) = 0 ∧ (¯ δ m ( x ) − y ) ∈ W u (cid:1)(cid:17) . As usual, by quantifying over coefficients, the axiom scheme (DL) can be expressedin the language L δ .When T = RCF, the theory RCF ∗ δ corresponds to CODF (which is consistent).When T is either ACVF ,p , RCVF, p CF d or the RV-theory of C (( t )) or R (( t )), the FRANC¸ OISE POINT ( † ) consistency of the theory T ∗ δ follows by results in [7, Corollary 3.8, Proposition 3.9](and see also [2, Theorem 3.3.2]). In [7], we showed how to embed a differentialtopological field ( K, δ ) which satisfied a hypothesis called (Hypothesis (I)) (see [7,Definition 2.21]), similar to largeness in this topological context into a differentialextension model of the scheme (DL). Largeness is a notion introduced by Pop [14]). In[2], instead of assuming a largeness hypothesis, we worked with henselian topologicalfields. (Note that henselian fields are large [14].) In the annex, we will indicate thehypothesis we need to make these embedding proofs work (see section 5).An immediate consequence of the axiomatisation is the density property of differ-ential points in open subsets of models of T ∗ δ . Lemma 2.9 ([7, Lemma 3.17]) . Let K δ | = T ∗ δ . Let O be an open subset of K n . Thenthere is a ∈ K such that ¯ δ n − ( a ) ∈ O .Proof: Let ¯ u := ( u , · · · , u n − ) ∈ O . We consider the differential polynomial δ n − ( x ).The corresponding algebraic polynomial is x n − . Then we consider the differentialequation δ n − ( x ) = u n − ; its separant is 1 and so we can apply the scheme (DL) andget a differential solution close to the algebraic solution ¯ u . So there is a ∈ K such that δ n − ( a ) = u n − and ¯ δ n − ( a ) ∈ O . ✷ Under assumption ( A ) on the L -theory T , different model-theoretic properties trans-fer from T to T ∗ δ , as shown by the following results. Theorem 2.10. [7, Theorem 4.1] [2]
The theory T ∗ δ admits quantifier eliminationrelative to the field sort in L δ . Using this quantifier elimination result, one can easily show the following two transferresults. (A proof of the second result may be found in [2, Appendix A.0.5].)
Proposition 2.11 ([7, Corollary 4.3]) . The theory T ∗ δ is NIP , whenever T is NIP . Proposition 2.12 (Chernikov) . The theory T ∗ δ is distal, whenever T is distal. Let us recall the definition of open core (in a general setting) (see also [3]).
Definition 2.13.
Let
K | = T , let ˜ L be an expansion of L and let ˜ T be the correspondingexpansion of T . Let ˜ K be an ˜ L -expansion of K . Then ˜ K has L -open core if every ˜ L -definable open subset is L -definable. An extension of ˜ T has L -open core if every modelof that extension has L -open core.Let S be a collection of sorts of L eq . We let L S denote the restriction of L eq to thehome sort together with the new sorts in S . Theorem 2.14. [2, Theorem 4.0.5]
Suppose that T admits elimination of imaginariesin L S and that the theory T ∗ δ has L -open core. Then the theory T ∗ δ admits eliminationof imaginaries in L S δ . Finally let us recall a result on the existence of a fibered dimension function ondefinable subsets (on the home sort) in models of T ∗ δ (which uses that T ∗ δ admitsquantifier elimination). EFINABLE GROUPS IN TOPOLOGICAL FIELDS WITH A GENERIC DERIVATION 9
Definition 2.15. An L -structure M is called equationally bounded if for each definableset S ∈ M m +1 such that for every ¯ a ∈ M m , S ¯ a is small, there exist finitely many L ( M )-terms f ( x , . . . , x m , y ) , . . . , f r ( x , . . . , x m , y ) such that for every ¯ a ∈ M m , there exists1 ≤ i ≤ r with f i (¯ a, y ) = 0 and S ¯ a ⊂ { b ∈ M : f i (¯ a, b ) = 0 } .Using a notion of independence built in from the algebra of all terms (t-independence),following [4], [10], one can define on Def( K δ ), when K δ | = T ∗ δ a dimension function(dim δ ) [8, Definition 2.3]. In order to show that this dimension is fibered, one usesthe closure operator cl over K which is defined by: a ∈ cl ( A ) if and only if there isa differential polynomial Q ∈ K h A i{ X } \ { } such that Q ( a ) = 0. Then associatedwith this closure operator, one has a notion of independence and dimension for tuplesof elements. Let ˜ K δ be a | K | + -saturated extension of K δ . Define for ¯ a = ( a , . . . , a n )a tuple in ˜ K , cl-dim(¯ a ) := max {| B | : B ⊆ K h ¯ a i , B is cl − independent } . Then one shows that, for X ∈ Def( K δ ) (see [8, Lemma 2.11]):dim δ ( X ) = max { cl − dim(¯ c ) : c ∈ X ( ˜ K ) } . One then proves that any model of T ∗ δ is equationally bounded, which entails that dim δ is a fibered dimension function. Note that there are infinite definable subsets X withdim δ ( X ) = 0. Proposition 2.16. [8, Corollary 3.10]
Let K δ | = T ∗ δ , then there is a dimension function dim δ that defines a fibered dimension function on Def( K δ ) . This dimension function has been further investigated in [1] and one can check thatfor ¯ a a tuple of elements in ˜ K that cl -dim(¯ a )=inf { dim δ ( ϕ ( K )) : ϕ (¯ x ) ∈ tp (¯ a/K ) } .3. The open core property in models of T ∗ δ From now on, let K be a model of an L -open theory T and let K δ its expansion by aderivation δ . In this section to a L δ -definable set X , we will associate an L -definable setwhere the differential prolongation of X is dense. Such result was already shown in [2]using a characterization of continuous L δ -correspondences (with L -definable domain). Notation 3.1.
Under assumption ( A ( ii )), any quantifier-free (relative to the field sort) L δ -definable set X ⊆ K n is of the form ∇ − m ( Y ) for a quantifier-free L -definable set Y ⊆ K n ( m +1) (quantifier-free relative to the field sort). Indeed, let x = ( x , . . . , x n ) be a tu-ple of field sort variables. By assumption on the language L , any L δ -term t ( x ) is equiv-alent, modulo the theory of differential fields, to an L δ -term t ∗ ( δ m ( x ) , . . . , δ m n ( x n ))where t ∗ is an L -term, for some ( m , . . . , m n ) ∈ N n . Therefore, by possibly addingtautological conjunctions like δ k ( x i ) = δ k ( x i ) if needed, we may associate with any L δ -formula ϕ ( x ) without field sort quantifiers, an equivalent L δ -formula (modulo thetheory of differential fields) of the form ϕ ∗ (¯ δ m ( x )) where m ∈ N and ϕ ∗ is an L -formulawithout field sort quantifiers. The formula ϕ ∗ arises by uniformly replacing every oc-currence of δ m ( x i ) by a new variable y mi in ϕ with the natural choice for the order ofvariables ϕ ∗ ( y , . . . , y m , . . . , y n , . . . , y mn ). Therefore, if X is defined by ϕ , letting Y bethe set defined by ϕ ∗ gives that X = ∇ − m ( Y ). Definition 3.2 (Order) . Let X ⊆ K n be an L δ -definable set. Let ¯ d = ( d , . . . , d n ) ∈ N n and let Z ⊂ K d + ... + d n such that X = ∇ − d ( Z ) The order of X , denoted by o ( X ) , ( † ) is the smallest integer m such that m = max ≤ i ≤ n d i and X = ∇ − d ( Z ) . Let ϕ bea field-sort quantifier-free L δ -formula such that X = { ( a , . . . , a n ) ∈ K n : K | = ϕ ∗ (¯ δ d ( a ) , . . . , ¯ δ d n ( a n )) } ; we say that d i (also denoted by d x i ) is the order of x i in ϕ , ≤ i ≤ n . Note that o ( X ) = 0 if and only if X is L -definable. Definition 3.3. [2, Definition 4.0.2] Let X ⊆ K n be a non-empty quantifier-free L δ -definable set (relative to the field sort). Let m be a positive integer and let Z ⊆ K ( m +1) n be an L -definable set such that:(1) x ∈ X if and only if ∇ m ( x ) ∈ Z and(2) Z = ∇ m ( X ).Then we call ( X, Z, m ) as above a linked triple.
Proposition 3.4. [2, Proposition 4.0.3]
The theory T ∗ δ has L -open core if and only iffor every model K δ , for every L δ -definable set X ⊂ K n , there is an integer m and an L -definable set Z ⊆ K ( m +1) n , such that ( X, Z, m ) is a linked triple. In addition, if T ∗ δ has L -open core, then there is a linked triple of the form ( X, Z, o ( X )) . Therefore, when T ∗ δ has L -open core, we will associate two dimension functionsto X , the first one dim δ ( X ) (see Proposition 2.16) and the second one dim ∗ ( X ) :=min { dim( Z ) m +1 : ( X, Z, m ) is a linked triple } .In [2, Theorem 6.0.7], we showed that T ∗ δ has L -open core under following hypotheseson T . In the ordered case, the field sort is L -definably complete and in the valued case,the value group sort Γ ∞ is L -definably complete. Furthermore, in the valued case, weassume the following on the value group:( † ) either there is a model K ′ of T for which Γ( K ′ ) is a divisible ordered abeliangroup in which every infinite L -definable set has an accumulation point;( †† ) or there is a model K ′ of T with Γ( K ′ ) = Z .This last hypotheses enabled us to show that for every L -definable open set V ⊆ K n ,any L -definable function f : V → Γ ∞ is continuous almost everywhere [2, Proposition2.6.11].3.1. Prolongations.
At the beginning of this subsection, we simply assume that K δ is a differential field of characteristic 0. We introduce some notations on prolongationswhich will be used later on for the L -open core property in models of T ∗ δ . Lemma 3.5. [2, Lemma 3.1.4]
Let x = ( x , . . . , x n ) be a tuple of variables and y be asingle variable. Let P ∈ K { x, y } be a differential polynomial such that k = ord y ( P ) > . There is a sequence of rational differential functions ( f Pi ) i > such that for every a ∈ K n +1 and b ∈ KK | = [ P ( a, b ) = 0 ∧ s P ( a, b ) = 0] → δ k + i ( b ) = f Pi ( a, b ) . In addition, each f Pi is of the form f Pi ( x, y ) = Q i ( x, y ) s P ( x, y ) ℓ i , EFINABLE GROUPS IN TOPOLOGICAL FIELDS WITH A GENERIC DERIVATION 11 where ℓ i ∈ N , ord y ( Q i ) = ord y ( P ) andord x j ( Q i ) = ( ord x j ( P ) + i if ord x j ( P ) > − otherwiseWe call the sequence ( f Pi ) i > the rational prolongation along P . Notation 3.6.
Let x = ( x , . . . , x m ) be a tuple of variables. For an integer d > , wedefine a new tuple of variables x ( d ) which extends x by d new variables that is, x ( d ) := ( x , . . . , x m , x m +1 , . . . , x m + d ) , with the convention that if d = 0 , then x (0) = x , When ¯ x = (¯ x , . . . , ¯ x ℓ ) is a tuple oftuples of variables, we let ¯ x [ d ] := (¯ x ( d ) , . . . , ¯ x ℓ ( d )) . Note that if d = 0 and ¯ x is not asingleton, then ¯ x [ d ] and ¯ x ( d ) are different.Let x = ( x , . . . , x n ) and y be a variable with | y | = 1 . Let P ∈ K { x, y } be adifferential polynomial of order m and let ( f Pi ) i > be its rational prolongation along P .Let ¯ y be such that | ¯ y | = m + 1 . We denote by λ dP (¯ x [ d ] , ¯ y, y m +1 , . . . , y m + d ) the L -formula: P ∗ (¯ x, ¯ y ) = 0 ∧ s ∗ P (¯ x, ¯ y ) = 0 ∧ d ^ i ≥ y m + i = ( f Pi ) ∗ (¯ x [ d ] , ¯ y ) . Given a tuple a := ( a , . . . , a n ) and an element b , we will call the tuple (¯ a [ d ] , ¯ b, b m +1 , . . . , b m + d ) satisfying the formula λ dP (¯ a [ d ] , ¯ b, b m +1 , . . . , b m + d ) the rational prolongation of ( a, b ) (along P ). When d = 0 , we set λ P (¯ x [0] , ¯ y ) to be the formula P ∗ (¯ x, ¯ y ) = 0 ∧ s ∗ P (¯ x, ¯ y ) = 0 . We will make the following abuse of notation. When P ∈ K [ x, y ] with x =( x , . . . , x m ) and | y | = 1, we can view P as ˜ P ∗ where ˜ P ∈ K { x, y } with ord x ( ˜ P ) = m .We will still denote by λ dP ( x ( d ) , y ( d )) the formula λ d ˜ P .We will use the following notation concerning projections. Notation 3.7.
For non-zero natural numbers n, k n , we let π k : K n → K k denote the projection onto the first k coordinates and π ( k ) : K n → K denote the pro-jection onto the k th coordinate. For natural numbers n, m, ℓ, k i with ≤ i ≤ ℓ ≤ n and k i m + 1 , we let π [ k ,...,k ℓ ] : K n ( m +1) → K ℓ ( m +1) denote the projection sendingthe i th -block of m + 1 -coordinates to the first k i -coordinates of such block, that is, π [ k ,...,k ℓ ] (( x , , . . . , x ,m +1 , . . . , x n, , . . . x n,m +1 )) = ( x , , . . . , x ,k , . . . , x ℓ, , . . . x ℓ,k ℓ ) . For ≤ i ≤ n , we will allow in π [ k ,...,k n ] the possibility for k i to be equal to , inwhich case the corresponding subtuple ( x , , . . . , x ,k i ) is simply empty. Therefore thenotation π [0 ,..., ,k ℓ ] : K n ( m +1) → K m +1 denotes the projection sending the ℓ th -block of m + 1 -coordinates to the first k ℓ -coordinates of such block, that is, π [0 ,..., ,k ℓ ] (( x , , . . . , x ,m +1 , . . . , x n, , . . . x n,m +1 )) = ( x ℓ, , . . . x ℓ,k ℓ ) . Note that π k : K n ( m +1) → K k coincides with π [ k ] : K n ( m +1) → K k , k ≤ m = 1. ( † ) The main result of this section is the following proposition, where we follow thesame convention as in Proposition 2.6, namely an open subset of K is a finite set ofpoints. Proposition 3.8.
Let
K | = T ∗ δ and let X ⊆ K n be an L δ -definable set. Then there is d ∈ N and a finite family { Y i | i ∈ I } of L -definable subsets of K n ( d +1) such that X = [ i ∈ I ∇ − d ( Y i ) , and for each i ∈ I , Y i is either open, or equal to { ¯ δ d ( a ) } for some a ∈ X , or the graphof a continuous L -definable correspondence h i such that for some n-tuples ( d , . . . , d n ) where ≤ d i ≤ d + 1 , ≤ i ≤ n , ( d , . . . , d n ) , where d i := max { d i , } ,(1) π [ d , ··· ,d n ] ( Y i ) is open in K d × · · · × K d n ,(2) h i : π [ d , ··· ,d n ] ( Y i ) ⇒ K d +1 − d × · · · × K d +1 − d n , and(3) for every open subset U ⊆ K nd such that U ∩ π [ d , ··· ,d n ] ( Y i ) = ∅ , there is a ∈ π [1 ,..., ( U ∩ π [ d , ··· ,d n ] ( Y i )) such that ¯ δ d ( a ) ∈ Y i .We will denote S i ∈ I Y i by X ∗∗ and call it an L -open envelop of X (in K n ( d +1) ).Proof. Let m := o ( X ); since ∇ − m commutes with finite union, by Theorem 2.10, No-tation 3.1 and Lemma 2.6, we may assume that X is a finite union of subsets of theform ∇ − m ( Z i ), where Z i is an L -definable subset of K n ( m +1) which is either open,finite or, up to a permutation of coordinates, the graph of a continuous L -definable ℓ -correspondence f i : U i ⇒ K n ( m +1) − e , where U i ⊂ K e is a non-empty open L -definableset, 1 e < n ( m + 1). We will consider each Z i separately. From now on, we drop theindex i . The result is immediate for Z in the following cases: either Z is open and weapply Lemma 2.9, or dim( π [1 ,..., ( Z )) = 0 and so ∇ − m ( Z ) is finite.In the other cases, we proceed as follows. We associate with Z a finite branchingtree ( T, E ) with root labelled by Z = Z (0) and we denote the subset of the nodesat level t by T ( t ), t ∈ N . If v ∈ T ( t ), then we denote by Z ( v ) the correspondencelabelling the node v with the property that ∇ − Z ( v ) = S w ∈ T ( t +1) ,E ( v,w ) ∇ − Z ( w ) and Z ( v ) ⊂ K n ( m ( v )+1) , where m ( v ) has been chosen minimal such. We will show thateach branch is finite and at the end of each branch the correspondence which labelsthe vertex is one of the Y i , i ∈ I as in the statement of the proposition. Note that byKoenig’s lemma, we get a finite tree.To simplify notation instead of denoting a correspondence obtained at level t by Z ( v ) with v ∈ T ( t ), we will simply denote it by Z ( t ) and we will denote m ( v ) by m ( t ).Given a correspondence Z ( t ), we associate a tuple ¯ d ( t ) := ( d ( t ) , . . . , d n ( t )) minimalin the lexicographic ordering, such that for each 1 ≤ i ≤ n ,dim( π [0 ,..., ,d i ( t )+1] ( Z ( t ))) = d i ( t ) , placing ourselves in K n ( m ( t )+2) (see Notation 3.7).For d ∈ N , we use the notation d to mean max { d, } . On N n we also put a partialordering ≺ defined component-wise: ( d , . . . , d n ) ≺ ( ˜ d , . . . , ˜ d n ), if for all 1 ≤ i ≤ n , d i ≤ ˜ d i and for some 1 ≤ i ≤ n , d i < ˜ d i . EFINABLE GROUPS IN TOPOLOGICAL FIELDS WITH A GENERIC DERIVATION 13
Define g ( Z ( t )) := ¯ d ( t ) and k ( Z ( t )) := |{ i : 1 ≤ i ≤ n, d i ( t ) = m ( t ) + 1 } . By assumption on Z (0), we have 0 ≤ k ( Z (0)) < n and we will show that k ( Z ( t )) ≤ k ( Z ( t − t ≥
1. We decompose the tuple ¯ d ( t ) into two subtuples: ¯ d [ k ( Z ( t ))] ( t ) oflength k ( Z ( t )) which collects (respecting the order) all the components of ¯ d ( t ) equalto m ( t ) + 1 and ¯ d [ n − k ( Z ( t ))] such that ¯ d ( t ) = ( ¯ d [ k ( Z ( t ))] , ¯ d [ n − k ( Z ( t ))] ). We proceed byinduction on ( k ( Z ( t )) , ¯ d [ n − k ( Z ( t ))] ) . Note that when k ( Z ( t )) = 0, then ¯ d [ n − k ( Z ( t ))] =¯ d ( t ) = g ( Z ( t )) and in that case we will show that when t increases then ¯ d ( t ) decreases.To the tuple ¯ d ( t ), we associate the following n -tuple of projections, with the con-vention that d i ( t ) ∗ + 1 := min { d i ( t ) + 1 , m ( t ) + 1 } , 1 ≤ i ≤ n , π [ d ( t ) ∗ +1] , π [ d ( t ) ∗ +1 ,d ( t ) ∗ +1] , . . . , π [ d ( t ) ∗ +1 ,...,d n ( t ) ∗ +1] . Now let us describe how to get from Z ( t ) to Z ( t + 1). Since this step is similar tothe step getting from Z (0) ⊂ K n ( m +1) to Z (1), in the discussion below, we will set t = 0 and d i (0) = d i , 1 ≤ i ≤ n .If d = m + 1, π [ m +1] ( Z ) is an open subset of K m +1 , then we consider the leastindex i such that d i ≤ m , 1 < i ≤ n , and the set π [ m +1 ,...,m +1 ,d i +1] ( Z ). (Note that byassumption on Z , there is such an index i .)Or 0 ≤ d < m + 1, so π [ d +1] ( Z ) is a finite union of subsets W ,j , j ∈ J with J finite, definable by L -formulas of the form ϕ j ( x , y ) : Z S j A ,j ( x , y ) ∧ θ ,j ( x , y ) , ( † ) with | x | = d , | y | = 1, θ j ( x , y ) an L -formula defining an open subset of K d +1 and A ,j ⊆ K [ x , y ] a finite set of non-zero polynomials such that either A ,j ⊆ K [ x ] or A y ,j = { P ,j } and S j = ( ∂∂y P ,j ) R j for some R j ∈ K [ x ] \ { } (see Lemma 2.1). Notethat every A ,j = ∅ as otherwise we contradict the minimality of d . Moreover we mayassume that for some formula ϕ j ( x , y ), we have A ,j ∩ K [ x ] = ∅ (by the assumption π [ d ] ( Z ) is a non-empty open subset unless d = 0 and we use that dim coincides withthe acl-dimension (see section 2.4)). For each j ∈ J such that A ,j ∩ K [ x ] = ∅ , wemay define the rational prolongation of ( x , y ) along P ,j by the formula λ ℓP ,j ( x , y ( ℓ )), ℓ ≥
0, where y k = f P ,j k ( x , y ), 1 ≤ k ≤ ℓ and y = y (see Notation 3.6).In order to unify the notations, in case d = m + 1, we still define λ ℓ ( x ( ℓ )), ℓ ≥ x ( ℓ ) = x ( ℓ ) (namely putting no conditions on the prolongation of x ).We have λ ℓ ( K ) = π [ m +1] ( Z ) × K ℓ . In case A ,j ∩ K [ x ] = ∅ , we consider the subset { ¯ z ∈ Z : π [ d ] (¯ z ) ∈ W j } and weexpress it as a finite union of continuous correspondences ˜ Z j , which will have theproperty that dim( π [ d ] ( ˜ Z j )) < d . So for each such correspondence ˜ Z j , we have g ( ˜ Z j ) ≺ g ( Z ) and d [ n − k ( ˜ Z j )] ≺ d [ n − k ( Z )] . So we apply the induction hypothesis. So foreach ˜ Z j we obtained a finite number of correspondences as described in the statementof the proposition and we label the corresponding number of nodes at level 1 (that weconnect to the root) by these correspondences. ( † ) Either d = m + 1 and so π [0 ,d ] ( Z ) is an open subset of K m +1 . In this case weproceed, namely we consider π [ d ∗ +1 ,m +1 ,d ∗ +1] ( Z ), 0 ≤ d ≤ m + 1.Assume that d < m + 1 and consider π [ d ∗ +1 ,d +1] ( Z ). Again by Lemma 2.1, weobtain that π [ d ∗ +1 ,d +1] ( Z ) is a finite union of L -definable sets W ,j , j ∈ J , each ofwhich defined by a formula of the form: ϕ j ( x , x , y ) := Z S j A ,j ( x , x , y ) ∧ θ ,j ( x , x , y ) , ( † ) with | x | ≤ d ∗ + 1, | x | = d , | y | = 1, θ ,j ( x , x , y ) an L -formula defining an opensubset of K d ∗ +1 × K d +1 and A ,j ⊆ K [ x , x , y ] a finite non-empty set of non-zeropolynomials such that either A ,j ⊆ K [ x , x ] or A y ,j = { P ,j } and S = ( ∂∂y P ) R for some R ∈ K [ x , x ] \ { } . (The fact that A ,j = ∅ follows from the assumptionon the dimension of π [ d ∗ +1 ,d +1] ( Z ).) Moreover we may assume that for some formula ϕ j we have that A ,j ∩ K [ x , x ] = ∅ . For each such j ∈ J , we define the rationalprolongation of ( x , x , y ) along P ,j . Namely let λ ℓP ,j ( x ( ℓ ) , x , y ( ℓ )), ℓ ≥
0, where y k = f P ,j k ( x ( ℓ ) , x , y ), 1 ≤ k ≤ ℓ and y = y (see Notation 3.6). Note that letting x = ( x , . . . , x m ) and if 0 ≤ d ≤ m is maximum deg x d P ,j >
0, then x ( ℓ ) =( x , . . . , x d + ℓ ) ).Again if d = m + 1, we define λ ℓ ( x , x ( ℓ )), ℓ ≥
0, as the formula x ( ℓ ) = x ( ℓ ),putting no conditions on the prolongation of ( x , x , y ).In case A ,j ∩ K [ x , x ] = ∅ , we consider the subset { ¯ z ∈ Z : π [ d ∗ +1 ,d ] (¯ z ) ∈ W ,j } and we express it as a finite union of continuous correspondences ˜ Z ,j , which will havethe property that dim( π [ d ∗ +1 ,d ] ( ˜ Z ,j )) < d + d . So for each such correspondence ˜ Z ,j ,we have g ( ˜ Z ,j ) ≺ g ( Z ), d [ n − k ( ˜ Z ,j )] ≺ d [ n − k ( Z )] and we apply the induction hypothesis.Then we proceed with π [ d ∗ +1 ,d ∗ +1 ,d +1] ( Z ) in case d ≤ m , otherwise we proceed with π [ d ∗ +1 ,d ∗ +1 ,m +1 ,d +1] ( Z ). At step 1 ≤ k ≤ n when d k ≤ m , we denote the formulaswe obtained by ϕ kj , j ∈ J k finite, and if d k = m + 1 we don’t make any modificationsbut in order to uniformize notations we introduce λ ℓk ( x , . . . , x k − , x k ( ℓ )), ℓ ≥
0, as theformula x k ( ℓ ) = x k ( ℓ ), putting no conditions on the prolongation of ( x , . . . , x k − , x k ).Then we consider the projection on the next coordinate.Set for 1 ≤ i ≤ n :(1) in case d i ≤ m , ℓ i := m − d i ,(2) in case d i = m + 1, ℓ i := 0.Then define m (1) = m in case for all i , d i ≤ m , otherwise let m (1) = max i { m + n X j>i ℓ j : 1 ≤ i < n, d i = m + 1 } . Then, in the construction we replace Z by each of the following subsets ˜ Z as follows.Set ¯ z := ( z , . . . , z n ) with z i := ( z i , . . . , z i m ), and for 1 ≤ i ≤ n ,in case d i ≤ m , let u i := ( z i , . . . , z i d i ),in case d i = m + 1, let u i = z i . EFINABLE GROUPS IN TOPOLOGICAL FIELDS WITH A GENERIC DERIVATION 15
For each 1 ≤ i ≤ n , we pick one of the formulas ϕ ij ( u , . . . , u i ), j ∈ J i , with | u i | = d ∗ i + 1 and | u s | ≤ d ∗ s + 1, 1 ≤ s ≤ i , as occurring above and we define˜ Z := { ¯ z [ m (1) − m ] : ¯ z ∈ Z ∧ ^ ≤ i ≤ n λ ℓ i + m (1) − mP i,j ( u ( ℓ i + m (1) − m ) , . . . , u i ( ℓ i + m (1) − m ) } , making the convention that when d i = m + 1, the formula λ ℓ i + m (1) − mP i,j ( u ( ℓ i + m (1) − m ) , . . . , u i ( ℓ i + m (1) − m )) is replaced by λ ℓ i + m (1) − mj ( u , . . . , u i − , u i ( ℓ i + m (1) − m ))if i > i = 1 by the formula λ ℓ + m (1) − mj ( u ( ℓ + m (1) − m )). Also note that if d ≤ d ∗ i + 1 is maximal such that deg z sd P i,j >
0, then in the formula λ ℓ i + m (1) − mP i,j ( u ( ℓ i + m (1) − m ) , . . . , u i ( ℓ i + m (1) − m )), u s ( ℓ i ) = ( z s , . . . , z sd , z sd +1 , . . . , z s ( d + ℓ i + m (1) − m )).Recall that along the way we took off Z subsets of smaller dimension, so let usdenote Z the remaining subset. Then we get S ∇ − ( ˜ Z ) = ∇ − ( Z ).Then we apply Proposition 2.6 to decompose each ˜ Z into a finite union of corre-spondences to which we apply the preceding procedure. Let us denote one of thesecorrespondences by Z (1).Since there was nothing special going from Z to Z (1), replacing 0 by t (and 1 by t + 1), m by m ( t ), d i by d i ( t ), ℓ i by ℓ i ( t ), we have from the construction above, thatthe tuple ¯ d ( t ) ∈ N n associated with Z ( t ), t ≥
1, has the following properties:(1) if d i (0) < m + 1, then d i (1) ≤ d i (0), 1 ≤ i ≤ n ,(2) if d i ( t ) < d i ( t − d i ( t + 1) ≤ d i ( t ), 1 ≤ i ≤ n ,(3) if d i ( t ) > d i ( t − d i ( t −
1) = m ( t −
1) and for some i < j ≤ n , d j ( t ) < d j ( t − ≤ i ≤ n ,(4) d n ( t ) ≤ d n ( t − ≤ d n (0) = d n ≤ m + 1,We make a similar abuse of notation by denoting k ( Z ( t )) by k ( t ), and write the tuple¯ d ( t ) as two subtuples one: ¯ d [ k ( t )] ( t ) of length k ( t ) which collects (in increasing order)all the components of ¯ d ( t ) equal to m ( t ) + 1 and the other one ¯ d [ n − k ( t )] such that¯ d ( t ) = ( ¯ d [ k ( t )] , ¯ d [ n − k ( t )] ). (Recall that k ( t + 1) ≤ k ( t ) < n .)So by (4), for all t , d n ( t ) ≤ d n , by (1) , (2) that the only indices i , 1 ≤ i ≤ n , forwhich d i ( t + 1) > d i ( t ) are those where the projection onto the i th -block of variableshas the same dimension as the ambient space and by (3) in order that this dimensionincreases from step t to step t + 1, the dimension of the projection onto some j th -block of variables, i < j ≤ n has dimension strictly smaller than the dimension ofthe ambient space. Observe that if k ( t ) = 0, then for all i , d i ( t + 1) ≤ d i ( t ) and( d ( t + 1) , . . . , d n ( t + 1)) ≺ ( d ( t ) , . . . , d n ( t )) ≺ ( d , . . . , d n ), t ≥ k ( t ) = k ( t + 1) and wecannot have ¯ d [ n − k ( t ′ +1)] ≺ ¯ d [ n − k ( t ′ )] for infinitely many t ′ > t . So now suppose that k ( t ) = k ( t + 1) and ¯ d [ n − k ( t +1)] = ¯ d [ n − k ( t )] . By (3), we get that ¯ d [ k ( t +1)] = ¯ d [ k ( t )] . So¯ d ( t + 1) = ¯ d ( t ).Let us show that in this case, the differential points are dense in Z ( t + 1) and that wecan decompose Z ( t +1) into a finite union of correspondences ones for which we do havethe required description and the other ones ˜ Z ( t + 1) for which g ( ˜ Z ( t + 1)) ≺ g ( Z ( t )).Let ¯ z := ( z , . . . , z n ) ∈ Z ( t + 1) with z i := ( z i , . . . , z i m ( t ) ), 1 ≤ i ≤ n . When d i ( t ) ≤ ( † ) m ( t ), let u i := ( z i , . . . , z i d i ( t ) ) and v i := ( z i , . . . , z i d i ( t ) − ). When d i ( t ) = m ( t ) + 1, u i = z i = v i .We have π [ ¯ d ( t )] ( Z ( t )) = U × . . . × U n where U i ⊂ K d i ( t ) are open L -definable subsetsof dimension d i ( t ) (with the convention that if d i ( t ) = 0, then U i is a point), 1 ≤ i ≤ n .By assumption π [ ¯ d ( t +1)] ( Z ( t + 1)) = V × . . . × V n where ∅ 6 = V i ⊂ U i . Let us describea correspondence f sending ( v , . . . , v n ) ∈ V × . . . × V n to ( u ( ℓ ( t )) , . . . , u n ( ℓ n ( t )),where for each 1 ≤ i ≤ n , there is j ∈ J i such that λ ℓP i,j ( u ( ℓ ) , . . . , u i ( ℓ )) holds withthe following convention: when d i ( t ) = m ( t ) + 1, then λ ℓP i,j ( u ( ℓ ) , . . . , u i ( ℓ )) is replacedby λ ℓi ( u ( ℓ ) , . . . , u i − ( ℓ ) , u i ( ℓ )). Moreover we only keep those ( u ( ℓ ( t )) , . . . , u n ( ℓ n ( t )))which belongs to Z (0).The differential points are dense in π [ ¯ d ( t +1)] ( Z ( t + 1)) (Lemma 2.9), so we can pick a n -tuple of the form ( δ d ( t ) − ( a ) , . . . , δ d n ( t ) − ( a n )) close to ( v , . . . , v n ). Then we use thescheme ( DL ) and the continuity of the rational functions f P i,j k , 1 ≤ k ≤ ℓ , associatedwith P i,j , with i such that d i ( t ) ≤ m ( t ) and j ∈ J i . We proceed by induction on 1 ≤ i ≤ n , namely we assume that the differential points are dense in π [( i − m ( t )+1)] ( Z ( t + 1)with the convention that if i = 1, this is the empty projection. Now suppose that d i ( t ) ≤ m ( t ), then for ( u , . . . , u i − ) ∈ π [ d ( t ) ∗ +1 ,...,d i − ( t ) ∗ +1] ( Z ( t + 1)), u i ∈ K d i ( t )+1 , if K | = P i,j ( u , . . . , u i − , u i ) = 0 ∧ s P i,j ( u , . . . , u i − , u i ) = 0 , then there is a differential point ¯ δ d i ( t )+1 ( u ) close to u i , for some u ∈ K , such that K | = P i,j ( u , . . . , u i − , ¯ δ d i ( t )+1 ( u )) = 0 ∧ s P i,j ( u , . . . , u i − , ¯ δ d i ( t )+1 ( u )) = 0 . Let U be a basic open neighbourhood of ( z , . . . , z i ) ∈ π [ i ( m ( t )+1)] ( Z ( t +1). By induc-tion hypothesis, there is a differential tuple close to ( z , . . . , z i − ) and by scheme ( DL ),there is a differential tuple close to ( z , . . . , z i − , u i ). By the continuity of the functions f P i,j k (see Lemma 3.5), there is V be a basic open neighbourhood of ( z , . . . , z i − , u i )such that V ′ := V × f P i,j ( V ) × · · · × f P i,j m ( t ) − d i ( V ) ⊆ U. The rational functions f P i,j k , 1 ≤ k ≤ m ( t ) − d i , applied to ¯ δ d i ( t )+1 ( u ) give itssuccessive derivatives.Then by [2, Proposition 2.6.10], the correspondence f is continuous on an opensubset O almost equal to V × . . . × V n . It remains to apply induction to f ↾ (( V × . . . × V n ) \ O ). ✷ We apply the proposition above to obtain the open core property for T ∗ δ . Corollary 3.9.
Let K | = T ∗ δ and let X ⊆ K n be an L δ -definable set. Then there isan L -definable subset Z such that ( X, Z, o ( X )) is a linked triple. In particular T ∗ δ has L -open core.Proof: By the above proposition, given an L δ -definable set X , there is an associatedlinked triple ( X, X ∗∗ , d ) with X ∗∗ an open L -envelop for X and d ∈ N . In general thisinteger d will be larger than o ( X ) but, by Proposition 3.4, one can construct anotherlinked triple with d = o ( X ). ✷ EFINABLE GROUPS IN TOPOLOGICAL FIELDS WITH A GENERIC DERIVATION 17
The following corollary slightly improves [2, Proposition 6.1.1] where we showed thatone can associate with an L δ -definable ℓ –correspondence, an L -definable ℓ -correspondence.There we assumed the domain of the correspondence to be L -definable (but the sameproof works when it is only L δ -definable). Furthermore, here we want the domain ofthe correspondence to have the special form described in Proposition 3.8. One cancheck that indeed the previous proof easily adapts. Corollary 3.10.
Let K | = T ∗ δ . Let X ⊂ K n and assume it is L δ -definable. Let f : X ⇒ K d be an L δ -definable ℓ -correspondence with d, ℓ ≥ . Let X ∗∗ be an L -openenvelop of X . There is m ∈ N , an L -definable ℓ -correspondence F : X ∗∗ ⇒ K d suchthat for every x ∈ X f ( x ) = F (¯ δ m ( x )) . ✷ Definable groups in models of T δ First we will recall a few facts about definable groups and generics in the settingof o-minimal theories [13], [11], since most of these notions only uses properties of thedimension function that hold in our present setting.Then given an L δ -definable group G in a model of T ∗ δ , where T is an L -open theoryof topological fields, we will show that on large L -definable subset of an L -open envelopof G as defined in Proposition 3.8, one can recover an L -definable operation inducedby the L δ -definable group law of G .Finally we will use the work of K. Peterzil on the group configuration in o-minimalstructures [11]. In a complete o-minimal theory admitting elimination of imaginaries,he showed that a group configuration gives rise to a transitive action of a type-definablegroup on infinitesimal neighbourghoods. Without appealing directly to a group con-figuration but using the same strategy, we will associate to an L δ -definable group atype L -definable group (possibly in a higher dimensional space).In the last part of this section we will add sorts S of L eq to the language L in orderthat T admits elimination of imaginaries in L S , the restriction of L eq to L togetherwith S . Note that on the one hand, the expansion T S of the theory T in L S is againan L S -open theory of topological fields (see [2, Remark 2.2.1]) and on the other hand,by Corollary 3.9 and [2, Theorem 4.0.5], T S δ admits elimination of imaginaries.We will first place ourselves in any L -structure M (without our previous assumptionson the language L ) but we will assume that M is endowed with a fibered dimensionwhich has the exchange property and that this dimension is preserved under definablebijection.By definable group G := ( G, · ,
1) in M , we mean that the domain of G is an L -definable subset of some cartesian product M n of M and the graph of the groupoperation · is an L -definable subset of M n . Fact 4.1. [13]
Let G be a definable group in M . Any element of G is the product oftwo generic elements and given a generic element a of G over b , then b · a is genericin G . ( † ) Definition 4.1. [13] Let G be a definable group in M . A generic definable subset of G is a subset of G such that finitely many translates cover G .Note that it entails that a generic set has the same dimension than G and so itcontains a generic point of G . Fact 4.2. [13, Lemma 2.4]
Let G be a definable group of M . Let X be a definablesubset of G , almost equal to G , then X is generic. For the rest of this section, we will work in models of an L -open theory T of topo-logical fields. In particular, we assume that T satisfies hypothesis ( A ). Proposition 4.2.
Let
K | = T ∗ δ be sufficiently saturated, let G := ( G, f ◦ , f − , e ) be an L δ -definable group in K (possibly with parameters) and let G ∗∗ be an L -open envelopof G . Then there exist L -definable maps F − : G ∗∗ → G ∗∗ and F ◦ : G ∗∗ × G ∗∗ → G ∗∗ ,a large open subset V of G ∗∗ and Y a definable large open subset of G ∗∗ × G ∗∗ suchthat(1) the map F − : G ∗∗ → G ∗∗ (respectively F ◦ : G ∗∗ × G ∗∗ → G ∗∗ ) coincides ondifferential tuples with f − (respectively f ◦ ),(2) the map F − : V → V is a continuous idempotent map,(3) the map F ◦ : Y → V is continuous,(4) for any a ∈ V , if b is generic of V over a , then ( b, a ) ∈ Y and ( F − ( b ) , F ◦ ( b, a )) ∈ Y .Proof: The proof follows the same pattern as in [13, Proposition 2.5] and we willconstruct a large subset V of G ∗∗ with the required properties by steps.Assume that the domain of G is included in some K n and is an L δ -definable set.Then by Proposition 3.8, there is an L -open envelop G ∗∗ of G , described as follows.For some d ∈ N , there is a finite family { Y i | i ∈ I } of L -definable subsets of K n. ( d +1) such that G ∗∗ = S i ∈ I Y i , G = [ i ∈ I ∇ − d ( Y i ) , and for each i ∈ I , Y i is either open, or equal to { ¯ δ d ( a ) } for some a ∈ G , or the graphof a continuous L -definable correspondence h i such that for some n-tuple ( d , . . . , d n ):(1) 1 d j < d + 1, 1 ≤ j ≤ n ,(2) π [ d , ··· ,d n ] ( Y i ) is open in K d × · · · × K d n ,(3) h i : π [ d , ··· ,d n ] ( Y i ) ⇒ K d +1 − d × · · · × K d +1 − d n , and(4) for every open subset U ⊆ K nd such that U ∩ π [ d , ··· ,d n ] ( Y i ) = ∅ , there is a ∈ π [1 ,..., ( U ∩ π [ d , ··· ,d n ] ( Y i )) such that ¯ δ d ( a ) ∈ Y i .Moreover letting f − the inverse function on G , by Corollary 3.10, there is a definable L -function F − on G ∗∗ which coincides with f − on differential points. Similarly given f ◦ the group law on G × G there is a definable L -function F ◦ on G ∗∗ × G ∗∗ whichcoincides with f ◦ on differential points. (We also use here that a 1-correspondence isa function).Let I := { i ∈ I : dim( Y i ) = dim( G ∗∗ ) } . By [2, Proposition 2.6.10], F − ◦ h i iscontinuous on a large open subset U i of π [ d , ··· ,d n ] ( Y i ). EFINABLE GROUPS IN TOPOLOGICAL FIELDS WITH A GENERIC DERIVATION 19
Let ˜ Y i := h i ( U i ) for i ∈ I and V := [ i ∈ I ˜ Y i . Claim 4.3.
Let V ′ := { ¯ x ∈ V : F − ( F − (¯ x )) = ¯ x } . Then V ′ is a large definablesubset of V . Proof of Claim:
Suppose on the contrary that we can find an open subset U ⊂ V where F − ( F − (¯ x )) = ¯ x . Then choose in U an element ¯ u ∇ for some ¯ u ∈ G . Since F − and f − coincide on differential points, we get a contradiction. ✷ From now on, we allow ourselves to replace V by V ′ , namely we will assume that F − ◦ F − is the identity on V .Using again [2, Proposition 2.6.10], F ◦ ◦ ( h i , h j ) is continuous on a large open subset O i,j of π [ d , ··· ,d n ] ( Y i ) × π [ d , ··· ,d n ] ( Y j ). Let ˜ Y i,j := ( h i , h j )( O i,j ) for i, j ∈ I and Y := [ i,j ∈ I ˜ Y i,j . Claim 4.4.
Let V := { ¯ y ∈ V : ∀ ¯ a ∈ G ∗∗ L -generic of G ∗∗ over ¯ y , (¯ a, ¯ y ) ∈ Y } . Then V is definable and almost equal to V . Proof of Claim:
The set V is L -definable by Fact 2.1. By the way of contradiction,assume there is a relatively open subset U of V ⊂ G ∗∗ such that if ¯ y ∈ U , we havethat { ¯ a ∈ G ∗∗ : (¯ a, ¯ y ) ∈ Y } is not almost equal to G ∗∗ . In particular, we may choosein U a generic element ¯ y ∈ V of G ∗∗ such that the set { ¯ a ∈ G ∗∗ : (¯ a, ¯ y ) ∈ Y } is not almost equal to G ∗∗ . So its complement would contain an open subset U (¯ y ).Choose ¯ b ∈ G ∗ generic over ¯ y in that open set. Then (¯ b , ¯ y ) is generic in G ∗∗ × G ∗∗ by equation 2.3.1 but would not belong to Y , a contradiction. ✷ Claim 4.5.
Let a, b ∈ G and choose a ∇ L -generic over b ∇ . Then F ◦ ( a ∇ , b ∇ ) is L -generic over b ∇ and so it belongs to V . Proof of Claim:
By hypothesis, dim( G ∗∗ ) = dim( a ∇ /b ∇ ). Let us show that dim( F ◦ ( a ∇ , b ∇ ) /b ∇ ) =dim( a ∇ /b ∇ ). By construction, we have that F ◦ and F − coincide respectively on G ∇ with f ◦ and f − respectively. So F ◦ ( a ∇ , b ∇ ) = f ◦ ( a, b ) ∇ , F − ( b ∇ ) = f − ( b ) ∇ and F ◦ ( F ◦ ( a ∇ , b ∇ ) , F − ( b ∇ )) = a ∇ .So, acl ( b ∇ , F ◦ ( a ∇ , b ∇ )) ⊂ acl ( b ∇ , a ∇ ) ⊂ acl ( F ◦ ( a ∇ , b ∇ ) , F − ( b ∇ ) , b ∇ ) ⊂ acl ( F ◦ ( a ∇ , b ∇ ) , b ∇ ). ✷ Claim 4.6.
Let a, b ∈ G and choose a ∇ L -generic over b ∇ . Then F − ( a ∇ ) is L -genericover b ∇ and so it belongs to V . Proof of Claim:
By hypothesis, dim( G ∗∗ ) = dim( a ∇ /b ∇ ). Let us show that dim( F − ( a ∇ ) /b ∇ ) =dim( a ∇ /b ∇ ). By construction, we have that F − coincide on G ∇ with f − . So F − ( a ∇ ) = f − ( a ) ∇ and F − ( F − ( a ∇ )) = F − ( f − ( a ) ∇ ) = f − ( f − ( a ) ∇ ) = a ∇ . So, acl ( b ∇ , a ∇ ) ⊂ acl ( b ∇ , f − ( a ) ∇ ) ⊂ acl ( b ∇ ) , a ∇ ). ✷ ( † ) Let ˜ Y (¯ z ) := { (¯ x, ¯ y ) ∈ Y : (¯ y, ¯ z ) ∈ Y & F ◦ (¯ y, ¯ z ) ∈ V &(¯ x, F ◦ (¯ y, ¯ z )) ∈ Y & F ◦ (¯ x, F ◦ (¯ y, ¯ z )) = F ◦ ( F ◦ (¯ x, ¯ y ) , ¯ z ) } Let V ′′ := { ¯ z ∈ V : ˜ Y (¯ z ) is almost equal to Y } . The set V ′′ is definable since being almost equal is an L -definable property and allthe other data is L -definable. Claim 4.7. V ′′ is almost equal to V . Proof of Claim:
We proceed by contradiction. If not there would exist ¯ z ∈ V and anopen neighbourhood U of ¯ z such that for any element ¯ u ∈ U , the set ˜ Y (¯ u ) is not almostequal to Y , which means that there exists an open subset W of Y containing (¯ x, ¯ y )where one of the following statement fails: (¯ y, ¯ u ) ∈ Y , F ◦ (¯ y, ¯ u ) ∈ V , (¯ x, F ◦ (¯ y, ¯ u )) ∈ Y , or if everything else hold, that F ◦ (¯ x, F ◦ (¯ y, ¯ u )) = F ◦ ( F ◦ (¯ x, ¯ y ) , ¯ u ) fails.We may choose such ¯ u ∈ V ∩ U of the form ¯ u ∇ for some ¯ u ∈ G ( G ∇ is dense in G ∗∗ ). We choose (¯ a ∇ , ¯ b ∇ ) ∈ W as follows. First we choose ¯ b ∇ L -generic over ¯ u ∇ (inparticular ¯ b ∇ ∈ V ). By Claim 4.4, (¯ b ∇ , ¯ u ∇ ) ∈ Y and by Claim 4.5, F ◦ (¯ b ∇ , ¯ u ∇ ) ∈ V .(Note that f ◦ ( b, u ) ∇ = F ◦ (¯ b ∇ , ¯ u ∇ ) . ) Then we choose ¯ a ∇ L -generic over ¯ b ∇ and over f ◦ ( b, u ) ∇ . So (¯ a ∇ , ¯ b ∇ ) ∈ Y and (¯ a ∇ , f ◦ ( b, u ) ∇ ) ∈ Y . Since G is a group and F ◦ and f ◦ coincide on differential points, we get that F ◦ (¯ a ∇ , F ◦ (¯ b ∇ , ¯ u ∇ )) = F ◦ ( F ◦ (¯ a ∇ , ¯ b ∇ ) , ¯ u ∇ ). ✷ Let V ′′ := { ¯ z ∈ V : { ¯ x ∈ V : ( F − (¯ x ) , ¯ z ) ∈ Y &(¯ x, F ◦ ( F − (¯ x ) , ¯ z )) ∈ Y & F ◦ (¯ x, F ◦ ( F − (¯ x ) , ¯ z )) = ¯ z } is almost equal to V } . The set V ′′ is definable since being almost equal is an L -definable property. Claim 4.8. V ′′ is almost equal to V . Proof of Claim:
Suppose not then there would exist an open subset U of V over which { ¯ x ∈ V : ( F − (¯ x ) , ¯ z ) ∈ Y & (¯ x, F ◦ ( F − (¯ x ) , ¯ z )) ∈ Y & F ◦ (¯ x, F ◦ ( F − (¯ x ) , ¯ z )) = ¯ z } is notalmost equal to V .So we can find an element of the form ¯ z ∇ ∈ U , ¯ z ∈ G and ¯ x ∇ ∈ V genericover ¯ z ∇ . By Claim 4.6, F − ( x ∇ ) is generic over ¯ z ∇ and so it belongs to V and( F − (¯ x ∇ ) , ¯ z ∇ ) ∈ Y . By Claim 4.5, F ◦ ( F − (¯ x ∇ ) , ¯ z ∇ ) is generic over ¯ z ∇ . Now let usshow that ¯ x ∇ is generic over F ◦ ( F − (¯ x ∇ ) , ¯ z ∇ ).We first note that acl (¯ z ∇ , ¯ x ∇ ) = acl (¯ x ∇ , F ◦ ( F − (¯ x ∇ ) , ¯ z ∇ )).So dim(¯ x ∇ , ¯ z ∇ ) = dim(¯ x ∇ , F ◦ ( F − (¯ x ∇ ) , ¯ z ∇ )) . By equation 2.3.1,dim(¯ x ∇ , ¯ z ∇ ) = dim(¯ x ∇ / ¯ z ∇ ) + dim(¯ z ∇ ) anddim(¯ x ∇ , F ◦ ( F − (¯ x ∇ ) , ¯ z ∇ )) = dim(¯ x ∇ /F ◦ ( F − (¯ x ∇ ) , ¯ z ∇ )) + dim( F ◦ ( F − (¯ x ∇ ) , ¯ z ∇ )) . EFINABLE GROUPS IN TOPOLOGICAL FIELDS WITH A GENERIC DERIVATION 21
So, dim(¯ x ∇ /F ◦ ( F − (¯ x ∇ ) , ¯ z ∇ )) = dim(¯ x J / ¯ z ∇ ), so ¯ x ∇ is generic over F ◦ ( F − (¯ x ∇ ) , ¯ z ∇ )and therefore (¯ x ∇ , F ◦ ( F − (¯ x ∇ ) , ¯ z )) ∈ Y . Since F ◦ and F − coincide with f ◦ and f − ondifferential points, we get F ◦ (¯ x ∇ , F ◦ ( F − (¯ x ∇ ) , ¯ z ∇ )) = f ◦ (¯ x ∇ , f ◦ ( f − (¯ x ) ∇ , ¯ z ∇ )) = ¯ z ∇ . ✷ Since both V ′′ , V ′′ are L -definable and almost equal to G ∗∗ (see Claims 4.7, 4.8), V ′′ ∩ V ′′ is L -definable and almost equal to G ∗∗ . So it can be expressed as a finiteunion of continuous correspondences by Proposition 2.6. Let V be the union of thosewith the same L -dimension as G ∗∗ . Claim 4.9.
Let V := V ∩ F − ( V ). Then V is L -definable, relatively open in G ∗∗ andalmost equal to G ∗∗ . Moreover F − maps V to V and is continuous. Proof of Claim:
It remains to show that F − maps V to V . An element of V can bewritten as ¯ a ∈ V and F − (¯ b ) for ¯ b ∈ V . Since V ⊂ V , we get that F − ( F − (¯ b )) = ¯ b . ✷ By construction, V is a relatively open definable large subset of G ∗∗ . We define Y := { (¯ x, ¯ y ) ∈ ( V × V ) ∩ Y : F ◦ (¯ x, ¯ y ) ∈ V } . Note that Y is large in V × V . If wechoose ¯ x ∇ ∈ V generic, then ¯ y ∇ ∈ V generic over ¯ x ∇ . Then F ◦ (¯ x ∇ , ¯ y ∇ ) is generic over¯ x ∇ by Claim 4.5 and so belongs to V . Note that on Y , the map F ◦ is continuous.By Claim 4.9, V is a large definable relatively open subset of G ∗∗ . By the paragraphabove Y is a large definable relatively open subset of G ∗∗ × G ∗∗ and F ◦ is continuouson Y (statement (3) of the proposition. Statement (1) of the proposition follows byconstruction, (2) is Claim 4.9 and (4) is proven in the same way as we did with V and Y (see Claim 4.4). ✷ Now let us recall the notion of infinitesimal neighbourhoods in the context of L -opentheories T of topological fields. Definition 4.10. [11, Definition 2.3] Let
M | = T and let a ∈ M n . Then the M -infinitesimal neighbourhood of a ∈ M n is the partial type consisting of all formulaswith parameters in M that define an open subset of M n containing a . Given N an | M | + -saturated extension of M , we denote by µ a ( N ) its realization in N . Let X be a M -definable subset of M n containing a , then µ a ( X ) = µ a ( N ) ∩ X ( N ).We will also use the notation u ∼ M u ∈ µ ( N ) and use the term forsuch elements u , M -infinitesimals.If a is generic in X , then µ a ( X ) is independent of the choice of X , namely we havethe analog of [11, Fact 2.4], using property 2.3.1 of the dimension function in modelsof T . For convenience of the reader, we prove it below. Lemma 4.11. [11, Fact 2.4]
Let X be an A -definable subset of M n and a a genericelement of X over A . Then for any A -definable set Y , if a ∈ Y , then µ a ( X ) ⊂ µ a ( Y ) .In particular if dim( X ) = dim( Y ) , then µ a ( X ) = µ a ( Y ) .Proof: Consider X ∩ Y . Then since a is generic in X , a ∈ Int X ( X ∩ Y ). Since thetopology is definable, there exists an open A -definable set U containing a such that U ∩ X ⊂ X ∩ Y . So, U ∩ X = U ∩ X ∩ Y . It follows that µ a ( X ) = µ a ( X ∩ Y ) and ( † ) so µ a ( X ) ⊂ µ a ( Y ). If dim( X ) = dim( Y ), then a is also generic in Y and the reverseinclusion holds, namely µ a ( Y ) ⊆ µ a ( X ). ✷ The above lemma allows us to introduce the following notation.
Notation 4.12. [11, Definition 2.5] Given a ∈ M n , A ⊂ M and X an A -definablesubset of M n with a generic in X over A , we denote µ M ( a/A ) (or µ ( a/A )) the set µ a ( X ). Notation 4.13. [11, Definition 2.10] Let
M | = T and let O be an open subset of M n .Let p , p : O → M be two maps and let y ∈ O . Then p ∼ y p means that for someopen neighbourhood U ⊂ O of y , p ↾ U = p ↾ U . Lemma 4.14. [11, Lemma 2.11]
Let M be a sufficiently saturated model of T . Let V ⊂ M n be a L -definable open subset in M ; let F := { p b : V → V : b ∈ V } be a familyof L -definable bijections of V . Let x be a generic element of V and let a ∈ V genericover { x } . Then there exist definable open subsets W containing x and U containing a such that for every a ′ , a ′′ ∈ U and y ∈ W , if p a ′ ∼ y p a ′′ , then p a ′ ↾ W = p a ′′ ↾ W .Proof: Denote by [ a ] x := { u ∈ V : p u ∼ x p a & p u ∈ F } . Let a ∈ [ a ] x be generic over { a , x } . Let W be an open neighbourhood of x such that p a ↾ W = p a ↾ W . Recallthat the topology on K is definable and let ¯ d be parameters such that W = ¯ χ ( M, ¯ d ).We may assume by shrinking W if necessary that ¯ d is independent from a , a, x . Wecan express by an L -formula in a , a, x, ¯ d the following property: p a ∼ x p a → p a ↾ W = p a ↾ W . So it continues to hold for all a ′ in an open neighbourhood U of a and this lastproperty can be expressed by an L -formula in a , x and ¯ d : ∀ a ′ ∈ U ( p a ∼ x p a ′ → p a ↾ W = p a ′ ↾ W ) . Since a and x are independent, this formula continues to hold in an open neigh-bourhood U of a and an open neighbourhood W of x . Then U := U ∩ U and W := W ∩ W are the sought neighbourhoods. ✷ Theorem 4.15.
Let S be sorts in L eq such that T admits elimination of imaginariesin L S . Let K | = T ∗ δ and assume that K is sufficiently saturated. Let G := ( G, f ◦ , f − , e ) be an L δ -definable group in (the field sort of ) K (over a subset of parameters). Let G ∗∗ be an L -open envelop of G . Then there exists a type L -definable topological group H (over some parameters) with dim( H ) = dim( G ∗∗ ) .Proof: We keep the same notations as in Proposition 4.2; in particular we will use F − , F ◦ , V and Y . In order to simplify notations we will not indicate the parameters we areworking with and simply use L ; however we will indicate all the additional parameters.Let a ∈ G ∗∗ be L -generic and let a be L -generic over a . Then by Proposition4.2 (4), we have: ( a , a ) ∈ Y and ( F − ( a ) , F ◦ ( a , a )) ∈ Y . By Proposition 4.2 (3), a ∈ V . Claim 4.16.
Let us show that a = F ◦ ( a , a ) is L -generic in V . Proof of Claim:
This is similar to the proof of Claim 4.5. We have that
EFINABLE GROUPS IN TOPOLOGICAL FIELDS WITH A GENERIC DERIVATION 23 acl ( a , F ◦ ( a , a )) ⊂ acl ( F ◦ ( a , a ) , a , F − ( a )) ⊂ acl ( a , a ) ⊂ acl ( F ◦ ( a , a ) , a ). So,by equation (2.3.1) dim( a , a ) = dim( a /a ) + dim( a ) =dim( a , F ◦ ( a , a )) = dim( F ◦ ( a , a ) /a ) + dim( a ) . By assumption dim( a /a ) = dim( G ∗∗ ) = dim( a ) = dim( a ).Therefore, F ◦ ( a , a ) is generic over a and so generic in G ∗∗ . ✷ So for each 1 ≤ i ≤
3, the map V → V : u F ◦ ( a i , u ) is a bijection on V by Claim4.8 and the definition of V . Note that this is a L -definable property of each a i and sothis will hold for any a ′ i ∈ µ ( a i ), 1 ≤ i ≤ V around each a i , 1 ≤ i ≤
3, using the followingnotations.(1) Denote for a ′ ∈ µ ( a ), p a ′ : V → V : u F ◦ ( a ′ , u ) and by P := { p a ′ : a ′ ∈ µ ( a ) } .(2) Denote for a ′ ∈ µ ( a ), q a ′ : V → V : u F ◦ ( a ′ , u ) and by Q := { q a ′ : a ′ ∈ µ ( a ) } .(3) Denote for a ′ ∈ µ ( a ), h a ′ : V → V : u F ◦ ( a ′ , u ) and by H := { h a ′ : a ′ ∈ µ ( a ) } .Let x ∈ V be L -generic over { a , a } . In particular µ ( x ) = µ ( x/ { a , a } ). As in thegroup configuration theorem [11, Theorem 3.4], we consider the following subgroup H of the group Sym ( µ ( x )) of permutations on µ ( x ) generated by: { p − a ′ ◦ p a ′′ : a ′ , a ′′ ∈ µ ( a ) } . Claim 4.17. ∀ p ∈ P ∀ q ∈ Q ∃ h ∈ H q ◦ p = h on µ ( x ). Proof of Claim:
Let a ′ ∈ µ ( a ) be such that p = p a ′ , let a ′ ∈ µ ( a ) be such that q = q a ′ . The map F ◦ is continuous on Y , so for every neighbourhood W of a there exists W a neighbourhood of a and W a neighbourhood of a such that F ◦ ( W , W ) ⊂ W .First let us show that F ◦ ( a , F ◦ ( a , x )) = F ◦ ( F ◦ ( a , a ) , x ). By Claim 4.7, we knowthat ˜ Y ( x ) is almost equal to Y . Since x is L -generic over { a , a } , we get that( a , a ) ∈ ˜ Y ( x ). So, ( a , F ◦ ( a , x )) ∈ Y and F ◦ ( a , F ◦ ( a , x )) = F ◦ ( F ◦ ( a , a ) , x ).Furthermore, ( a , a ) ∈ Int Y ( ˜ Y ( x )). So for a ′ ∈ µ ( a ) and a ′ ∈ µ ( a ), we get that( a ′ , a ′ ) ∈ Y ( x ) and so F ◦ ( a ′ , F ◦ ( a ′ , x )) = F ◦ ( F ◦ ( a ′ , a ′ ) , x ).Now we can express by a L ( { a , a } )-formula (in x ) the property that ( a , a ) ∈ Int Y ( ˜ Y ( x )). So for any x ′ ∈ µ ( x ) = µ ( x/ { a , a } ), since x is generic in V , we getthat ( a , a ) ∈ Int Y ( ˜ Y ( x ′ )). Therefore, F ◦ ( a , F ◦ ( a , x ′ )) = F ◦ ( F ◦ ( a , a ) , x ′ ). By thesame reasoning as above, it also holds for a ′ , a ′ in place of a , a . ✷ Claim 4.18. ∀ h ∈ H ∀ q ∈ Q ∃ p ∈ P q ◦ p = h on µ ( x ). Proof of Claim:
Let a ′ ∈ µ ( a ) be such that q = q a ′ , let a ′ ∈ µ ( a ) be such that h = h a ′ . Then define a map p on u ∈ µ ( x ) as follows p ( u ) := F ◦ ( F ◦ ( F − ( a ′ ) , a ′ ) , u ).Note that F − ( V ) = V and F ◦ ( a , F ◦ ( F − ( a ) , a )) = a by Claim 4.8 ( a is L -generic).Since F ◦ and F − are continuous, it holds in a neighbourhood of a , respectively a . ✷ ( † ) Claim 4.19. ∀ h ∈ H ∀ p ∈ P ∃ q ∈ Q q ◦ p = h on µ ( x ). Proof of Claim:
Let a ′ ∈ µ ( a ) be such that p = p a ′ , let a ′ ∈ µ ( a ) be such that h = h a ′ . Then define a map q on u ∈ µ ( F ◦ ( a , x ) by q ( u ) := F ◦ ( F ◦ ( a ′ , F − ( a ′ )) , u ). Notethat F − ( V ) = V and F ◦ ( F ◦ ( a , F − ( a )) , a ) = a by Claim 4.8 ( a is L -generic and a is L -generic over a ). Since F ◦ and F − are continuous, it holds in a neighbourhoodof a , respectively a . ✷ Claim 4.20. ∀ p ∈ P ∀ p ∈ P ∀ p ∈ P ∃ p ∈ P p − p = p − p on µ ( x ). Proof of Claim:
We apply the previous claims in order to write first p as q − ◦ h with q ∈ Q and h ∈ H by Claim 3.8. Then write p as q − ◦ h with q ∈ Q by Claim3.10 and finally p = q − ◦ h with h ∈ H by Claim 3.8. Composing these maps, weget p p − p = q − h ∈ P by Claim 3.9. ✷ Claim 4.21.
Let a be a fixed element of µ ( a ) L -generic in V over a . Then group H is equal to { p − a ◦ p a ′ : a ′ ∈ µ ( a ) } . Proof of Claim:
We apply the previous claim with p = p a . ✷ Claim 4.22.
The action of the group H is transitive on µ ( x ). Proof of Claim:
Since x ∈ V , F − ( x ) ∈ V and since x is generic over a , ( a , x ) ∈ Y .By Claim 4.7, F ◦ ( F ◦ ( a , x ) , F − ( x )) = a . So given x ′ ∈ µ ( x ) and x ′ ∈ µ ( F ◦ ( a , x )),we may define a ′ := F ◦ ( x ′ , F − ( x ′ )) and this element belongs to µ ( a ).Then we re-apply the same reasoning to x ′′ ∈ µ ( x ) and x ′ , and define a ′′ := F ◦ ( x ′ , F − ( x ′′ )). Then a ′′ ∈ µ ( a ). Finally consider p − a ′′ ◦ p a ′ ; by Claim 4.21, itbelongs to H and p − a ′′ ◦ p a ′ ( x ′ ) = x ′′ . ✷ Finally by Lemma 4.14, there is a L -definable open neighbourhoods W ⊂ V of x and U ⊂ V of a such that if p a ′ ∼ x p a ′′ , then p a ′ ↾ W = p a ′′ ↾ W . For a ′ ∈ U ,let [ a ′ ] x := { a ′′ ∈ U : p a ′ ∼ x p a ′′ } = { a ′′ ∈ U : p a ′ ↾ W = p a ′′ ↾ W } . Now supposethat p − a p ′ a ( u ) = p − a p ′ a ( u ), namely F ◦ ( F − ( a ) , F ◦ ( a ′ , u )) = F ◦ ( F ◦ ( F − ( a ) , a ′′ ) , u ).So F circ ( a, F ◦ ( F − ( a ) , F ◦ ( a ′ , u ))) = F ◦ ( a, F ◦ ( F − ( a ) , F ◦ ( a ′ , u ))) which implies that F ◦ ( a ′ , u ) = F ◦ ( a ′ , u ). So, two elements of H are equal on a neighbourhood of x , thenthey coincide on U by Lemma 4.14. This will allow us to identify H := { p − a ◦ p a ′ : a ′ ∈ µ ( a ) } with µ ( a ) /E where µ ( a ) is type-definable over M and E is a definableequivalence relation on V , defined by E ( a ′ , a ′′ ) if and only if p a ′ ↾ W = p a ′′ ↾ W .The group law is definable since the action of the elements of H on µ ( x ) is definedusing the L -definable functions F ◦ and F − . Finally since H acts transitively on µ ( x )(Claim 4.22) its dimension is equal to dim( x ) = dim( V ) = dim( G ∗∗ ). Furthermore H a topological group, namely the group laws are continuous. (It follows from the factthat F − is continuous on V and F ◦ is continuous on Y .) ✷ . Remark 4.23.
Note that we may associate with H a dim( G ∗∗ )-group configuration[11, Definition 3.1]. 5. Annex: Largeness
In the following proposition, we put an extra-hypothesis besides largeness on adifferential topological field that we will call c-largeness, in order to embed a differential
EFINABLE GROUPS IN TOPOLOGICAL FIELDS WITH A GENERIC DERIVATION 25 topological field (
K, δ ) endowed with a definable topology into a differential extensionmodel of the scheme (DL).A topological field K is c-large if it is large and the following holds. We consideran embedding K ֒ → K (( s , · · · , s n − )) ֒ → K ∗ with K ∗ a non principal ultrapower of K , s , . . . , s n − algebraically independent over M and s i ∼ K
0, 0 ≤ i ≤ n − K (( s , . . . , s n − )) with a valuation v , trivial on K and with0 < v ( s ) < v ( s ) < · · · < v ( s n − ). Let M v be the maximal ideal of K [[ s , . . . , s n − ]],consisting of elements of strictly positive valuation. Then the additional condition weask is the following: M v embeds into the K -infinitesimal elements of K ∗ . Proposition 5.1.
Let K δ be a c-large topological differential field. Then we can embed K δ into a model of the scheme ( DL ) and with the same L -theory as K .Proof: We only show the main step. As classically done in the construction of exis-tentially closed models, one first enumerate the existential formulas (with parametersin the ground field K , belonging to a certain family) one wants to satisfy in an ex-tension and then one redo the construction ω times in order to get an existentiallyclosed model (with respect to that family of formulas) containing K as the union ofelementary extensions of K .Consider a differential polynomial p ( x ) in one variable in K { x } of order n > p ∗ (¯ a ) = 0 & s ∗ p (¯ a ) = 0 for some ¯ a := ( a , . . . , a n ) ∈ K n +1 . Then we wantto find an L -elementary extension of K and an extension of δ such that in that el-ementary extension there is a differential solution of p ( x ) = 0 with ¯ δ n ( b ) ∼ K ¯ a .We first consider an ultrapower K ∗ of K which is | K | + -saturated and in that ultra-power n elements s , . . . , s n − with s ∼ K i ≥ s i ∼ K ( s ,...,s i − )
0. Then s , . . . , s n − are algebraically independent over K . Since K is large, we have that K ⊂ K (( s , . . . , s n − )) ⊂ K ∗∗ for some ultrapower of K . Furthermore, we may assumethat s i , 0 ≤ i ≤ n −
1, are still K -infinitesimals in K ∗∗ .Set ¯ s n := ( s , · · · , s n − ). We rewrite ¯ a as (¯ a n , a n ) with | ¯ a n | = n , | a n | = 1 and p ∗ (¯ x ) as p ∗ (¯ x n , x n ) with | ¯ x n | = n , | x n | = 1. We consider the polynomial in x n : p ∗ (¯ a n + ¯ s n , x n ). We endow the field K (( s , . . . , s n − )) with a valuation v , trivial on K and with 0 < v ( s ) < v ( s ) < · · · < v ( s n − ). (So the value group is isomorphic to Z n with the lexicographic order.) We are in the conditions of Hensel’s Lemma since v ( p ∗ (¯ a n + ¯ s n , a n )) > v ( s ∗ p (¯ a n + ¯ s n , a n )) = 0. So there is b ∈ K (( s , · · · , s n − ))such that v ( b − a n ) > p ∗ (¯ a n + ¯ s n , b ) = 0 & s ∗ p (¯ a n + ¯ s n , b ) = 0 . By c-largeness, we have b ∼ K a n . We extend δ on K ( s , · · · , s n − ) by setting δ ( a i + s i ) = a i +1 + s i +1 , 0 ≤ i < n − δ ( a n − + s n − ) = b ∈ K ∗ . Thanks to equation(5.0.1), δ ( b ) is completely determined and belongs to the subfield of K ( s , · · · , s n − )of K ∗ . We set K = K ( s , · · · , s n − ) and extend δ on the relative algebraic closureof K in K ∗ . We apply Lowenheim-Skolem theorem to embed K in an L -elementaryextension ˜ K of K of the same cardinality and extend δ on ˜ K . ✷ References [1] Brouette Q., Cubides Kovacsics P., Point F., Density of definable types and elimination ofimaginaries in models of CODF, The journal of Symbolic Logic 84 (2019), no. 3, 1099-1117. ( † ) [2] Cubides Kovacsics P., Point F., Topological fields with a generic derivation, arXiv:1912.07912(december 17, 2019).[3] Dolich A., Miller C., Steinhorn C., Structures having o-minimal open core, Trans. Amer.Math. Soc. 362, number 3, 2010, 1371-1411.[4] van den Dries L., Dimension of definable sets, algebraic boundedness and Henselian fields,Annals of Pure and Applied Logic, vol. 45, 1989, 189-209.[5] van den Dries L., Tame topology and O-minimality, London Mathematical Society LectureNote Series 248, 1998.[6] Flenner J., Relative decidability and definability in Henselian valued fields, The Journal ofSymbolic Logic 76, number 4, 2011, 1240-1260.[7] Guzy N., Point F., Topological differential fields, Ann. Pure Appl. Logic 161 (2010), no. 4,570-598.[8] Guzy N., Point F., Topological differential fields and dimension functions, The Journal ofSymbolic Logic vol. 77, no. 4, Dec. 2012, 1147-1164.[9] Johnson W.A., Fun with fields, PhD thesis, University of Berkeley, spring 2016.[10] Marczewski, Independence and homomorphisms in abstract algebras, Fundamenta Mathe-maticae, vol. 50 (1961/1962), 45-61.[11] Peterzil, K., An o-minimalistic view of the group configuration, arXiv:1909.09994v1 (22 sep2019).[12] Pillay A., First order topological structures and theories, J. Symbolic Logic 52 (1987), no. 3,763-778.[13] Pillay A., On groups and fields definable in o-minimal structures, Journal of Pure and AppliedAlgebra 53 (1988) 239-255.[14] Pop, F., Henselian implies large, Ann. of Math. (2) 172 (2010), no. 3, 2183-2195.[15] Simon, P., Walsberg, E., Tame topology over dp-minimal structures, Notre-Dame J. FormalLogic 60 (2019) no.1, 61-76. Department of Mathematics (De Vinci), UMons, 20, place du Parc 7000 Mons, Belgium
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