Topological fields with a generic derivation
aa r X i v : . [ m a t h . L O ] D ec TOPOLOGICAL FIELDS WITH A GENERIC DERIVATION
PABLO CUBIDES KOVACSICS AND FRANC¸ OISE POINT ( † ) Abstract.
We study a class of tame theories T of topological fields and their extension T ∗ δ by a generic derivation. The topological fields under consideration include henselianvalued fields of characteristic 0 and real closed fields. For most examples, we show thatthe associated expansion by a generic derivation has the open core property (i.e., there areno new open definable sets). In addition, we show various transfer results between tameproperties of T and T ∗ δ , including relative elimination of field sort quantifiers, NIP, distalityand elimination of imaginaries, among others. As an application, we derive consequences forthe corresponding theories of dense pairs. In particular, we show that the theory of pairsof real closed fields (resp. of p -adically closed fields and real closed valued fields) admits adistal expansion. This gives a partial answer to a question of P. Simon. Contents
Introduction 11. Main results 22. Open expansions of topological fields 43. Theories of topological fields with a generic derivation 164. Transfer of elimination of imaginaries 285. Applications to dense pairs 306. Open core 33Appendix A. Classical transfers 37References 39
Introduction
The study of topological fields with a derivation has been traditionally divided in twomain branches. The first branch, as studied in [1, 32, 34, 35], treats the case where somecompatibility between the derivation and the topology is assumed ( e.g. , continuity). Thesecond branch, as studied in [14, 15, 29, 38, 41], deals with the case where no such compatibilityis required but rather a generic behaviour of the derivation occurs. An example of such ageneric behaviour arises in existentially closed ordered differential fields, a class studied andaxiomatized by Singer in [38]. Each branch seems to tackle different aspects of differentialfields and has its own applications.The purpose of this article is to further develop the study of generic derivations and showthat many tame properties of theories of topological fields transfer to their expansions by suchderivations. Examples of the topological fields under consideration include real closed fields
Mathematics Subject Classification.
Primary 12L12, 12J25, 12H05; Secondary 13N15, 03C60.
Key words and phrases.
Topological fields, differential fields, generic derivations, elimination of imaginaries,open core.( † ) Research director at the Fonds National de la Recherche Scientifique (FNRS-FRS). and henselian valued fields of characteristic 0. We adopt a uniform treatment and developmentof such topological fields in the spirit of Mathews [25] and Pillay [28], which we considerinteresting on its own. As an application of generic derivations, we derive consequences forthe corresponding theories of dense pairs of topological fields (as studied in [2, 12, 23, 33, 43],to mention a few), supporting the idea that this framework is a useful tool to study such pairsof structures.The following section gathers a more detailed overview of our main results.1. Main results
The article is divided into two main parts. The first part is devoted to the study a particularclass of theories of topological fields which we call open theories of topological fields . Informally,an open theory of topological fields is a first order topological theory of fields in the sense ofPillay [28] (i.e., the topology is uniformly definable) in which definable sets are finite booleancombinations of Zariski closed sets and open sets. This being said, we will allow multi-sortedstructures in our setting and restrict the above conditions to the field sort. The formaldefinition will be given in Section 2. Examples include complete theories of henselian valuedfields of characteristic 0 and the theory of real closed fields.We show various tameness properties for open theories of topological fields including thefact that the topological dimension defines a dimension function in the sense of van den Dries[42] (later Corollary 2.4.5) and that they eliminate the field sort quantifier ∃ ∞ (Proposition2.4.1). Of special interest for us is a cell decomposition theorem analogous to the recentcell decomposition theorem proven for dp-minimal topological structures by P. Simon and E.Walsberg in [37]. This corresponds to the following theorem: Theorem (Later Theorem 2.7.1) . Let T be an open theory of topological fields and K be amodel of T . Let X be a definable subset of K n . There are finitely many definable subsets X i with X = S X i such that X i is, up to permutation of coordinates, the graph of a definablecontinuous m i -correspondence f : U i ⇒ K n − d i , where U i is a definable open subset of K d i , forsome d i n , m i ≥ . Correspondences are simply multi-valued functions. A crucial input of the proof of theprevious theorem consists in showing that a definable correspondence on an open set is con-tinuous almost everywhere (i.e., outside of a set of lower dimension). This is the content ofProposition 2.6.10. When the topology on a model K of T is given by a valuation and Γ isthe value group of K , a similar result is proven for Γ-valued correspondences (see Proposition2.6.11). The proof presented here closely follows Simon and Walsberg’s argument, adaptingit to the present setting. It is worthy to point out that, in contrast with other cell decom-positions results for topological fields such as in Mathews [25], Simon and Walsberg’s proofis almost purely combinatorial and does not make use of an implicit function theorem ondefinable functions.The results of this first part will also play an essential role in the second part of the article.Let T be a first order topological L -theory of fields, again in the sense of Pillay. Let L δ denote the extension of L by a symbol δ for a derivation, and T δ be the theory T together withaxioms stating that δ is a derivation on the field sort. The second part of the article focuses onthe study of models of an L δ -extension T ∗ δ of T δ . Informally, models of T ∗ δ satisfy the followingproperty: for any unary differential polynomial P , if the ordinary polynomial associated with P has a regular solution a , then one can find differential solutions of P arbitrarily close to OPOLOGICAL FIELDS WITH A GENERIC DERIVATION 3 a . A derivation δ on a model K of T is called generic if ( K, δ ) is a model of T ∗ δ . The aboveproperty implies that the derivation is highly non-continuous.When T is the theory of real closed fields, the theory T ∗ δ corresponds to the theory of closedordered differential fields CODF as originally introduced and axiomatized by M. Singer in [38].The idea behind CODF has been generalized to many different contexts including work by M.Tressl [41] and N. Solancki [39] in the framework of large fields, and by N. Guzy and the secondauthor in [14, 15]. As in [14, 15], we will closely follow Singer’s original axiomatization. Themain difference in the present setting with respect to previous work is the explicit allowanceof multi-sorted languages. This permits us to include complete theories of henselian valuedfields of characteristic 0 by studying them in a multi-sorted language as defined by J. Flenner[11], where they admit relative quantifier elimination.Most of our results concerning topological fields with a generic derivation are proven in theparticular case when T is an open theory of topological fields as defined in part 1. For sucha theory T , we show several transfer results from T to T ∗ δ . Some of these results, such asthe transfer of quantifier elimination, NIP or distality, were known in the one-sorted case andwe present adapted arguments in the multi-sorted setting. New results include the followingtransfer of elimination of imaginaries, whose proof is based on an unpublished argument ofM. Tressl in the case of CODF. Theorem (Later Theorem 4.0.5) . Let T be an open theory of topological fields. Let G be acollection of sorts of L eq and L G denote the restriction of L eq to the sorts in G . Suppose that T admits elimination of imaginaries in L G and that the theory T ∗ δ has L -open core. Then thetheory T ∗ δ admits elimination of imaginaries in L G δ . Recall that T ∗ δ has L -open core if every open L δ -definable set is already L -definable. Weprovide a general criterion, both in the ordered and valued case, to show that T ∗ δ has L -opencore (see later Theorem 6.0.7). Here the cell decomposition theorem proven in the first partplays a crucial role. Using this criterion, we show that the theory T ∗ δ has L -open core in thefollowing cases: Theorem (Later Theorem 6.0.8) . Let T be either: ACVF ,p , RCVF , p CF d or the L RV -theoryof a henselian valued field of characteristic 0, as defined in [11] , with value group either a Z -group or a divisible ordered group. Then, the theory T ∗ δ has L -open core. As a consequence of the previous two theorems we obtain the following corollary:
Corollary (Later Corollary 4.0.7) . Let L G denote the geometric language of valued fields asdefined in [16] . The theories (ACVF ,p ) ∗ δ , RCVF ∗ δ and p CF ∗ δ have elimination of imaginariesin L G δ . One can also use the above transfer of elimination of imaginaries to give another proof ofthe fact that CODF has elimination of imaginaries. This result was first proved by the secondauthor in [29] and later reproved in [6]. The argument here presented corresponds to Tressl’sargument.Last but not least, we illustrate how the theory T ∗ δ provides a useful setting to study densepairs of models of a one-sorted open L -theory of topological fields T . Let L P be the expansionof L by a unary relation P and let T P be the theory of dense elementary pairs of models of T . If K | = T ∗ δ , then the pair ( K, C K ), where C K is the subfield of constants of K , is a denseelementary pair of models of T (see later Lemma 5.2.2). Using this observation, we derivevarious consequences for the theory T P . Among them, we show that if T ∗ δ has L -open core, CUBIDES KOVACSICS AND POINT then T P has L -open core (Theorem 5.2.4), providing a new proof that the theory T P has L -open core when T is either RCF, ACVF ,p , p CF d and RCVF. We also deduce that thetheory T P admits a distal expansion (namely T ∗ δ ) whenever T is a distal theory (see laterCorollary 5.2.6). In particular, we show that T P admits a distal expansion when T is RCF, p CF d and RCVF. It is worthy to note that even when T is a distal theory, the theory T P isnot in general distal [17]. Our result gives a positive answer to a particular case of a questionof P. Simon who asked if the theory of dense pairs of an o-minimal structure (extending agroup) has a distal expansion (see [27] for a discussion). T. Nell provided a positive answerin the case of ordered vector fields [27]. Our result extends to pairs of real closed fields.The paper is laid out as follows. Open theories of topological fields are studied in Section2: dimension properties are considered in Section 2.4; correspondences are studied in Sections2.6 and 2.6.1; and the cell decomposition theorem is presented in Section 2.7. Topologicalfields with a generic derivation are introduced in Section 3: consistency results are presentedin Section 3.3; relative quantifier elimination is given in Section 3.4 and its consequences aregathered in Section 3.5. In Section 4 we show the transfer of elimination of imaginaries underthe assumption of the open core property. The applications to dense pairs are presented inSection 5. Finally, the open core property is studied in Section 6. Some transfer proofs whichwere known in the one-sorted case (such as the transfer of NIP and distality) are gathered inthe Appendix. Acknowledgements.
We would like to thank Marcus Tressl for encouraging discussionsand for sharing with us his proof strategy to show elimination of imaginaries in CODF. It isprecisely his strategy what we adapted in the the general setting. We would also like to thankArno Fehm and Philip Dittmann for interesting discussions around henselian valued fields,and the Institute of Algebra of the Technische Universit¨at Dresden for its hospitality duringa visit of the second author in May 2019.2.
Open expansions of topological fields
Preliminaries.
Model theory.
We will follow standard model theoretic notation and terminology. Lower-case letters like a, b, c and x, y, z will usually denote finite tuples and we let ℓ ( x ) denote thelength of x . We will sometimes use ¯ x to denote a tuple ¯ x = ( x , . . . , x n ) where each x i isa tuple. Let L be a possibly multi-sorted language and M be an L -structure. For a sort S in L , we let S ( M ) denote the elements of M of sort S . For a single variable x , we let S x denote the sort of the variable x . Given a tuple of variables x = ( x , . . . , x n ) we let S x ( M ) = S x ( M ) × · · · × S x n ( M ). For an L -formula ϕ ( x ) we let ϕ ( M ) denote the set { a ∈ S x ( M ) : M | = ϕ ( a ) } . We let L ( M ) denote the extension of L by constants for all elements in M . By an L -definableset of M we mean definable with parameters, that is, of the form ϕ ( M ) for an L ( M )-formula ϕ . Given a complete L -theory T we let U denote a monster model of T .We let acl denote the model-theoretic algebraic closure operator on M . Given a sort S in L , we let acl S denote the model-theoretic algebraic closure restricted to S , that is, for anysubset C ⊆ S ( M ), we let acl S ( C ) = acl( C ) ∩ S ( M ). Note that acl S is a closure operator on S ( M ). OPOLOGICAL FIELDS WITH A GENERIC DERIVATION 5
For a subset X ⊆ R × T where R and T are finite products of sorts in L , and for a ∈ R ,the fiber of X over a is denoted by X a := { b ∈ T : ( a, b ) ∈ X } .2.1.2. Topological fields.
Throughout this article, every topological field will assumed to benon-discrete and Hausdorff.Let K be a field and τ be a topology on K making it into a topological field. The topologicalclosure of a set X ⊆ K n will be denoted by X and its interior by Int( X ). The frontier of X ,denoted Fr( X ), is equal to the set X \ X . The topological dimension of a non-empty subset X ⊆ K n , denoted dim( X ), is defined as the maximal ℓ n such that there is a projection π : K n → K ℓ such that Int( π ( X )) = ∅ (and equal to − X = ∅ ).We let L ring denote the language of rings {· , + , − , , } and L field := L ring ∪ { − } denote thelanguage of fields. We treat every field is an L field -structure by extending the multiplicativeinverse to 0 by 0 − = 0. Let L be a (possibly multi-sorted) language extending the languageof rings and suppose M is an L -structure. We say τ is an L -definable field topology if there isan L -formula χ τ ( x, z ) with x a single variable of field sort such that { χ τ ( M, a ) : a ∈ S z ( M ) } is a basis of neighbourhoods of 0. For example, if M is an ordered field and the order is L -definable, then the order topology on M is an L -definable field topology. Similarly, if ( M, v ) isa valued field and the relation { ( x, y ) ∈ M : v ( x ) v ( y ) } is L -definable, then the valuationtopology on M is an L -definable field topology.When K is a dp-minimal field, the following result of W. Johnson [21] guarantees theexistence of a definable field topology Theorem ([21, Theorem 1.3]) . Let ( K, + , · , · · · ) be an infinite field, possibly with extra struc-ture. Suppose K is dp-minimal but not strongly minimal. Then K can be endowed with anon-discrete Hausdorff definable field topology such that any definable subset of K has finiteboundary. Furthermore, the topology is always induced either by a non-trivial valuation or anabsolute value. For more on dp-minimal fields, we refer to reader to [20] and [21].2.2.
Open expansion of topological fields.
We will work in a first-order setting of topo-logical fields which follows the same spirit of [42, Section 2], [25] and [14]. The main newingredient of the present account is that we explicitly allow multi-sorted structures.Let L r be a relational extension of L field . For the rest of the article we will work in apossibly multi-sorted language L extending L r such that L and L r coincide on the field sort,and every new sort is a sort in L eq r . When L is multi-sorted and K is an L -structure, we willabuse of notation and identify K with the field sort and write any other sort of K as S ( K )(for S a sort in L ).Let K be a field of characteristic 0 endowed with an L -structure and an L -definable fieldtopology. Let T be its L -theory. Any such theory will be called an L -theory of topologicalfields . They are first order theories of topological structures in the sense of Pillay [28]. Wewill further impose the following two conditions on T which will be hereafter referred to as assumption ( A ):(i) T eliminates field sort quantifiers in L and(ii) for every tuple x of field sort variables, every field sort quantifier free L -formula ϕ ( x )is equivalent modulo T to a formula _ i ∈ I ^ j ∈ J i P ij ( x ) = 0 ∧ θ i ( x ) CUBIDES KOVACSICS AND POINT where I is a finite set, each J i is a finite set (possibly empty), P ij ∈ Q [ x ] \ { } and θ i ( x ) defines an open set in every model of T .Any L -theory of topological fields satisfying assumption ( A ) will be called an open L -theory oftopological fields . Note that any open L -theory of topological fields T is a complete L -theory.For K a model of T , when L is one-sorted and both the relations of L r and their complementare interpreted in K by the union of an open set and a Zariski closed set, such a model K isalso a topological L -field as defined in [14].2.2.1. Examples.
The theory T = Th( K ) is an open L -theory of topological fields in thefollowing cases:(1) when K is a real closed field and L is L of the language of ordered fields L field ∪ { < } .The definable topology is given by the order topology. We use in this case RCF for T .(2) When K is an algebraically closed valued field of characteristic 0 and L is the one-sorted language of valued fields L div = L field ∪ { div } . The definable topology corre-sponds to the valuation topology. We use in this case ACVF ,p for T , where p is aprime number or 0.(3) When K is a real closed valued field and L = L ovf is the language of ordered valuedfields L of ∪{ div } . We use in this case RCVF for T . The definable topology correspondsto both the order and the valuation topology (which coincide).(4) When K is a p -adically closed field of p -rank d and L is L p,d := L field ∪ { div , c , · · · , c d } ∪ { P n : n > } as defined in [31]. The definable topology corresponds to the valuation topology. Weuse in this case p CF d for T .(5) When K is a henselian valued field of characteristic 0 and L is the multi-sorted L RV -language as defined in [11] (having L field in the field-sort). The definable topologycorresponds again to the valuation topology. Examples include classical fields such as C (( t )), R (( t )) and more generally any Hahn power series field k (( t Γ )), where k is a fieldof characteristic 0 and Γ is an ordered abelian group.2.2.1. Remark. If T = Th( K ) is an open L -theory of topological fields, then T eq is an L eq -open theory of topological fields. In fact, if ( K , L ′ ) is an extension by definitions of a reductof ( K , L eq ), then the L ′ -theory of K is an L ′ -open theory of topological fields. For example,if K is an algebraically closed valued field of characteristic 0, its theory in the two-sortedlanguage of valued fields with a new sort for the value group also satisfies assumption ( A ).2.2.2. Remark.
Observe that most but not all theories in Examples 2.2.1 are dp-minimal.Indeed, while theories in (1)-(4) are dp-minimal, there are various henselian fields of equichar-acteristic 0 which are not dp-minimal. By a result of F. Delon [9] combined with results of Y.Gurevich and P. H. Schmitt [13], the Hahn valued field k (( t Γ )) is NIP if and only if k is NIP(as a pure field). Even assuming NIP, by a result of A. Chernikov and P. Simon in [7], when k is algebraically closed, the field k (( t Γ )) is dp-minimal if and only if Γ is dp-minimal. However,there are ordered abelian groups which are not dp-minimal, as follows by a characterizationof pure dp-minimal ordered abelian groups due to F. Jahnke, P. Simon and E. Walsberg in[20, Proposition 5.1].2.2.3. Question.
Is there any open L -theory of topological fields whose topology does notcome from an order or a valuation?In the remainder of the section we prove various tameness properties of open theoriesof topological fields. To begin with, we will prove that such theories eliminate the field OPOLOGICAL FIELDS WITH A GENERIC DERIVATION 7 sort quantifier ∃ ∞ and are algebraically bounded in the sense of [42, Definition 2.6]. Thisimplies that acl K induces a dimension function on definable sets in the sense of [42] whichwe show that agrees with the topological dimension (Section 2.4). In particular, when L is a one-sorted language, T is a geometric theory in the sense of [2]. We will finish thesection by showing that definable functions (and more generally definable correspondences)are continuous almost everywhere and that definable sets are finite unions of correspondencesas in the cell decomposition theorem proved by Simon and Walsberg for non-strongly minimaldp-minimal fields in [37, Proposition 4.1] (see Section 2.7).We start with some notation preliminaries together with some basic but crucial lemmasfrom commutative algebra.2.3. Some auxiliary lemmas from commutative algebra.
Through this section, K willdenote any field of characteristic 0.Let x = ( x , . . . , x n ) be a tuple of variables and y be a single variable. We will needto present Zariski closed subsets of K n +1 as finite unions of locally Zariski closed sets withfurther properties on formal derivatives. It will thus be useful to work with presentations ofideals rather than with the ideals themselves. Let us introduce some notation.Throughout Section 2.3, we let A be a finite subset of K [ x, y ] and R ∈ K [ x, y ]. We let A y := { P ∈ A | deg y ( P ) > } , deg y ( A ) := max { deg y ( P ) | P ∈ A} , A y max := { P ∈ A y | deg y ( P ) = deg y ( A ) } , and d A := X P ∈A y max deg y ( P ) . We let the L ring ( K )-formula Z A ( x, y ) be ^ P ∈A P ( x, y ) = 0 . Thus, the algebraic subset of K n +1 defined by A corresponds to Z A ( K ). For P ∈ K [ x, y ], welet Z P denote Z { P } . For an element R ∈ K [ x, y ] we let Z R A ( x, y ) be(2.3.1) Z A ( x, y ) ∧ R ( x, y ) = 0 . Lemma.
The locally Zariski closed subset Z R A ( K ) is the union of finitely many sets Z S B ( K ) such that(1) |B y | ;(2) S ∈ K [ x, y ] ;(3) deg y ( B ) deg y ( A ) .Proof. If |A y | = 1 there is nothing to prove, so suppose that |A y | >
1. By induction, it sufficesto show that Z R A ( K ) is the union of finitely many sets of the form Z S B ( K ) with S ∈ K [ x, y ]and B such that deg y ( B ) deg y ( A ) and |B y | < |A y | . We proceed by a second induction on d A . Let P ∈ A y max and P ∈ A y be such that P = P . If P = P mi =0 c i ( x ) y i with c i ∈ K [ x ],by Euclid’s algorithm (see [19, Lemma 2.14]), there is a positive integer ℓ and Q, R ∈ K [ x, y ]such that c ℓm P = P Q + R . In addition, deg y ( R ) < deg y ( P ). Setting B := A ∪ { c m } , wehave Z R A ( K ) = Z c m R A ( K ) ∪ Z R B ( K ) . CUBIDES KOVACSICS AND POINT
Letting A := ( A \ { P } ) ∪ { R } , we have that Z c m R A ( K ) = Z c m R A ( K ). Since d A < d A , theresult for Z c m R A follows by induction, and hence for Z c m R A . For Z R B ( K ), setting B = ( A \ { P } ) ∪ { c m , m − X i =0 c i ( x ) y i } , we have Z R B ( K ) = Z R B ( K ). Since d B < d B d A , the result for Z R B ( K ) follows by induction,which completes the proof. (cid:3) Lemma.
Suppose that |A y | = 1 . Then, the locally Zariski closed set Z R A ( K ) is a finiteunion of sets of the form Z S B ( K ) where S ∈ K [ x, y ] and either B ⊆ K [ x ] , or B y = { P } and ∂∂y P divides S .Proof. Let P be the unique element of A y and write it as P ( x, y ) = P mi =0 c i ( x ) y i for c i ∈ K [ x ].We proceed by induction on deg y ( A ) = deg y ( P ). First, note that for A = A ∪ { ∂∂y P } and S = ( ∂∂y P ) R we have Z R A ( K ) = Z S A ( K ) ∪ Z R A ( K ) . Since Z S A ( K ) has already the desired form, it suffices to show that Z R A is the union of finitelymany locally closed sets as in the statement. By Euclid’s algorithm, there is a positive integer ℓ and Q, R ∈ K [ x, y ] such that c ℓm P = ( ∂∂y P ) Q + R with deg y ( R ) < deg y ( ∂∂y P ). Setting A := A ∪ { c m } , we have Z R A ( K ) = Z c m R A ( K ) ∪ Z R A ( K ) . As before, letting A := ( A \ { P } ) ∪ { R } , we have that Z c m R A ( K ) = Z c m R A ( K ). Moreover, d A < d A . By Lemma 2.3.2, Z c m R A ( K ) is the union of finitely many locally closed sets Z S B with |B y | = 1, S ∈ K [ x, y ] and d B d A < d A , so the result follows by induction for eachsuch set. It remains to show the result for Z R A ( K ). Setting A := ( A \ { P } ) ∪ { c m , m − X i =0 c i ( x ) y i , m − X i =1 ic i ( x ) y i − } , we have Z R A ( K ) = Z R A ( K ). In addition, deg y ( A ) < deg y ( A ). The result follows again byLemma 2.3.2 and the induction hypothesis. (cid:3) Corollary.
Every locally Zariski closed subset of K n +1 can be written as a union ofsets of the form Z S B ( K ) where S ∈ K [ x, y ] and either B ⊆ K [ x ] , or B y = { P } and ∂∂y P divides S .Proof. Direct consequence of Lemmas 2.3.2 and 2.3.3. (cid:3)
Corollary.
Let T be an open L -theory of topological fields and K be a model of T .Then every L -definable subset of K n +1 is defined by _ j ∈ J Z S j A j ( x, y ) ∧ θ j ( x, y ) where x = ( x , . . . , x n − ) , y a single variable, and for each j ∈ J , θ j is an L ( K ) -formula thatdefines an open subset of K n , S j ∈ K [ x, y ] , and either(1) A j ⊆ K [ x ] \ { } or(2) A j ⊆ K [ x, y ] \ { } , A yj = { P j } and ∂∂y P j divides S j .Proof. Follows from assumption ( A ) and Corollary 2.3.4. (cid:3) OPOLOGICAL FIELDS WITH A GENERIC DERIVATION 9
Topological dimension and the algebraic closure.
Through this section, we let T be an open L -theory of topological fields and K be a model of T .Recall that an integral domain D is algebraically bounded (in the sense of [42]) if for everydefinable subset X ⊆ D n +1 there exist non-zero polynomials P , . . . , P m ∈ D [ x , . . . , x n , y ]such that for every a ∈ D n , if X a is finite, then X a ⊆ Z P i ( a,y ) ( D ) for some i ∈ { , . . . , m } .Being algebraically bounded implies that the algebraic dimension algdim, in the sense ofvan den Dries [42, Lemma 2.3], defines a dimension function on definable subsets of D [42,Proposition 2.7]. Given a definable set X ⊆ K n , algdim( X ) is the maximal integer k forwhich there is a ∈ X ( U ) such that the field extension K ( a ) | K has transcendence degree k .When acl K has the exchange property, we let dim acl K denote the induced dimension function.2.4.1. Proposition.
The field sort of every model of T is algebraically bounded. The algebraicdimension algdim on the field sort coincides with dim acl K and defines a dimension functionin the sense of [42] . In particular, T eliminates the field sort quantifier ∃ ∞ . When L is aone-sorted language, T is thus a geometric theory.Proof. Since open sets are infinite, algebraic boundedness follows directly from assumption( A ). It also follows from assumption ( A ) that algdim coincides with dim acl K . That algdimdefines a dimension function as defined in [42] follows by [42, Proposition 2.15]. The remainingproperties are straightforward. (cid:3) Lemma.
Let P ( x ) ∈ K [ x ] \ { } with x = ( x , . . . , x n ) . Then dim( Z P ( K )) < n .Proof. By induction on n . For n = 1, we have that Z P ( K ) is finite and the result is clear.Suppose the result holds for all k < n + 1, let y be a single variable and P ∈ K [ x, y ]. Write P as P di =0 c i ( x ) y i where c i ∈ K [ x ]. Suppose for a contradiction that Z P ( K ) contains an open set U × V where U ⊆ K n and V ⊆ K . For a ∈ U , if W di =1 c i ( a ) = 0, then the fiber Z P ( K ) a wouldbe finite, contradicting that it contains the infinite set V . Therefore, for every a ∈ U we have V di =0 c i ( a ) = 0. But this contradicts the induction hypothesis since for every i ∈ { , . . . , d } U ⊆ { a ∈ K n : d ^ i =0 c i ( a ) = 0 } ⊆ Z c j ( K ) ⊆ K n , where c j = 0 . (cid:3) Corollary.
Let P ∈ K [ x ] \{ } with x = ( x , . . . , x n ) . Then the set D ( P ) := K n \ Z P ( K ) is open and dense in K n with respect to the ambient topology. (cid:3) Proposition.
For every n > and every definable subset X ⊆ K n , dim( X ) =dim acl K ( X ) .Proof. Suppose X is defined over C ⊆ K and that dim acl K ( X ) = k . Let a ∈ X ( U ) be suchthat ( a i , . . . , a i k ) is an algebraically independent tuple over C . Let I = { i , . . . , i k } and π I : K n → K k be its corresponding projection. Letting π ( x ) := ( x i , . . . , x i k ), by assumption( A ), π I ( X ) is defined by _ i ∈ I ^ j ∈ J i P ij ( π I ( x )) = 0 ∧ θ i ( π I ( x )) , where each P ij is a non-zero polynomial with coefficients in C and θ i ( π I ( x )) defines an opensubset of K k . If for every i ∈ I , the set J i = ∅ , then ( a i , . . . , a i k ) would be algebraicallydependent over C . Thus, there is i ∈ I such that J i = ∅ . This shows Int( π I ( X )) = ∅ andtherefore dim acl K ( X ) dim( X ). Conversely, suppose dim( X ) = d and let π I : K n → K d be such that π I ( X ) has non-empty interior. It suffices to show that the open set Int( π I ( X )) contains (in U ) an algebraically independent tuple. This follows by compactness and Corollary2.4.3. (cid:3) Corollary.
The topological dimension satisfies the following properties for definablesets
X, Y ⊆ K n :(1) dim( X ) = 0 if and only if X is finite and non-empty,(2) dim( X ∪ Y ) = max(dim( X ) , dim( Y )) .(3) dim(Fr( X )) < dim( X ) = dim( X ) ,(4) dim is additive, that is: for a non-empty definable set X ⊆ K m + n and d ∈ { , , . . . , n } , dim [ a ∈ X ( d ) X a = dim( X ( d )) + d, where X ( d ) = { a ∈ K m : dim X a = d } and is definable.Proof. Points (1), (2) and (4) follows by Proposition 2.4.1. Point (3) follows by Proposition2.4.1 and [42, Proposition 2.23]. (cid:3)
Uniform structures.
In Sections 2.6 and 2.7 we will closely follow various results from[37]. For the reader’s convenience, and to make easier the comparison with [37], we will recallpart of their setting and notation.A basis for a uniform structure on a set A is a collection B of subsets of A satisfying thefollowing:(1) the intersection of the elements of B is the diagonal of A ;(2) if U ∈ B and ( x, y ) ∈ U , then ( y, x ) ∈ U ;(3) for all U, V ∈ B there is W ∈ B such that W ⊆ U ∩ V ;(4) for all U ∈ B there is a W ∈ B such that W ◦ W ⊆ U , where W ◦ W = { ( x, z ) ∈ A : ( ∃ y ∈ A )( x, y ) ∈ W, ( y, z ) ∈ W } . The collection B induces a topology on A by setting as a neighbourhood basis for a ∈ A , thecollection { U [ a ] : U ∈ B} where U [ a ] := { x ∈ A : ( a, x ) ∈ U } . Suppose M is an L -structure for some first-order language L . Let S be a sort in L andsuppose S ( M ) = A . We say B is a definable uniform structure on S (or a definable basis fora uniform structure on S ) if there is an L -formula ϕ ( x, y, z ) with ℓ ( x ) = ℓ ( y ) = 1 variables ofsort S such that B = { ϕ ( M, c ) : c ∈ D ⊆ S z ( M ) } for some 0-definable set D .Let K be a field endowed with an L -structure and an L -definable topology τ . Let χ τ ( x, z )be an L -formula defining a basis of open neighbourhoods of 0. The collection B K = { W c : c ∈ S z ( K ) } where W c := { ( a, b ) ∈ K : K | = χ τ ( a − b, c ) } is a definable uniform structure on K having τ as its induced topology.We will also need to equip certain ordered abelian groups extended by an infinitely largeelement ∞ with a definable uniform structure. Let Γ := (Γ , + , − , , < ) be an ordered abeliangroup and Γ ∞ := Γ ∪ {∞} for ∞ a new element satisfying, for all γ ∈ Γ: • γ < ∞ , • γ + ∞ = ∞ + γ = ∞ , • ∞ + ∞ = ∞ , OPOLOGICAL FIELDS WITH A GENERIC DERIVATION 11 • −∞ = ∞ .Let L og = { + , − , , < } be the language of ordered groups and L ∞ og be L og extended by a newconstant symbol ∞ . Let L be a language extending L ∞ og and Γ ∞ be an ordered abelian groupequipped with an L -structure. For γ ∈ Γ ∞ we let | γ | denote γ if γ > − γ otherwise.Consider the following L og -definable family B Γ := { W γ,ξ : γ, ξ ∈ Γ , < γ, ξ } where W γ,ξ := { ( x, y ) ∈ Γ ∞ : | x − y | < γ ∨ ( x = ∞ ∧ ξ < y ) ∨ ( y = ∞ ∧ ξ < x ) } Lemma.
Let Γ be an ordered abelian group which is either divisible or discrete. Then,the collection B Γ is a definable uniform structure on Γ ∞ .Proof. Conditions (1)-(3) are straightforward and hold for any group Γ.For condition (4), suppose first Γ is divisible and fix two strictly positive elements γ, ξ ∈ Γ.Let γ ′ be such that 0 < γ ′ < γ and ξ ′ := ξ + γ ′ . The reader may check that W γ ′ ,ξ ′ ◦ W γ ′ ,ξ ′ ⊆ W γ,ξ .Now suppose Γ is discrete and let 1 denote the minimal strictly positive element of Γ. Then,for any strictly positive elements γ, ξ ∈ Γ we have that W ,ξ ◦ W ,ξ ⊆ W γ,ξ . (cid:3) The induced topology on Γ ∞ by B Γ is the order topology extended by open sets of the form( γ, ∞ ] for every γ ∈ Γ.2.6.
Almost continuity of definable correspondences.
In the absence of finite Skolemfunctions, we need to deal with the more general concept of correspondence which we nowrecall (see also [37, Section 3.1]).2.6.1.
Definition. A correspondence f : E ⇒ K ℓ consists of a definable set E together witha definable subset graph( f ) of E × K ℓ such that0 < |{ y ∈ K ℓ : ( x, y ) ∈ graph( f ) }| < ∞ , for all x ∈ E. The set { y ∈ K ℓ : ( x, y ) ∈ graph( f ) } is also denoted by f ( x ). For a positive integer m , we say f is an m -correspondence if | f ( x ) | = m for all x ∈ E . The correspondence f is continuous at x ∈ E if for every open set V ⊆ K ℓ containing f ( x ), there is an open neighbourhood U of x such that f ( U ) ⊆ V .Note that a 1-correspondence can be trivially identified with a function. The followinglemma is a reformulation of [37, Lemma 3.1].2.6.2. Lemma.
Let U ⊆ K n be open and let f : U ⇒ M ℓ be a continuous m -correspondence.Every a ∈ U has a neighbourhood V such that there are continuous functions g , . . . , g m : V → M ℓ such that graph( g i ) ∩ graph( g j ) = ∅ when i = j and graph( f | V ) = graph( g ) ∪ · · · ∪ graph( g m ) . In addition, if f is definable, we can further choose V and the functions g i to be definable. (cid:3) Convention.
Let f : U ⊆ K m ⇒ K n be a correspondence. If m = 0, we identifygraph( f ) with a finite subset of K n . If U is an open subset and n = 0, then we identifygraph( f ) with the set U .2.6.4. Convention.
Given a definable set X , we say that a property holds almost everywhereon X if there is a definable subset Y ⊆ X such that dim( X \ Y ) < dim( X ) and the propertyholds on Y . The following result is a reformulation of [37, Proposition 3.7] in which we isolate thecomponents of its proof in an axiomatic way. Recall that a family of sets F is said to be directed if for every A, B ∈ F there is C ∈ F such that A ∪ B ⊆ C .2.6.5. Proposition ([37, Proposition 3.7]) . Let T be an L -theory of topological fields (notnecessarily satisfying assumption ( A ) ) and K be a model of T . Suppose K satisfies thefollowing properties(1) if A is a definable open subset of K n and B is a definable subset of A which is densein A , then the interior of B is dense in A ; in particular, dim( A \ B ) < dim( A ) .(2) if A, A , . . . , A k are definable subsets of K n , A is open and A = S ki =1 A i , then, Int( A i ) = ∅ for some i k .(3) if C ⊆ K m + n is a definable set such that the definable family { C a : a ∈ K m } is adirected family and S a ∈ K m C a has non-empty interior, then, there is a ∈ K m suchthat C a has non-empty interior.(4) There is no infinite definable discrete subset of K n .Then, for V ⊆ K n a definable open set, every definable correspondence f : V ⇒ K ℓ is contin-uous on an open dense subset of V , and thus is continuous almost everywhere on V .Proof. The proof is a word by word analogue after replacing [37, Lemma 2.6] by condition(1), [37, Corollary 2.7] by condition (2), [37, Lemma 3.5] by condition (3) and [37, Lemma3.6] by condition (4). (cid:3)
We will now show that all four conditions in Proposition 2.6.5 hold for open theories oftopological fields. Note that condition (2) already follows from Corollary 2.4.5.For the remaining of this section we assume T is an open L -theory of topological fields andlet K be a model of T .2.6.6. Lemma. If A is a definable open subset of K n and B is a definable subset of A whichis dense in A , then the interior of B is dense in A . In particular, dim( A \ B ) < dim( A ) .Proof. It suffices to show that Int( B ) = ∅ . By assumption (A) , B is defined by a formula ofthe form W i ∈ I V j ∈ J i P ij ( x ) = 0 ∧ θ i ( x ), where P ij ∈ K [ x ] \ { } and θ i ( x ) defines an open set inevery model of T . Suppose that for every i ∈ I , the set of indices J i is non-empty. Therefore,the algebraic dimension of B is strictly less than n . Since the topological and the algebraicdimension coincide, dim( B ) < n . By Corollary 2.4.5, dim( A \ B ) = n which implies there isan open subset U ⊆ A disjoint from B . This contradicts the density of B in A . Then, theremust be i ∈ I such that J i = ∅ , hence Int( B ) = ∅ . (cid:3) Lemma.
Suppose C ⊆ K m + n is a definable set inducing a definable family { C a : a ∈ K m } which is directed. If S a ∈ K m C a has non-empty interior, then there is a ∈ K m such that C a has non-empty interior.Proof. Let ϕ ( x, y ) with ℓ ( x ) = m and ℓ ( y ) = n be an L ( K )-formula defining C . Let Y ⊆ K n denote the definable set S a ∈ K m C a . By hypothesis, Int( Y ) = ∅ . Since the family { C a : a ∈ K m } is directed, we may assume there are infinitely many different C a in the family (asotherwise the result follows directly from Corollary 2.4.5).For y = ( y , . . . , y n ), we let ˜ y denote the tuple ( y , . . . , y n − ). By Corollary 2.3.5 applied tothe formula ϕ ( x, y ) with respect to the variable y n , ϕ ( x, y ) is equivalent to a finite disjunction W i ∈ I ϕ i where each ϕ i is of the form Z S i A i ( x, y ) ∧ θ i ( x, y ) where θ i ( K ) defines an open subsetof K m + n and either(1) A i ⊂ K [ x, ˜ y ], or OPOLOGICAL FIELDS WITH A GENERIC DERIVATION 13 (2) A y n i = { P i } and ∂∂y n P i divides S i .Collect all the subformulas of the disjunction of form (1) (resp. form (2)) and denote by ϕ ( x, y ) (resp. ϕ ( x, y )) their disjunction. We have that ϕ ( x, y ) = ϕ ( x, y ) ∨ ϕ ( x, y ).Note that if A i = ∅ for some i ∈ I , then each fiber C a contains the open set θ i ( a, K ) andhas therefore non-empty interior. Thus, we may assume that A i = ∅ for all i ∈ I . We proceedby induction on n . Let d be the maximum of the degrees (in y n ) of the polynomials occurringin all A i ’s.Assume n = 1. Suppose first that A i ⊆ K [ x ] for some i ∈ I . Then, the fiber C a containsan open set whenever Z S A i ( a, K ) = ∅ . If Z A i ( a, K ) = ∅ for every a ∈ K m , then we removethe corresponding member from the disjunction. Therefore, we are left with the case where ϕ ( x, y ) = ϕ ( x, y ). We show this case cannot happen. First, note that in this situation eachfiber C a has finite cardinality bounded by d | I | . Since the family { C a : a ∈ K m } is directed,there is a ∈ K m such that ϕ ( a , K ) = Y . But this contradicts that Y contains an open set(and is thus infinite). This concludes the case n = 1.Now assume n >
1. Let π : K n → K n − denote the projection onto the first n − a, u ) ∈ K m × K n − we denote by C a,u the fiber ( C a ) u = { b ∈ K : ( a, u, b ) ∈ C } .By the form of each formula ϕ i , each fiber C a,u either contains a non-empty open subset oris finite (and bounded by d | I | ). We uniformly partition the projection π ( C a ) of each fiber C a into sets π ( C a ) and π ( C a ) where π ( C a ) := { u ∈ K n − : C a,u contains an open set } and π ( C a ) := { u ∈ K n − : | C a,u | d | I |} . Since the definition is uniform, we have π ( Y ) = π ( [ a ∈ K m C a ) = [ a ∈ K m π ( C a ) = [ a ∈ K m π ( C a ) ∪ [ a ∈ K m π ( C a ) . As Y contains an open set, so does π ( Y ). Therefore, by Corollary 2.4.5, either π ( Y ) := [ a ∈ K m π ( C a ) \ ( [ a ∈ K m π ( C a ) ) or π ( Y ) := [ a ∈ K m π ( C a ) , contains an open set.2.6.8. Claim.
The set π ( Y ) must contain an open set. Suppose for a contradiction dim( π ( Y ) ) < n −
1. Partition Y into Y ∪ Y where Y i = { ( u, b ) ∈ K n : u ∈ π ( Y ) i } for i = 1 , . By Corollary 2.4.5, Y or Y contains an open subset. By construction, we have that π ( Y ) = π ( Y ) and, by assumption, dim( π ( Y ) ) < n −
1. Therefore dim( Y ) < n . Hence we musthave that dim( Y ) = n . By the additivity of the dimension function (Corollary 2.4.5), theremust be u ∈ π ( Y ) such that the fiber ( Y ) u is infinite. In particular, there are k ∈ N and a , . . . , a k ∈ K m such that S kj =1 C a j ,u has cardinality bigger than d | I | . Since the family { C a,u : a ∈ K m } is directed, there is a ∈ K m such that S kj =1 C a j ,u ⊆ C a,u , which contradictsthe fact that the fiber C a,u has cardinality smaller or equal than d | I | . This completes theclaim.Consider the directed family { π ( C a ) : a ∈ K m } . By the claim, π ( Y ) = S a ∈ K m π ( C a ) contains a non-empty open set. Therefore, by induction, there is a ∈ K m such that π ( C a )
14 CUBIDES KOVACSICS AND POINT contains an open subset, say U ⊆ K n − . For each i ∈ I , set U i := { u ∈ U : dim( Z S i A i ( a, u, K ) ∩ θ i ( a, u, K )) = 1 } . Given that U ⊆ π ( Y ) , we have that U = S i ∈ I U i . Then, by Corollary 2.4.5, there is i ∈ I such that U i contains an open set, say V ⊆ U i . By the definition of U i , the set Z S i A i ( a, u, K ) ∩ θ i ( a, u, K ) is infinite for every u ∈ V . Thus, since A i = ∅ , we must have A i ⊆ K [ x, ˜ y ]. But in this situation Z A i ( a, K ) = K n . Indeed, consider B = A i as a set ofpolynomials in K [ x, ˜ y ]. Then Z B ( a, K ) ⊆ K n − contains an open set, namely, V . By Lemma2.4.2, Z B ( a, K ) = K n − , and thus Z A i ( a, K ) = K n (as a subset of K n ). Therefore, the fiber C a contains the open set { ( u, b ) ∈ V × K : S i ( a, u, b ) = 0 ∧ θ i ( a, u, b ) } . This is indeed open, as the set defined by the formula S i ( x, ˜ y, y n ) = 0 ∧ θ i ( x, ˜ y, y n ) defines anon-empty open subset of K m + n by Corollary 2.4.3. (cid:3) Lemma.
There is no infinite definable discrete subset of K n .Proof. The proof goes by induction on n exactly as the proof of [37, Lemma 3.6]. The basecase follows directly straightforward by assumption ( A ). The inductive case follows word byword the proof in [37]. (cid:3) Proposition.
Suppose T is an open L -theory of topological fields and let K be amodel of T . For a definable open set V ⊆ K n , every definable correspondence f : V ⇒ K ℓ iscontinuous almost everywhere.Proof. This follows from Proposition 2.6.5 where conditions (1)-(4) correspond respectivelyto Lemma 2.6.6, Corollary 2.4.5, Lemma 2.6.7 and Lemma 2.6.9. (cid:3)
Almost continuity of definable functions to the value group.
In this section we let K be a model of an open L -theory of topological fields, where L is a language extending thelanguage of valued fields L div containing a sort for the value group Γ ∞ . We will prove a resultanalogous to Proposition 2.6.10 for definable functions f : U ⊆ K n → Γ ∞ , where U is an openset. We will show such a result when:( † ) there is a model K ′ of T for which Γ( K ′ ) is a divisible ordered abelian group in whichevery infinite L -definable set has an accumulation point;( †† ) there is a model K ′ of T with Γ( K ′ ) = Z .Observe that when Γ( K ′ ) is a pure divisible ordered abelian group, every infinite definableset contains a non-empty interval, and therefore ( † ) is satisfied. Similarly, when Γ( K ′ ) is apure Z -group, then ( †† ) is satisfied. This covers most examples listed in Examples 2.2.1. Inaddition, by Lemma 2.5.1, we have an L -definable uniform structure on Γ ∞ in both contexts.Since Γ ∞ is totally ordered, every Γ ∞ -valued definable correspondence can be definablydecomposed into finitely many definable Γ ∞ -valued functions. Thus, we do not need to workwith definable correspondences but only with definable functions.2.6.11. Proposition. K be a model of an open L -theory of topological fields, where L is alanguage extending the language of valued fields L div containing a sort for the value group Γ ∞ . Assume T satisfies either ( † ) or ( †† ) . Then, for every L -definable open set V ⊆ K n , any L -definable function f : V → Γ ∞ is continuous almost everywhere.Proof. Since the property stated in the proposition is an elementary property, we may sup-pose K is a model of T as in ( † ) (resp. ( †† )). The proof follows the same strategy as in[37, Proposition 3.7]. However, since there are two uniform structures at play, namely the OPOLOGICAL FIELDS WITH A GENERIC DERIVATION 15 uniform structure on K and the uniform structure on Γ ∞ , we include a proof for the reader’sconvenience. We let B K be the uniform structure of K and B Γ be the uniform structure of Γ(as defined in 2.5). Suppose for a contradiction there are a definable open set V ⊆ K n and adefinable function f : V → Γ ∞ which is discontinuous at every point in V . Let n be minimalwith this property.Assume n = 1. Consider the definable set B ⊆ B Γ × V of pairs ( W, a ) such that for each openneighbourhood U of a , there exists b ∈ U , such that ( f ( a ) , f ( b )) / ∈ W . Observe that the familyof fibers { B W : W ∈ B Γ } is a directed definable family. Indeed, for any W , W ∈ B Γ , taking W ∈ B Γ such that W ⊆ W ∩ W , we obtain B W ∪ B W ⊂ B W . By assumption on V and f , V = S W ∈B Γ B W . Therefore, by Lemma 2.6.7, there is W ∈ B Γ such that Int( B W ) = ∅ . Let O be an open definable subset of B W . Consider the definable set f ( O ) ⊆ Γ ∞ . If f ( O ) is finite,then f would be constant (and hence continuous) on an open definable subset of O , whichcontradicts the assumption. So suppose f ( O ) is infinite. If Γ satisfies ( † ), then f ( O ) has anaccumulation point γ ∈ Γ ∞ . If Γ = Z , f ( O ) must be either coinitial or cofinal. By considering − f , we may assume without loss of generality f ( O ) is cofinal. In this case, set γ = ∞ . Let W ′ ∈ B Γ be such that W ′ ◦ W ′ ⊆ W and consider the set A = { a ∈ O : ( f ( a ) , γ ) ∈ W ′ } . Inboth cases, by the choice of γ , the set A is an infinite definable subset of O and thereforecontains a non-empty open subset O ′ . For a ∈ O ′ , since O ′ ⊂ B W , there is b ∈ O ′ suchthat ( f ( a ) , f ( b )) / ∈ W . However ( f ( a ) , γ ) ∈ W ′ and ( f ( b ) , γ ) ∈ W ′ , so ( f ( a ) , f ( b )) ∈ W , acontradiction.Suppose n >
2. We may assume V is an open box of the form V × V , where V ⊆ K and V ⊆ K n − . For each ¯ y ∈ V , let f ¯ y : V → Γ ∞ , where f ¯ y := f ( t, ¯ y ). By the minimalityof n , f ¯ y is continuous almost everywhere for each ¯ y ∈ V . Thus, by additivity of dimension(Corollary 2.4.5), dim( { ( t, ¯ y ) : f ¯ y is discontinuous at t } < n. By replacing V and V by smaller open definable subsets, we may suppose f ¯ y is continuous on V for every ¯ y ∈ V . Defining B ⊆ B Γ × V as for n = 1, we may assume that there is W ∈ B Γ such that B W has non-empty interior and for every a ∈ B W and every neighbourhood U of a there is b ∈ U such that ( f ( a ) , f ( b )) / ∈ W . Let W ′ ∈ B Γ be such that W ′ ◦ W ′ ⊂ W . Considerthe definable family C ⊆ B K × ( V × V ) formed of pairs ( O, ( t, ¯ b )) such that if t ′ ∈ O [ t ], then( f ¯ y ( t ) , f ¯ y ( t ′ )) ∈ W ′ . Similarly as for n = 1, the definable family { C O : O ∈ B K } is directedand covers V × V . By Lemma 2.6.7, there is an O ∈ B K such that C O has non-empty interior.Possibly replacing V × V by smaller open subsets, we assume both that V × V ⊂ B O and V × V ⊆ O . Fix t ∈ V and let f t : V → Γ ∞ be the function f t ( y ) = f ( t, y ). By theminimality of n , there is ¯ z ∈ V such that f t is continuous at ¯ z . By continuity, possiblyshrinking V , we may assume that ( f t (¯ y ) , f t (¯ z )) ∈ W ′ for all ¯ y ∈ V . Now, if ( s, ¯ y ) ∈ V × V ,then ( f ( t, ¯ y ) , f ( t, ¯ z )) ∈ W ′ . In addition, since s ∈ O [ t ] and ( s, ¯ y ) ∈ O , we obtain that( f ( t, ¯ y ) , f ( s, ¯ y )) ∈ W ′ . Therefore, ( f ( t, ¯ z ) , f ( s, ¯ y )) ∈ W . This contradicts the assumption asthere are arbitrary close elements ( s, ¯ y ) to ( t, ¯ z ) such that ( f ( t, ¯ z ) , f ( s, ¯ y )) / ∈ W . (cid:3) Remark.
One may also replace in the above proposition the assumption that K is a model of an open L -theory of topological fields by assuming that the theory of K isdp-minimal. As shown in [37], all conditions used in the proof are also satisfied under theassumption of dp-minimality.2.7. Cell decomposition.
We finish this section with the cell decomposition theorem foropen L -theories of topological fields. It is an exact analogue of the cell decomposition [37,Proposition 4.1] proved for dp-minimal fields. Theorem.
Let T be an open L -theory of topological fields and K be a model of T . Let X be a definable subset of K n . There are finitely many definable subsets X i with X = S X i such that X i is, up to permutation of coordinates, the graph of a definable continuous m i -correspondence f : U i ⇒ K n − d i , where U i is a definable open subset of K d i , for some d i n , m i ≥ .Proof. One can argue exactly as in the proof of [37, Proposition 4.1] after replacing [37,Lemma 2.3] by Proposition 2.4.4, [37, Corollary 2.7] by Corollary 2.4.5 and [37, Proposition3.7] by Proposition 2.6.10. We include here an alternative argument.We proceed by induction on ( n, dim( X )). If n = 1, the statement is directly implied byassumption (A) . If dim( X ) = 0, then X is finite and the statement also holds. This shows thebase case ( n,
0) for each n . Suppose the result has been shown for all ( k, m ) with k < n + 1and that dim( X ) > x = ( x , . . . , x n ), y be a single variable and ϕ ( x, y ) be an L ( K )-formula defining X ⊆ K n +1 . By Corollary 2.3.4, X is a finite union of sets defined by formulas of the form(2.7.2) Z S A ( x, y ) ∧ θ ( x, y )where S ∈ K [ x, y ] and either A ⊂ K [ x ], or A y = { P } and ∂ y P ) divides S . Without loss ofgenerality, we may assume X is defined by a formula as in (2.7.2). If A = ∅ , then X is openand we are done, so suppose A 6 = ∅ . We split in cases depending on whether A ⊆ K [ x ] or A y = { P } . Case 1:
Suppose
A ⊆ K [ x ]. Let B denote A as a subset of K [ x ], so that Z B ( K ) is a subsetof K n . By induction hypothesis, the set Z B ( K ) decomposes into finitely many { Y i } i ∈ I each ofwhich is the graph of a continuous definable correspondence g i : V i ⇒ K n − m i . For each i ∈ I ,consider the set W i := ( V i × K ) ∩ θ ( K ) and the correspondence h i : W i ⇒ K n − m i defined by h i ( a, b ) := g i ( a ). Each W i is non-empty and open. Moreover, h i is continuous and definable.It is easy to check that X is the union over I of the graphs of the correspondences h i , up topermutation of variables. Case 2:
Suppose deg y ( P ) = d and consider for each i ∈ { , . . . , d } the definable set Y i := { a ∈ K n : ∃ = i yϕ ( a, y ) } . Let π : K n +1 → K n denote the projection onto the first n coordinates. By the case assump-tion, π ( X ) = S i d Y i . It suffices to show the result for each π − ( Y i ) ∩ X . So fix some i ∈ { , . . . , d } . By induction hypothesis, Y i is a finite union of sets { Z j } j ∈ J each of which isthe graph of a continuous definable ℓ j -correspondence g j : V j ⇒ K n − m j . Once more, it sufficesto show the result for π − ( Z j ) ∩ X for each j ∈ J , so fix j ∈ J . If m j = 0, then V j is finiteand therefore π − ( Z j ) ∩ X is finite too, so we are done. Suppose m j > iℓ j -correspondence h j : V j ⇒ K n − m j +1 whose graph precisely corresponds to π − ( Z j ) ∩ X . Itremains to take care of continuity. By Proposition 2.6.10, h j is continuous on an open densesubset U j of V j . Since the set π − ( V j ) ∩ X is the union of π − ( U j ) ∩ X and π − ( V j \ U j ) ∩ X , itsuffices to show the result for these two sets. The former is the graph of a continuous definablecorrespondence. For the latter, note that dim( V j \ U j ) < dim( U j ), which implies thatdim( π − ( V j \ U j ) ∩ X ) < dim( π − ( V j ) ∩ X ) dim( X ) . Thus, the result holds for π − ( V j \ U j ) ∩ X by the induction hypothesis. (cid:3) Theories of topological fields with a generic derivation
Let T be an L -theory of topological fields and L δ be the language L extended by a symbolfor a derivation (in the field sort). The second part of the article focuses on the study of an OPOLOGICAL FIELDS WITH A GENERIC DERIVATION 17 L δ -extension T ∗ δ of T . The derivation δ of any model of T ∗ δ will be called a generic derivation .Every such a derivation is highly non-continuous. The theory T ∗ δ will be defined in Section 3.2.In Section 3.3 we show various examples of theories T for which the theory T ∗ δ is consistent.Then, in Sections 3.4 and 3.5 we further investigate the theory T ∗ δ when T is an open L -theoryof topological fields, showing that many tame properties transfer from T to T ∗ δ .Before defining T ∗ δ , let us fix some notation and recall the needed background on differentialalgebra.3.1. Differential algebra background.
Let (
K, δ ) be a differential field of characteristic 0,that is, a field K of characteristic 0 endowed with an additive morphism δ : K → K whichsatisfies Leibnitz’s rule δ ( ab ) = δ ( a ) b + aδ ( b ). Such a function is called a derivation on K . Welet C K denote the field of constants of K , namely, C K := { a ∈ K : δ ( a ) = 0 } . It is a subfieldof K .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 ele-ment a = ( a , . . . , a n ) ∈ K n , we will write ¯ δ m ( a ) to denote the element (¯ δ m ( a ) , . . . , ¯ δ m ( a n )) ∈ K n ( m +1) . For notational clarity, we will sometimes use ∇ m instead of ¯ δ m , especially concern-ing the image of subsets of K n . For example, when A ⊆ K , we will use the notation ∇ m ( A )for { ¯ δ m ( a ) : a ∈ A } instead of ¯ δ m ( A ). Likewise for A ⊆ K n , ∇ m ( A ) := { ¯ δ m ( a ) : a ∈ A } ⊆ K n ( m +1) . We will always assume our tuples of variables are ordered, as for example x = ( x , . . . , x n ).Moreover, as a convention, given a variable y , the tuple ( x, y ) is ordered such that y is biggerthan x n (and similarly when y is an ordered tuple of variables).Given x = ( x , . . . , x n ), we let K { x } be the ring of differential polynomials in n + 1 differ-ential indeterminates x , . . . , x n over K , namely it is the ordinary polynomial ring in formalindeterminates δ j ( x i ), 0 i n , j ∈ N , with the convention δ ( x i ) := x i . We extend thederivation δ to K { x } by setting δ ( δ i ( x j )) = δ i +1 ( x j ). By a rational differential function wesimply mean a quotient of differential polynomials.3.1.1. Order and separant of a differential polynomial.
For P ( x ) ∈ K { x } and 0 i n , welet ord x i ( P ) denote the order of P with respect to 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 the order of P be ord( P ) := max { ord x i ( P ) : 0 i n } . Similarly, for a finite subset A of K { x } , we letord x i ( A ) := max { ord x i ( P ) : P ∈ A} and ord( A ) := max { ord( P ) : P ∈ A} . For R ∈ K { x } , we write ord x i ( A , R ) for ord x i ( A ∪ { R } ).Suppose ord( P ) = m . For ¯ x = (¯ x , . . . , ¯ x n ) a tuple of variables with ℓ (¯ x i ) = m + 1, welet P ∗ ∈ K [¯ x ] denote the corresponding ordinary polynomial such that P ( x ) = P ∗ (¯ δ m ( x )). This way we avoid expressions like (¯ δ m ) − ( A ) which might lead to confusion, and simply write ∇ − m ( A ). 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(3.1.1) 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 ofseparant to arbitrary polynomials with an ordering on their variables 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 ) by declaring 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 with theextended version for ordinary polynomials, i.e. , s P ∗ = s ∗ P .3.1.2. Minimal differential polynomials.
Let F ⊆ K be an extension of differential fields.Recall that an ideal I of F { x } is a differential ideal if for every P ∈ I , δ ( P ) ∈ I . For a ∈ K ,let I ( a, F ) denote the set of differential polynomials in F { x } vanishing on a . The set I ( a, F )is a prime differential ideal of F { x } . Let P ∈ I ( a, F ) be a differential polynomial of minimaldegree among the elements of I ( a, F ) having minimal order. Any such differential polynomialis called a minimal differential polynomial of a over F . Let h P i denote the differential idealgenerated by P and I ( P ) := { Q ( x ) ∈ F { x } : s ℓP Q ∈ h P i for some ℓ ∈ N } .3.1.2. Lemma. If P is a minimal differential polynomial for a over F , then I ( a, F ) = I ( P ) Proof.
See [24, Section 1]. (cid:3)
Rational prolongations.
We define an operation on K { x } sending P P δ as follows:for P written as in (3.1.1) P ( x ) P δ ( x ) = d X i =0 δ ( c i ( x ))( δ m ( x n )) i . A simple calculation shows that(3.1.3) δ ( P ( x )) = P δ ( x ) + s P ( x ) δ m +1 ( x n ) . Lemma-Definition.
Let x = ( x , . . . , x n ) be a tuple of variables and y be a singlevariable. Let P ∈ K { x, y } be a differential polynomial such that m = ord y ( P ) > . There isa 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 m + 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 , where ℓ i ∈ N , ord y ( Q i ) = ord y ( P ) and ord 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 . OPOLOGICAL FIELDS WITH A GENERIC DERIVATION 19
Proof.
It suffices to inductively define the polynomials Q i . By (3.1.3), if δ ( P ( x, y )) = 0 weobtain that δ m +1 ( y ) = − P δ ( x, y ) s P ( x, y ) , Setting Q = − P δ , the rational differential function f = Q s P satisfies the required property.Now suppose Q i has been defined and that f i = Q i ( s P ) ℓi satisfies δ m + i ( y ) = f i ( x, y ). By applying δ on both sides we obtain δ m + i +1 ( y ) = δ ( Q i ( x, y )) s P ( x, y ) − Q i ( x, y ) δ ( s P ( x, y )) s P ( x, y ) ℓ i . By replacing instances of δ m + i ( y ) in δ ( Q i ( x, y )) and δ ( s P ( x, y )) by f i ( x, y ), we obtain inthe numerator a differential polynomial of order m with respect to y . Setting Q i +1 as suchnumerator shows the result. The last assertion is a straightforward calculation. (cid:3) Notation.
For an integer d > x , we define the tuple ofvariables x ( d ) by induction on d as follows: x (0) := x and x ( d + 1) := ( x ( d ) , u d ), where u d isa variable. We will assume that if x and y are disjoint tuples of variables, then x ( d ) and y ( d )are also disjoint.3.1.6. Notation.
Let x = ( x , . . . , x n ) and y be a single variable. Let P ∈ K { x, y } be adifferential polynomial of order m and let ( f Pi ) i > be its rational prolongation along P . Let¯ x = (¯ x , . . . , ¯ x n ) where ¯ x i = ( x i , . . . , x im ) and ¯ y = ( y , . . . , y m ). For every d >
0, we let λ dP (¯ x ( d ) , ¯ y ( d )) be the L ( K )-formula: P ∗ (¯ x, ¯ y )) = 0 ∧ s ∗ P (¯ x, ¯ y )) = 0 ∧ d ^ i > y m + i = ( f Pi ) ∗ (¯ x ( i ) , ¯ y ) . Kolchin closed sets.
Let x be a tuple of variables with ℓ ( x ) = n . Similarly as in Section2.3, for a finite subset A of K { x } and R ∈ K { x } , we let Z A ( x ) denote the L δ ( K )-formula ^ P ∈A P ( x ) = 0 , and Z R A ( x ) denote the L δ ( K )-formula ^ P ∈A P ( x ) = 0 ∧ R ( x ) = 0 . Recall that a subset X ⊆ K n is called Kolchin closed if there is a finite subset
A ⊆ K { x } such that X = Z A ( K ). It is called locally Kolchin closed if X = Z R A ( K ) for some R ∈ K { x } .For the rest of Section 3.1.4, we let x = ( x , . . . , x n ), y be a single variable, A be a finitesubset of K { x, y } and R ∈ K { x, y } . We let k A := ( − A ⊆ K { x } min { ord y ( Q ) : Q ∈ A \ K { x }} otherwise.For an integer k > −
1, we let A ( k ) := { P ∈ A : ord y ( Q ) = k } .3.1.7. Lemma.
The set Z R A ( K ) is the union of finitely many sets Z S B B ( K ) such that ord y ( S B ) ord y ( A , R ) , ord x i ( B , S B ) ord x i ( A , R ) for each i n , and either(1) ord y ( B ) < ord y ( A ) or(2) ord y ( B ) = ord y ( A ) , B (ord y ( B )) = { P B } and s P B divides S B . Proof.
Letting m = ord( A ) and d = ord y ( A ), the result follows by Corollary 2.3.4 applied tothe polynomial ring K [¯ δ m ( x ) , ¯ δ d ( y )] with respect to the variable δ d ( y ). Note that since wework in an ordinary polynomial ring, the order in x i cannot increase. (cid:3) Lemma.
The set Z R A ( K ) is the union of finitely many sets Z S B B ( K ) such that ord y ( S B ) ord y ( A , R ) , ord x i ( B , S B ) ord x i ( A , R ) for each i n , and either(1) B ⊆ K { x } or(2) B ( k B ) = { P B } and s P B divides S B .Proof. We proceed by induction on ord y ( A ). If ord y ( A ) = −
1, then
A ⊆ K { x } and there isnothing to show. By Lemma 3.1.7, we may suppose there is P ∈ A such that A (ord y ( A )) = { P } and s P divides R . Letting D = A \ { P } , we have that ord y ( D ) < ord y ( A ). Thus, byinduction, Z R D ( K ) = [ j ∈ J Z S j B j where J is a finite set and for each j ∈ J the set Z S j B j satisfies the conclusion of the lemma.The result follows by setting C j := B j ∪ { P } , T j := s P S j and noting that Z R A ( K ) = [ j ∈ J Z T j C j ( K ) . We let the reader verify that the order bounds hold for this family of locally Kolchin closedsets. Note that if B j ⊆ K { x } then P C j = P , and otherwise P C j = P B j . (cid:3) Lemma.
Suppose there is a unique P ∈ A ( k A ) and that ord y ( P ) < ord y ( A ) . Then,there is a finite subset B of K { x, y } such that(1) ord y ( B ) = ord y ( P ) ,(2) ord x i ( B ) ord x i ( A ) + ord y ( A ) − ord y ( P ) , for each i n ,and if s P divides R , then Z R A ( K ) = Z R B ( K ) .Proof. For notational simplicity, let m = ord( A ), d = ord y ( A ) and k = ord y ( P ). Consider,for each Q ∈ A with ord y ( Q ) > k , the rational differential function T ( x, y ) = Q ∗ (¯ δ m ( x ) , ¯ δ k ( y ) , f P ( x, y ) , . . . , f Pd − k ( x, y )) , which arises by replacing all occurrences of δ k + i ( y ) by the rational differential function f Pi ( x, y )for each 1 i d − k . By Lemma-Definition 3.1.4, f Pi = Q i s ℓiP with ord y ( Q i ) k and for each1 j n , ord x j ( Q i ) = ( ord x j ( P ) + i if ord x j ( P ) > − T by the required power of s P , we obtain adifferential polynomial ‹ Q ( x, y ) with ord y ( ‹ Q ) k andord x j ( ‹ Q ) ord x j ( A ) + d − k = ord x j ( A ) + ord y ( A ) − ord y ( P )for each 1 j n . Define B as B := { Q ∈ A : ord y ( Q ) k } ∪ { ‹ Q : Q ∈ A , ord y ( Q ) > k } . By construction, B satisfies conditions (1)-(2). The last statement follows by Lemma-Definition3.1.4. (cid:3) OPOLOGICAL FIELDS WITH A GENERIC DERIVATION 21
Lemma.
The set Z R A ( K ) is the union of finitely many sets Z S B B ( K ) such that ord y ( S B ) ord y ( A , R ) , ord x i ( B , S B ) ord x i ( A , R ) + ord y ( A , R ) for each i n , and either • B ⊆ K { x } or • there is a unique P ∈ B of non-negative order in y , ord y ( P B ) ord y ( A , R ) and s P B divides S B .Proof. We proceed by induction on ord y ( A ). If ord y ( A ) = − y ( A ) >
0. By Lemma 3.1.8, we may suppose that there is P ∈ A suchthat A ( k A ) = { P } and s P divides R . If k A = ord y ( A ), then we are done. Otherwise, if k A < ord y ( A ), by Lemma 3.1.9 there is a finite subset C of K { x, y } such that Z R A ( K ) = Z R C ( K )and(1) ord y ( C ) = ord y ( P ),(1) ord x i ( C ) ord x i ( A ) + ord y ( A ) − ord y ( P ), for each 1 i n ,Since ord y ( C ) < ord y ( A ) we can apply the induction hypothesis to Z R C ( K ). So suppose Z R C ( K )is a finite union of locally Kolchin closed sets Z s B B ( K ) as in the statement. Let us show thateach Z S B B ( K ) satisfies the needed bounds with respect to Z R A ( K ). First, by (1) and (2)ord x i ( B , S B ) ord x i ( C , R ) + ord y ( C , R ) (ord x i ( A , R ) + ord y ( A , R ) − ord y ( P )) + ord y ( P )= ord x i ( A , R ) + ord y ( A , R ) . Similarly, ord y ( S B ) ord y ( C , R ) ord y ( A , R ) . Finally, if k B >
0, then ord y ( P B ) ord y ( C , R ) ord y ( A , R ) . (cid:3) The theory T ∗ δ . Let T be an L -theory of topological fields. Let L δ be the language L extended by a unary field sort function symbol δ . Denote by T δ the L δ -theory T togetherwith 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.
Let x be a tuple of field sort variables and w be a tuple of variables of othersorts. Let ϕ ( x, w ) be a field sort quantifier free L δ -formula. Then, there is an L -formula ψ such that T δ | = ∀ x ∀ w ( ϕ ( x, w ) ↔ ψ (¯ δ m ( x ) , w )) . Note that we use here the assumption that the restriction of L to the field sort is a relationalextension of L field . We define the order of ϕ as the minimal integer m such that ϕ is equivalentto ψ (¯ δ m ( x ) , w ) for a field sort quantifier free L -formula ψ . Even if ψ is not unique, we willdenote some (any) such L -formula by ϕ ∗ .We will now describe a scheme of L δ -axioms generalizing the axiomatization of closedordered differential fields (CODF) given by M. Singer in [38]. Let χ τ ( x, z ) be an L -formula providing a basis of neighbourhoods of 0. Abusing of notation, when x is a tuple of field sortvariables x = ( x , . . . , x n ) we let χ τ ( x, z ) denote the formula n ^ i =1 χ τ ( x i , z ) . Definition.
The L δ -theory T ∗ δ is the union of T δ and the following scheme of axioms(DL): given a model K of T δ , K satisfies (DL) if for every differential polynomial P ( x ) ∈ K { x } with ℓ ( x ) = 1 and ord x ( P ) = m >
1, for field sort variables y = ( y , . . . , y m ) it holds in K that ∀ z Ä ( ∃ y ( P ∗ ( y ) = 0 ∧ s ∗ P ( y ) = 0) → ∃ x Ä P ( x ) = 0 ∧ s P ( x ) = 0 ∧ χ τ (¯ δ m ( x ) − y, z ) ää . As usual, by quantifying over coefficients, the axiom scheme (DL) can be expressed in thelanguage L δ .The theory RCF ∗ δ is CODF.3.3. Consistency.
The main result of this section is Theorem 3.3.2 which shows that if T is a complete theory of henselian valued fields of characteristic 0, then T ∗ δ is consistent. As aconsequence we obtain the consistency of T ∗ δ for all theories T described in Examples 2.2.1. Forsome of such theories, the consistency of T ∗ δ has already been proved. Indeed, the consistencyof CODF was proved in [38] and was later generalized in [14] to a broader class of theories.Although we will follow a very similar strategy to the known proofs, our argument is basedon henselizations rather than using explicitly a notion of largeness (or topological largeness)for the fields under consideration.Let us start by a general criterion to show that T ∗ δ is consistent.3.3.1. Proposition.
Let T be a complete L -theory of topological fields and χ τ ( x, z ) be the L -formula defining a basis of neighbourhoods of 0. Suppose that for every model K of T andevery derivation δ on K the following holds( ∗ ) for every P ∈ K { x } ( ℓ ( x ) = 1 ) of order m > for which there is a ∈ K m +1 such that P ∗ ( a ) = 0 and s ∗ P ( a ) = 0 , there is a differential field extension ( F, δ ) of ( K, δ ) suchthat F is in addition an L -elementary extension of K and there is b ∈ F such that P ( b ) = 0 , s P ( b ) = 0 and for every c ∈ S z ( K ) F | = χ τ (¯ δ m ( b ) − a, c ) . Then, for every model K of T and every derivation δ on K , there is an extension K ≺ L L and an extension of δ to L making ( L, δ ) into a model of T ∗ δ . In particular, T ∗ δ is consistent.Proof. Fix some model K of T and some derivation δ on K . We use the following two stepconstruction to build ( L, δ ). Step 1:
We construct an L -elementary extension K ≺ L F K and a derivation on F K ex-tending δ as follows. Let ( P i ) i<λ be an enumeration of all differential polynomials P i ∈ K { x } with ord( P i ) = m i > a i ∈ K m i +1 such that P ∗ i ( a i ) = 0 and s ∗ P i ( a i ) = 0.Consider the following chain ( F i , δ i ) i<λ defined by(i) F := K ,(ii) ( F i +1 , δ i +1 ) is given by condition ( ∗ ) with respect to ( F i , δ i ) and P i , that is, F i ≺ L F i +1 , δ i +1 extends δ i and there is b i ∈ F i +1 such that P ( b i ) = 0, s P ( b i ) = 0 and for every c ∈ S z ( K ) F i +1 | = χ τ (¯ δ m ( b i ) − a i , c ) . OPOLOGICAL FIELDS WITH A GENERIC DERIVATION 23
Let F K := S i<λ F i and, abusing notation, let δ denote the union of the derivations δ i . Observethat indeed K ≺ L F K and ( K, δ ) ⊆ ( F K , δ ) is an extension of differential fields. Step 2:
Define a chain ( L i ) i<ω where ( L , δ ) := ( K, δ ), and L i +1 corresponds to the dif-ferential field ( F L i , δ ) obtained in Step (1) with respect to ( L i , δ ), so that L i ≺ L L i +1 . Let L := S i<ω L i and again, abusing of notation, δ denote the union of their derivations. By con-struction, K ≺ L L . It remains to show that L satisfies the axiom scheme (DL). Let P ∈ L { x } be a differential polynomial of order m > a ∈ L m +1 such that P ∗ ( a ) = 0and s ∗ P ( a ) = 0. Fix c ∈ S z ( L ) and let i < ω be such that a ∈ L m +1 i , c ∈ S z ( L i ) and P ∈ L i { x } .By Step (1), there is b ∈ L i +1 ⊆ L such that L i +1 | = χ τ (¯ δ m ( b ) − a, c ) , which shows the result. (cid:3) Theorem.
Let T be a complete L div -theory of henselian valued fields of characteristic0. For every model K of T and every derivation δ on K , there is an extension K ≺ L L andan extension of δ to L making ( L, δ ) into a model of T ∗ δ . In particular, T ∗ δ is consistent.Proof. By Proposition 3.3.1, it suffices to show condition ( ∗ ) above defined. Let χ τ ( x, z ) bethe L div -formula v ( x ) > v ( z ) ∧ z = 0 and let ( K, v ) be a model of T equipped with a derivation δ . Let Γ v denote the value group of ( K, v ). Suppose P ∈ K { x } is a differential polynomial oforder m > a = ( a , . . . , a m ) ∈ K m +1 such that P ∗ ( a ) = 0 and s ∗ P ( a ) = 0.Let t = ( t , . . . , t m ) be a tuple of new variables and consider the (ordinary) polynomial Q ( x ) = P ∗ ( a − t , . . . , a m − − t m − , x )in K ( t , . . . t m − )[ x ]. Let w : K ( t ) → Z n ∞ denote the t -adic valuation, that is, the iteratedcomposition of the t i -adic valuation such that 0 < w ( t ) ≪ w ( t ) ≪ · · · ≪ w ( t m ). Let Z n −→× Γ v denote the lexicographic extension of Γ v by Z n and v t : K ( t ) → ( Z n −→× Γ v ) ∞ denotethe composite valuation which sends a polynomial R ( t ) = P i ∈ I a i t i (in multi-index notation)to the pair ( w ( R ) , v ( a w ( R ) )).Note that w is a coarsening of v t . Let F = ( K ( t ) , w ) h and L = ( K ( t ) , v t ) h be theircorresponding henselizations. Without loss of generality, we may suppose that F ⊆ L andthat for all a ∈ F w ( a ) > ⇔ v t ( a ) > Γ v . Let us show that there is c ∈ F such that Q ( c ) = 0 and w ( c − a m ) >
0. The reduction ‹ Q of Q in F corresponds to P ∗ ( a , . . . , a m − , x ) ∈ K [ x ]. By assumption, ‹ Q ( a m ) = 0 and ∂∂x ‹ Q ( a m ) = 0. Then, by Hensel’s lemma, there is c ∈ F such that Q ( c ) = 0 and w ( c − a m ) > v t ( c − a m ) > Γ v ). This implies both that c / ∈ K and that ∂∂x Q ( c ) = 0. Weextend δ to the subfield K ( t , . . . , t m − , c ) ⊆ F by inductively setting(1) δ ( t i ) = δ ( a i ) + t i +1 − a i +1 for 0 i < m − δ ( t m − ) = c .Note that since Q ( c ) = 0 and ∂∂x Q ( c ) = 0, the derivative of c is already determined by therational prolongation f Q ( c ). Setting b := a − t , we have that P ( b ) = P ∗ (¯ δ m ( b )) = P ∗ ( a − t , . . . , a m − − t m − , c ) = Q ( c ) = 0 . Similarly, s P ( b ) = 0. In addition, for every e ∈ K × (3.3.3) v t (¯ δ m ( b ) − a ) = min { v t ( t ) , . . . , v t ( t m − ) , v t ( c − a m )) > v ( e ) . Extend the derivation from K ( t , . . . , t m − , b ) to L (such an extension always exists by [22,Theorem 5.1]). Let K ∗ be a saturated L div -elementary extension of K and c , . . . , c m ∈ K ∗ be such that Γ v < v ( c ) ≪ v ( c ) ≪ . . . ≪ v ( c m ) . Let g : ( K ( t , . . . , t m ) , v t ) → ( K ∗ , v ) be the (unique) L div -embedding over K sending t i → c i .Then g extends to an L div -embedding h : ( L, v t ) → ( K ∗ , v ). Equip K ( c , . . . , c m ) with theinduced derivation δ from h and extend it to K ∗ . Then P ( h ( b )) = 0, s P ( h ( b )) = 0 and since h is an embedding of valued fields it follows from (3.3.3) v (¯ δ m ( h ( b )) − a ) > v ( e ) , for every e ∈ K × , which completes the result. (cid:3) Corollary.
Let T be an L div -complete theory of henselian valued fields of characteristic0 and K be a model of T . Let ( K, L ) be an extension by definitions of some reduct of ( K, L eqdiv and let T ′ be the complete L -theory of some (any) model of T . Then the L δ -theory ( T ′ ) ∗ δ isconsistent. (cid:3) Corollary.
Let T be any theory from Examples 2.2.1. Then T ∗ δ is consistent.Proof. Except for CODF, all examples in Examples 2.2.1 correspond to theories of henselianvalued fields of characteristic 0 are extensions by definitions of a reduct of their L eqdiv ex-pansions. Thus, the result follows by Corollary 3.3.4. Note that the consistency of RCVF ∗ δ implies the consistency of CODF, as every model of RCVF ∗ δ is a model of CODF (the valuationtopology and the order topology induce the same topology on any model of RCVF). (cid:3) Remark.
Let (
K, v ) be a valued field of characteristic 0 endowed with a derivation δ . Let ( K h , v ) be the henselization of ( K, v ). Note that the derivation extends (uniquely)to K h . Let T be the theory of ( K h , v ). Theorem 3.3.2 implies that ( K, v, δ ) embeds as an L div ,δ -structure into a model of T ∗ δ .3.3.7. Remark.
Note that if T ∗ δ is consistent, then every model K of T embeds as an L -structure into a model of T ∗ δ . Indeed, take a model K ′ of T ∗ δ . Then the reduct of K ′ to L is a model of T , and since T is complete K ≡ L K ′ . The result follows by Keisler-Shelah’stheorem. A similar argument will be used later in Section 5.2.3.4. Relative quantifier elimination.
For the rest of Section 3 we let T be an open L -theory of topological fields and assume T ∗ δ is a consistent theory. As shown in Corollary 3.3.5,all theories T listed in Examples 2.2.1 satisfy such an assumption.We will need the following classical consequence of the axiom scheme (DL).3.4.1. Lemma ([14, Lemma 3.17]) . Let K be a model of T ∗ δ . Let O be an open subset of K n .Then there is a ∈ K such that ¯ δ n − ( a ) ∈ O . (cid:3) Theorem.
The theory T ∗ δ eliminates field sort quantifiers in L δ .Proof. Let Σ denote the set of field sort quantifier-free L δ -formulas, x, y be field sort tuplesof variables with ℓ ( y ) = 1 and ϕ ( x, y ) be a formula in Σ. Let K , K be two models of T ∗ δ and b i ∈ K ℓ ( x ) i be tuples which have the same Σ-type (i.e., they satisfy the same formulas in Σ).Let F i denote the differential subfield of K i generated by b i . The assumption on the tuples b and b implies there is an isomorphism of differential fields σ : F → F fixing Q and sending¯ δ ℓ ( b ) to ¯ δ ℓ ( b ) for every ℓ >
0. Moreover, by elimination of field sort quantifiers in T , wemay suppose F i algebraically closed in K i . Suppose there is a ∈ K such that K | = ϕ ( b , a ).We must show that there is c ∈ K such that K | = ϕ ( b , c ). Let m be the order of ϕ and OPOLOGICAL FIELDS WITH A GENERIC DERIVATION 25 ϕ ∗ (¯ x, ¯ y ) be the field sort quantifier-free L -formula such as in Notation 3.2.1. By assumption( A ), the formula ϕ ∗ (¯ x, ¯ y ) is equivalent to a finite disjunction of formulas of the form(3.4.3) ^ i ∈ I P i (¯ x, ¯ y ) = 0 ∧ θ (¯ x, ¯ y ) , where P i ∈ Q [¯ x, ¯ y ] \ { } , I possibly empty and θ (¯ x, ¯ y ) defines an open set (in every modelof T ). As existential quantifiers commute with disjunctions, we may suppose ϕ ∗ is alreadya conjunction as in (3.4.3). We split into cases depending on whether a is differentiallytranscendental over F or not: Case 1:
Suppose a is differentially algebraic (but not algebraic) over F . Let P ∈ F { x } be a minimal differential polynomial for a over F order k >
1. Since P is minimal, we musthave both k m and that s P ( a ) = 0. For d = m − k , we have then(3.4.4) K | = λ dP (¯ δ m + d ( b ) , ¯ δ m ( a )) . Let P σ (resp. P σi ) denote the corresponding polynomial over F in which every coefficientof P (resp. P i ) is replaced by its image under σ . Since I ( a, F ) = I ( P ) (by Lemma 3.1.2), P i ∈ I ( P ) for each i ∈ I . Our assumption on b and b implies that P σi ∈ I ( P σ ). Therefore,it suffices to show that there is c ∈ K such that K | = P σ ( c ) = 0 ∧ s P σ ( c ) = 0 ∧ θ (¯ δ m ( b ) , ¯ δ m ( c )) , as this will also imply that K | = V i ∈ I P i ( b , c ) = 0. By assumption ( A ) and (3.4.4), thereis ¯ e = ( e , . . . , e m ) ∈ K m +12 such that K | = λ dP σ (¯ δ m + d ( b ) , ¯ e ) (note that λ dP σ is an L ( K )-formula). Letting b e = ( e , . . . , e k ), the previous formula yields that e k + i = ( f P σ i ) ∗ (¯ δ m + i ( b ) , b e )for all 1 i d , where ( f P σ i ) i > is the rational prolongation of P σ . Since ( P σ ) ∗ ( b e ) = 0and s ∗ P σ ( b e ) = 0, the axiom scheme (DL) implies there is c ∈ K such that P σ ( c ) = 0 and s P σ ( c ) = 0. Moreover, by the continuity of the functions ( f P σ i ) ∗ , we may further suppose that θ (¯ δ m ( b ) , ¯ δ m ( c )) holds. This completes Case 1. Case 2:
Suppose b is differentially transcendental over F so I = ∅ . Since the set ϕ ∗ (¯ δ m ( b ) , K ) is an open subset of K m +12 , the result follows directly form Lemma 3.4.1. (cid:3) Consequences of quantifier elimination.
Corollary.
The theory T ∗ δ is complete.Proof. Let ϕ be an L δ -sentence. By Theorem 3.4.2, we may suppose ϕ has no variable offield sort. Therefore, since the constants of L belong to the subfield of constants of K , every L δ -term in ϕ is equal modulo T ∗ δ to an L -term. Then, ϕ is equivalent modulo T ∗ δ to an L -sentence and the result follows from the completeness of T . (cid:3) Let us recall some transfer results which (essentially) follow from Theorem 3.4.2.3.5.2.
Theorem ([15, Corollary 3.10]) . The L δ -definable subsets of the field sort can be en-dowed with a dimension function as defined by van den Dries in [42] . Theorem ([14, Corollary 4.3]) . If T is NIP, then T ∗ δ is NIP. Theorem (Chernikov) . If T is distal, then T ∗ δ is distal. Theorem.
The theory T ∗ δ eliminates the field sort quantifier ∃ ∞ . For the reader’s convenience, proofs of Theorems 3.5.3 and 3.5.4 will be given in the multi-sorted setting in the Appendix (the arguments are mutatis-mutandis , essentially the same).Theorem 3.5.5 was proven for CODF by the second author in [29] and the result is to our knowledge new in the general setting. Its proof is a little bit more involved and will be givenat the end of this section.It is worthy to mention that other model-theoretic properties such as the existence of primemodels or dp-minimality do not transfer from T to T ∗ δ . Indeed, M. Singer showed in [38] (seealso [30]) that CODF has no prime models (while RCF has) and Q. Brouette showed in histhesis [4] that CODF is not dp-minimal (while RCF is).Before, proving Theorem 3.5.5, let us start by showing some consequences of relative quan-tifier elimination on definable sets.3.5.6. Corollary.
Let K be a model of T ∗ δ and S be a sort of L different from the field sort.Then every L δ -definable subset X ⊆ S ( K ) n is L -definable.Proof. Let z = ( z , . . . , z n ) be S -sorted variables. By Theorem 3.4.2, there are a tuple x of field sort variables, a field sort quantifier free L δ ( K )-formula ϕ ( x, z ) (possibly with othernon-field sort parameters) and a ∈ K ℓ ( x ) , such that X is defined by ϕ ( a, z ). Since ϕ ( x, z ) hasno field quantifiers, X is also defined by ϕ ∗ (¯ δ m ( a ) , z ), where m is the order of ϕ . (cid:3) The following is a simple but important corollary of Theorem 3.4.2 that will be implicitlyused hereafter.3.5.7.
Corollary.
Every L δ -definable set X ⊆ K n is of the form ∇ − m ( Y ) for a field sortquantifier-free L -definable set Y ⊆ K n ( m +1) . (cid:3) Definition (Order) . Let X ⊆ K n be an L δ -definable set. The order of X , denotedby o ( X ), is the smallest integer m such that X = ∇ − m ( Y ) for some field-sort quantifier-free L -definable set Y ⊆ K n ( m +1) .Note that o ( X ) = 0 if and only if X is L -definable.We will finish by showing that T ∗ δ eliminates the field quantifier ∃ ∞ . To prove this weneed the following technical lemma, which is a parametric version of the density of differentialpoints (Lemma 3.4.1). This lemma will also play a crucial role in Section 6.1 to describe L δ -correspondences.3.5.9. Lemma.
Let K be a model of T ∗ δ . Let ϕ ( x, y ) be an L δ ( K ) -formula where x =( x , . . . , x n ) and y is a single variable. Let m be the order of ϕ . Then ϕ is equivalent toa finite disjunction of L δ ( K ) -formulas of the form Z S A ( x, y ) ∧ θ (¯ δ m ( x ) , ¯ δ m ( y )) , where θ is an L ( K ) -formula which defines an open subset of K ( n +1)( m +1) and either A ⊆ K { x } or A contains only one differential polynomial P of non-negative order in y and s P divides S . In addition, ord x i ( A , S ) m for i n and ord y ( A , S ) m .Proof. By Theorem 3.4.2 we may suppose ϕ has no field sort quantifiers. Consider the L ( K )-formula ϕ ∗ (¯ x, ¯ y ) where ¯ x = (¯ x , . . . , ¯ x n − ) with ℓ (¯ x i ) = m + 1 and ¯ y = ( y , . . . , y m ). Byassumption ( A ), ϕ ∗ (¯ x, ¯ y ) is equivalent to a disjunction of L -formulas of the form Z A (¯ x, ¯ y ) ∧ θ (¯ x, ¯ y ) , where θ defines an open subset of K ( n +1)( m +1) and A ⊆ K [¯ x, ¯ y ]. Define A ′ := { Q (¯ δ m ( x ) , ¯ δ m ( y )) : Q ∈ A} . OPOLOGICAL FIELDS WITH A GENERIC DERIVATION 27
By definition we have that ϕ is equivalent to the corresponding disjunction of L δ ( K )-formulasof the form Z A ′ ( x, y ) ∧ θ (¯ δ m ( x ) , ¯ δ m ( y )) . By Lemma 3.1.10, the formula Z A ′ ( x, y ) is equivalent (modulo T δ ) to a finite disjunction offormulas of the form Z S B B ( x, y ) such that ord y ( S B ) m , ord x i ( B , S B ) m for each 1 i n ,and either • B ⊆ K { x } or • there is a unique P ∈ B of non-negative order in y , ord y ( P B ) m and s P B divides S B .Then ϕ ( x, y ) is equivalent to the disjunction to the corresponding disjunction of L δ ( K )-formulas Z S B B ( x, y ) ∧ θ (¯ δ m ( x ) , ¯ δ m ( y )) . (cid:3) Lemma.
Let K be a model of T ∗ δ and X be an L δ -definable subset of K n +1 of order m . Then, for d = 2 m , there is an L -definable subset Y ⊂ K ( n +1)( d +1) such that(1) X = ∇ − d ( Y ) and(2) for every a ∈ K n and c ∈ K d +1 such that (¯ δ d ( a ) , c ) ∈ Y it holds that for every openneighbourhood W of c there is b ∈ K such that ¯ δ m ( b ) ∈ W and (¯ δ d ( a ) , ¯ δ d ( b )) ∈ Y .In particular, for every a ∈ K n such that if X a is finite, | X a | = | Y ¯ δ d ( a ) | .Proof. Let ϕ ( x, y ) be an L δ ( K )-formula of order m defining X where x = ( x , . . . , x n ) and y is a single variable.By Lemma 3.5.9, ϕ ( x, y ) is equivalent, modulo T ∗ δ , to a finite disjunction of the form _ j ∈ J Z S j A j ( x, y ) ∧ θ j (¯ δ m ( x ) , ¯ δ m ( y )) , where for each j ∈ J , θ j is an L ( K )-formula which defines an open subset of K ( n +1)( m +1) and either A j ⊆ K { x } or A j ⊆ K { x, y } , it only contains one differential polynomial P j of non-negative order in y and s P j divides S j . In addition, ord x i ( A j , S j ) m for 1 i n and ord y ( A j , S j ) m . For each j ∈ J , let ˜ θ j (¯ x ( m ) , ¯ y ( m )) be the L ( K )-formula θ (¯ x, ¯ y ) ∧ S ∗ j (¯ x ( m ) , ¯ y ) = 0. Note that ˜ θ j defines an open subset of K ( n +1)( d +1) . For each j ∈ J ,we define by cases an L ( K )-formula ψ j (¯ x ( m ) , ¯ y ( m )) depending on whether A ⊆ K { x } or not:( i ) if A j ⊆ K { x } then ψ j (¯ x ( m ) , ¯ y ( m )) is Z A j (¯ x ( m ) , ¯ y ) ∧ ˜ θ j (¯ x ( m ) , ¯ y ( m )) . ( ii ) otherwise, letting k j = ord y ( P j ) we define ψ j (¯ x ( m ) , ¯ y ( m )) as λ m − k j P j (¯ x ( m ) , ¯ y ) ∧ ˜ θ j (¯ x ( m ) , ¯ y ( m )) . Let ψ (¯ x ( m ) , ¯ y ( m )) be the disjunction W j ∈ J ψ j (¯ x ( m ) , ¯ y ( m )) and Y be the subset of K ( n +1)( d +1) defined by ψ . Let us show (1). The inclusion ∇ − d ( Y ) ⊆ X is clear. The converse follows bynoting that for each j ∈ JT ∗ δ | = ∀ x ∀ y ( Z S j A j ( x, y ) → λ m − k j P j (¯ δ d ( x ) , ¯ δ m ( y ))) . It remains to show (2).Fix a ∈ K n and c = ( c , . . . , c d ) ∈ K d +1 such that (¯ δ d ( a ) , c ) ∈ Y . Let j ∈ J be such that ψ j (¯ δ d ( a ) , c ) holds. We split in cases. If ψ j is as in ( i ), then the result follows from Lemma 3.4.1.So suppose ψ j is as in ( ii ). Let W be an open neighbourhood of c . Without loss of generality, we may suppose there is V an open neighbourhood of ¯ δ d ( a ) such that V × W ⊆ ˜ θ j ( K ). By thecontinuity of the functions f P ℓ i (see Lemma-Definition 3.1.4), we may shrink V to a smalleropen neighbourhood of ¯ δ d ( a ) and find an open neighbourhood W of ( c , . . . , c k j ) such that,letting U := V × W W × f P j ( U ) × . . . × f P j d − k j ( U ) ⊆ W. By the scheme (DL), we can find a differential tuple ¯ δ k j ( b ) ∈ W such that K | = P j (¯ δ m ( a ) , ¯ δ k j ( b )) = 0 ∧ ∂∂y k j P j (¯ δ m ( a ) , ¯ δ k j ( b )) = 0 . Since δ k j + i ( b ) = f P j i (¯ δ m + i ( a ) , ¯ δ k j ( b )) for each i ∈ { , . . . , d − k j } , we have that (¯ δ d ( a ) , ¯ δ d ( b )) ∈ V × W , and hence in ˜ θ ℓ ( K ). This shows that ψ j (¯ δ d ( a ) , ¯ δ d ( b )) holds, so (¯ δ d ( a ) , ¯ δ d ( b )) ∈ Y .The last statement follows directly from part (2) and the fact the topology is Hausdorff. (cid:3) We have now all tools to show Theorem 3.5.5.
Proof of Theorem 3.5.5:
Let K be a model of T ∗ δ . Let X ⊆ K n +1 be an L δ -definable set oforder m . Let d = 2 m and Y ⊆ K ( n +1)( d +1) be the L -definable set given by Lemma 3.5.10.Since T eliminates ∃ ∞ , there is a finite bound n Y such that for any n ( d + 1)-tuple of elements¯ e of K , either Y ¯ e is infinite or has cardinality n Y . By Lemma 3.5.10, if X a is finite for a ∈ K n , then | X a | = | Y ¯ δ d ( a ) | n Y , so the same bound shows the result for X . (cid:3) Transfer of elimination of imaginaries
In this section, following a proof strategy due to M. Tressl to show elimination of imagi-naries in CODF, we show how to transfer elimination of imaginaries from T to T ∗ δ under theadditional assumption on T ∗ δ of having L -open core . For background facts on the eliminationof imaginaries, we refer to [40, Section 8.4]. Let us start by recalling the definition of opencore. Throughout this section we let T be an open L -theory of topological fields and assume T ∗ δ is consistent. We let K be a model of T .4.0.1. Definition.
Let ˜ L be an extension of L and ˜ K be an ˜ L -expansion of K . We say ˜ K has L -open core if every ˜ L -definable open subset is L -definable. An ˜ L -theory ˜ T extending T has L -open core if every model of ˜ T has L -open core.We will use the following three properties satisfied by the topological dimension on L -definable sets X, Y ⊆ K n :(D1) dim( X ) = 0 if and only if X is finite and non-empty,(D2) dim( X ∪ Y ) = max(dim( X ) , dim( Y )),(D3) dim(Fr( X )) < dim( X ) = dim( X ).(see Corollary 2.4.5).Before proving the main theorem of this section, we will give a useful characterization ofthe L -open core property for T ∗ δ .4.0.2. Definition.
Let X ⊆ K n be a non-empty field sort quantifier-free L δ -definable set.Given a positive integer m and an L -definable set Z ⊆ K n ( m +1) , we call the triple ( X, Z, m )a linked triple if(1) X = ∇ − m ( Z ) and(2) Z = ∇ m ( X ). OPOLOGICAL FIELDS WITH A GENERIC DERIVATION 29
Note that the integer m occurring in a linked triple might be bigger than o ( X ). However,as the next proposition shows, in our setting one can always take m = o ( X ).4.0.3. Proposition.
The theory T ∗ δ has L -open core if and only if for every L δ -definable set X , there is an integer m and an L -definable set Z ⊆ K n ( m +1) , such that ( X, Z, m ) is a linkedtriple. In addition, if T ∗ δ has L -open core, for every X there is a linked triple of the form ( X, Z, o ( X )) .Proof. Let X ⊆ K n be an L δ -definable set. By Theorem 3.4.2, we may assume X is definedby a field-sort quantifier-free L δ ( K )-formula.( ⇒ ) Let Y ⊆ K ( o ( X )+1) n be an L -definable set such that X = ∇ − o ( X ) ( Y ). The subset ∇ o ( X ) ( X ) is both closed and L δ -definable and so it is L -definable by the L -open core. Considerthe L -definable set Z := Y ∩ ∇ o ( X ) ( X ). Since ∇ o ( X ) ( X ) ⊆ Z ⊆ ∇ o ( X ) ( X ), both properties(1) and (2) are easily shown. This also shows the last assertion of the proposition.( ⇐ ) It suffices to show that X is L -definable. By assumption there is an integer m andan L -definable set Z such that ( X, Z, m ) is a linked triple. Let π : K n ( m +1) → K n be theprojection sending each block of m + 1 coordinates to its first coordinate, that is, π ( x , , . . . , x ,m , x , , . . . , x ,m , . . . , x n, , . . . , x n,m ) = ( x , , x , , . . . , x n, ) . We leave as an exercise to show that X = π ( Z ). The results follows since, as Z is L -definable,so is π ( Z ). (cid:3) Lemma.
Let X ⊆ K n be field sort quantifier-free L δ -definable set. If m m then dim( ∇ m ( X )) dim( ∇ m ( X )) . Proof.
Let π : ∇ m ( X ) → K ℓ be a projection such that π ( ∇ m ( X )) has non-empty interior.Then, letting ρ denote the projection from ∇ m ( X ) onto ∇ m ( X ), we have that π ◦ ρ ( ∇ m ( X ))has non-empty interior. (cid:3) Let G be a collection of sorts of L eq . We let L G denote the restriction of L eq to the fieldsort together with the new sorts in G . Given an automorphism σ and a set X , we say that X is σ -invariant if σ fixes X setwise.4.0.5. Theorem.
Suppose that the L δ -theory T ∗ δ has L -open core and that T admits elimina-tion of imaginaries in L G . Then the theory T ∗ δ admits elimination of imaginaries in L G δ .Proof. Fix a sufficiently saturated model K of T ∗ δ . Let X ⊆ K n be a non-empty L δ -definableset. It suffices to show that X has a code in L G δ (that is, an element e ∈ G such that σ ( e ) = e if and only if σ ( X ) = X for every L δ -automorphism σ of K ). Observe that every L -definableset has a code in L G , and therefore a code in L G δ , as the L δ -automorphism group of K is asubgroup of the L -automorphism group of K . Consider the set ‹ X ⊇ X defined by ‹ X := ∇ − o ( X ) ( ∇ o ( X ) ( X )) . Since T ∗ δ has L -open core, the set ∇ o ( X ) ( X ) is L -definable. We proceed by induction ondim( ∇ o ( X ) ( X )). If dim( ∇ o ( X ) ( X )) = 0, then X is finite (by (D1)) and in particular L -definable, so it has a code in L G δ . Alternatively, one may use that every finite definable sethas a code modulo the theory of fields [24, Lemma 3.2.16]. To show the inductive step weneed the following claim:4.0.6. Claim. dim( ∇ o ( X ) ( ‹ X \ X )) < dim( ∇ o ( X ) ( X )) . Suppose the claim holds. Since o ( ‹ X \ X ) o ( X ), by Lemma 4.0.4, we have thatdim( ∇ o ( e X \ X ) ( ‹ X \ X )) dim( ∇ o ( X ) ( ‹ X \ X )) . Therefore, by Claim 4.0.6 and the induction hypothesis, let e be a code for ‹ X \ X . By theprevious observation, let e be a code for ∇ o ( X ) ( X ) (which is L -definable by the L -open corehypothesis). It is an easy exercise to show that e = ( e , e ) is a code for X .It remains to prove the claim. By the L -open core assumption and Proposition 4.0.3, let( X, Z, o ( X )) be a linked triple. Applying (D3), we havedim( ∇ o ( X ) ( ‹ X \ X )) = dim( ∇ o ( X ) ( ∇ − o ( X ) ( ∇ o ( X ) ( X )) \ X ))= dim( ∇ o ( X ) ( { x ∈ K n : ∇ o ( X ) ( x ) ∈ Z } \ X ))= dim( ∇ o ( X ) ( { x ∈ K n : ∇ o ( X ) ( x ) ∈ Z \ Z } )) dim( Z \ Z ) < dim( Z ) = dim( ∇ o ( X ) ( X )) . (cid:3) We will later show in Section 6 that T ∗ δ has L -open core for most L -theories T given inExamples 2.2.1 (including all henselian valued fields of characteristic 0 having a value groupwhich is either divisible or a Z -group). As a corollary we obtain the following.4.0.7. Corollary.
Let G denote the geometric language of valued fields. The theories ACVF ∗ δ , RCVF ∗ δ and p CF ∗ δ have elimination of imaginaries in L G δ .Proof. Let T be either ACVF, RCVF or p CF d . The theory T has elimination of imaginariesin L G by results of Haskell, Hrushovski and Macpherson for ACVF [16], of Mellor for RCVF[26], and of Hrushovski, Martin and Rideau for p CF d [18]. By Corollary 6.0.8, T ∗ δ has L -opencore. The result follows by Theorem 4.0.5. (cid:3) The fact that CODF has L -open core and eliminates imaginaries was first proved in [29] bydifferent methods. The following proof strategy is precisely Tressl’s unpublished argument.4.0.8. Corollary ([29]) . The theory
CODF has elimination of imaginaries in L δ .Proof. By Corollary 6.0.8, CODF has L -open core. The result follows by Theorem 4.0.5. (cid:3) Applications to dense pairs
The study of pairs of models of a given complete theory is a classical topic in modeltheory. Early results by A. Robinson [33] showed completeness (and model-completeness) ofthe theories of pairs of algebraically closed fields and dense pairs of real-closed fields, thatis, pairs in which the smaller field is dense in the larger one. In [23], A. Macintyre recastedRobinson’s results in an abstract setting which also encompassed dense pairs of p -adicallyclosed fields. In another direction, L. van den Dries [43] studied dense pairs of models of ano-minimal theory expanding the theory of ordered abelian groups, also generalizing some ofRobinson’s results.New developments have encompassed these results in different abstract frameworks. Twosuch frameworks are the theory of lovely pairs of geometric structures developed by A. Beren-stein and E. Vassiliev [2], and the theory of dense pairs of theories with existential matroidsdeveloped by A. Fornasiero [12]. In this section we will study the theory T P of dense pairs ofmodels of a one-sorted L -open theory of topological fields T . Our goal is to show that suchtheory is closely related with the theory T ∗ δ . In Section 5.1, we will define the theory T P and OPOLOGICAL FIELDS WITH A GENERIC DERIVATION 31 show how it fits into the two above mentioned abstract frameworks. In Section 5.2, we willshow how to use T ∗ δ to deduce properties of T P . Although most of the results gather in thissection concerning T P are known, the proofs and methods will put in evidence the interestingconnexion between the model theory of dense pairs and generic derivations.5.1. Dense pairs of models of T . Let us start by recalling Fornariero’s setting in [12].Given that the literature of the model theory of pairs is quite extensive, we will unify referencesand cite [12] even if particular cases of cited results where proven before by many differentauthors. Let T be a complete one-sorted geometric L -theory T extending the theory of fields(not necessarily an open L -theory of topological fields). Fornasiero considers more generallythe case where T admits an existential matroid (see [12, Definition 3.25]), but we will not needthis level of generality in the present paper. Let dim acl denote the dimension function inducedby the algebraic closure acl. Given a model M of T and a definable subset X ⊆ M , we saythat X is dense if X ∩ U = ∅ for every M -definable subset U of M such that dim acl ( U ) = 1(see [12, Definition 7.1]). Now we are ready to define the theory of dense pairs of models of T .We work in the language of pairs L P defined as L P := L ∪ { P } for P a new unary predicate.The theory T P of dense pairs of models of T is defined as the L P -theory of pairs ( K, P ( K ))such that K | = T , P ( K ) is acl-closed and dense in K (in the above sense). Equivalently(by [12, Lemma 7.4]), it corresponds to the L P -theory of pairs ( K, P ( K )) such that K | = T , P ( K ) ≺ L K and P ( K ) is dense in K . Among various model-theoretic results which areproven in [12] about the theory T P , what plays a crucial role in this section is the fact that T P is a complete theory [12, Theorem 8.3].Let us now recall the framework of lovely pairs of geometric theories introduced by A.Berenstein and E. Vassiliev [2]. Let M be a model of T and N ⊆ M be a proper subset of M . The pair ( M, N ) is said to be a lovely pair (see also [2, Definition 2.3]) if(1) N = acl( N ) and(2) for every A ⊆ M with acl( A ) = A and dim acl ( A ) ∈ N , and for every non-algebraictype q ∈ S ( A ),(a) there exists an element a ∈ N realizing q ,(b) there exists an element a ∈ M realizing q with a / ∈ acl( A ∪ N ).Berenstein and Vassiliev showed that all lovely pairs of models of T are elementarily equiv-alent [2, Corollary 2.9] and gave an explicit axiomatization of their common L P -theory whichwe will denote by T LP [2, Theorem 2.10]. They also showed that | T | + -saturated models of T LP are lovely pairs. It is not difficult to show that a lovely pair of models of T is a model of T P and therefore, in the light of the previous results, the theories T LP and T P coincide.Observe that when T is a one-sorted L -open expansion of topological fields, by Proposition2.4.1, T is geometric and therefore, T P is complete. Note also that in view of Part (2) ofProposition 2.4.4, the notion of density above defined coincides with the topological notion ofdensity.5.2. Dense pairs and generic derivations.
Throughout this section we suppose T is aone-sorted L -open expansion of topological fields for which T ∗ δ is consistent. The connectionof T P with the theory T ∗ δ arises via the field of constants C K of a model K of T ∗ δ . One canreadily observe that when K | = CODF, the pair ( K, C K ) is a dense pair of real-closed fields.The following lemma shows this holds in general for T ∗ δ . It was proven in [5, Corollary 1.7]under the assumption that the language L = L ring and that the theory T is a model-completetheory of large fields. It can also be deduced from [12, Lemma 7.4] and from [2, Lemma 2.5].For the convenience of the reader, we give a proof here, following the last two references. Lemma.
Let K be a model of T ∗ δ and C K be the constant subfield of K . Then ( K, C K ) is a model of T P and if K is | T | + -saturated, then ( K, C K ) is a lovely pair of models of T .Proof. Let K | = T ∗ δ . Then, a direct consequence of the scheme (DL) is that C K = K . Since C K is topologically dense in K [14, Lemma 3.12], C K is dense. So it remains to show that C K | = T . We apply Tarski-Vaught test. Let ϕ ( x, ¯ y ) be an L -formula and let ¯ b ∈ C K . Byhypothesis ( A ) on T , ϕ ( K, ¯ b ) is a finite union of finite subsets and open sets. Since C K is algebraically closed in K , either ϕ ( K, ¯ b ) ⊂ C K or contains an open subset. Since C K istopologically dense in K , we get the result. (cid:3) Lemma.
Every model ( K, F ) of T P has an L P -elementary extension ( K ∗ , F ∗ ) suchthat there is a generic derivation on K ∗ with constant field F ∗ .Proof. Since T P is complete, by Lemma 5.2.1, there is a model K ′ of T ∗ δ with constant field F ′ such that ( K ′ , F ′ ) ≡ L P ( K, F ). By Keisler-Shelah’s isomorphism theorem, there is a set I and an ultrafilter F on I such that( K, F ) L P ( K, F ) I / F =: ( K ∗ , F ∗ ) ∼ = L P ( K ′ ∗ , F ′ ∗ ) := ( K ′ , F ′ ) I / F < L P ( K ′ , F ′ ) . Since K ′ is a model of T ∗ δ , we have that K ′ ∗ is also a model of T ∗ δ with constant field F ′ ∗ = F I / F . Hence, the isomorphism ( K ∗ , F ∗ ) ∼ = L P ( K ′ ∗ , F ′ ∗ ) induces on K ∗ an L δ -structuremaking of K ∗ a model of T ∗ δ with constant field F ′ . (cid:3) Let us now show how the previous results allows us to transfer properties of T ∗ δ to T P .5.2.3. Corollary.
The theory T P eliminates ∃ ∞ .Proof. This follows directly from Theorem 3.5.5. (cid:3)
Proposition.
Assume that the theory T ∗ δ has L -open core. Then the theory T P has L -open core.Proof. By Lemma 5.2.2, let ( K ∗ , F ∗ ) be an L P -elementary extension such that K ∗ is a modelof T ∗ δ with constant field F ∗ . Let ϕ ( x, y ) be an L P -formula such that for a ∈ K | y | , ϕ ( x, a )defines the open set U . Then ϕ ( K ∗ , a ) defines an open set U ( K ∗ ) in ( K ∗ ) n . Since K ∗ is amodel of T ∗ δ and every L P -formula defines a set which is L δ -definable by replacing P ( t ) bythe formula δ ( t ) = 0, U ( K ∗ ) is L δ -definable. Now by assumption, we have that U ( K ∗ ) isdefinable by ψ ( x, c ) where ψ ( x, z ) is an L -formula and c ∈ ( K ∗ ) ℓ ( z ) . Then we have that( K ∗ , F ∗ ) | = ( ∀ x )( ϕ ( x, a ) ↔ ψ ( x, c )) , and quantifying over c , we have that( K, F ) | = ( ∃ c )( ∀ x )( ϕ ( x, a ) ↔ ψ ( x, c )) , which shows that U is L -definable. (cid:3) In Section 6 we will show that most theories T ∗ δ corresponding to T as in Examples 2.2.1have L -open core (see later Theorems 6.0.8 and 6.0.9). As a corollary we obtain the followingresult shown by Hieronymi and Boxall [3, Corollary 3.4] and Fornasiero [12, Theorem 13.11].5.2.5. Corollary.
Let T be either RCF , p CF d , RCVF or ACVF ,p . Then T P has L -opencore. (cid:3) We finish this section with some remarks on distality. By a result of P. Hieronymi and T.Nell in [17], the theory of dense pairs of an o-minimal expansion of an ordered group is notdistal. A natural question posed by P. Simon asks whether the theory of such pairs always
OPOLOGICAL FIELDS WITH A GENERIC DERIVATION 33 admits a distal expansion [27, Question 1]. In [27], T. Nell provided a positive answer tothe question for the theory of dense pairs of ordered vector spaces. A simple consequence ofour analysis is that the theory of dense elementary pairs of real closed fields admits a distalexpansion, namely, the theory CODF. More generally, the following is a direct consequenceof Theorem A.0.5 and Lemma 5.2.2.5.2.6.
Corollary.
If the theory T is distal, then T ∗ δ is a distal expansion of T P . In particular, T P admits a distal expansion when T is RCF , p CF d or RCVF . As a consequence of results of A. Chernikov and S. Starchenko in [8], definable relationsin models of T P satisfied the so called strong Erd˝os-Hajnal property (see [8, Definition 1.6,Theorem 6.10 (3)]).5.2.7. Corollary. If T is distal, then definable relations in models of T P satisfy the strongErd˝os-Hajnal property. This holds in particular for models of T P when T is RCF , p CF d or RCVF .Proof.
This follows from Corollary 5.2.6 and [8, Corollary 4.8]. (cid:3) Open core
We will prove in this section that T ∗ δ has L -open core for some theories T listed in Examples2.2.1. The proof strategy has two main steps. The first one consists in showing that continuous L δ -definable functions (and more generally continuous L δ -definable correspondences) are infact L -definable. The second one consists in associating to every closed L δ -definable set X ⊆ K n a continuous L δ -definable function which “measures” the distance of a point in K n to X . Combining both steps, one recovers X as the elements in K n of “distance 0”. We willcarry out these steps in the following two main contexts: • (Ordered) the definable topology τ on K comes from a total order; • (Valued) the definable topology τ on K comes from a valuation v : K → Γ ∞ . More-over we assume T satisfies either ( † ) or ( †† ) as defined in Section 2.6.1. For simplicity,we assume in this case both the value group and the valuation are part of the language L .The strategy described above was devised by M. Tressl for CODF. Various new ideas wereneeded to be included in order to adapt it to the present setting. In particular, the fact thatopen theories of topological fields do not necessarily have finite Skolem functions naturally ledus to consider the more general case of continuous definable correspondences. Furthermore,in the case of valued fields, continuous definable functions to the value group also needed tobe treated.Through the section, we let T be an open L -theory of topological fields and assume T ∗ δ isconsistent. We will add (Ordered) or (Valued) to indicate we are respectively in one of theabove contexts.6.0.1. Theorem.
Let K | = T ∗ δ . Let X ⊆ K n be an L -definable set and f : X ⇒ K be an L δ -definable ℓ -correspondence. If f is continuous, then it is L -definable. Theorem (Valued) . Let K be a model of T ∗ δ . Let X ⊆ K n be an L -definable set and f : X → Γ ∞ be an L δ -definable function. If f is continuous, then it is L -definable. The proof of the previous theorems will be given in Section 6.1.
In order to associate a function d X to every L δ -definable closed set X in the above cases,we further need the following results concerning definable completeness on either the field sortor the value group sort (when it applies). Recall that for a first order language L , a totallyordered L -structure M is L -definably complete if every bounded L -definable set X ⊆ M has an infimum and a supremum in M .6.0.3. Proposition (Valued) . Let K be a model of T ∗ δ . If Γ ∞ is L -definably complete, thenit is also L δ -definably complete.Proof. It is a straightforward consequence of Corollary 3.5.6. In fact, this also holds evenwithout assuming ( † ) or ( †† ). (cid:3) To show definable completeness in the ordered case we will use the following lemma whichis equivalent to having the open core for definable sets in one variable. It hints to a potentialproof of the open core which will follow directly from the axiomatization of T ∗ δ withoutspecifying what type of topology (ordered, valued, etc.) comes with the theory T .6.0.4. Lemma.
Let K be a model of T ∗ δ and X ⊆ K be an L δ -definable subset of order m .Then, for d = 2 m , there is an L -definable subset Y ⊂ K d +1 such that ( X, Y, d ) is a linkedtriple. In particular, if X is open then X is L -definable.Proof. By Lemma 3.5.10, there is an L -definable set Y ⊆ K d +1 with d = 2 m such that X = ∇ − d ( Y ) and Y = ∇ d ( X ) (this second property follows by Part (2) of Lemma 3.5.10).This shows that ( X, Y, d ) is a linked triple. Letting π : K d +1 → K denote the projectiononto the first coordinate, we have that π ( Y ) = X , so X is L -definable. This shows the laststatement of the lemma. (cid:3) Proposition (Ordered) . Let K be a model of T ∗ δ . If K is L -definably complete, thenit is also L δ -definably complete.Proof. Let X ⊆ K be an L δ -definable set. By Lemma 6.0.4, X is L -definable. Now, X isbounded since X is bounded and, in addition, X and X have the same supremum (resp. sameinfimum). (cid:3) We have all needed tools to associate the function d X to an L δ -definable set X and provethe L -open core for T ∗ δ in both contexts.6.0.6. Definition.
Let K be a model of T ∗ δ and X ⊆ K n be an L δ -definable set. • (Ordered) Assume K is L -definably complete. The function d X : K n → K is definedas d X ( a ) := inf b ∈ X n X i =1 ( a i − b i ) . • (Valued) Assume Γ ∞ is L -definably complete. The function d X : K n → Γ ∞ is definedas d X ( a ) := inf b ∈ X { v ( a − b ) , . . . , v ( a n − b n ) } . Propositions 6.0.5 and 6.0.3 ensure that the function d X is well defined in each case. As aconsequence we obtain:6.0.7. Theorem.
The theory T ∗ δ has L -open core, whenever • (Ordered) the field sort is L -definably complete; • (Valued) the value group sort is L -definably complete. OPOLOGICAL FIELDS WITH A GENERIC DERIVATION 35
Proof.
Let K be a model of T ∗ δ . As having L -open core is an elementary property, it sufficesto show that every closed L δ -definable set X ⊆ K n is L -definable. We split in cases. (Ordered) By Proposition 6.0.5, d X is a well-defined L -definable function. The function d X is continuous, so by Theorem 6.0.1, d X is L -definable. Since X is closed, X = { a ∈ K n : d X ( a ) = 0 } , and hence it is L -definable. (Valued) By Proposition 6.0.5, d X is a well-defined L -definable function. As in the pre-vious case, d X is continuous, so by Theorem 6.0.2, d X is L -definable. Finally, X being closedimplies that X = { a ∈ K n : d X ( a ) = ∞} . Hence, X is L -definable. (cid:3) Theorem.
Let T be one of the following theories: ACVF ,p , RCVF , p CF d or the L RV -theory of k (( t Γ )) for k a field of characteristic 0 and Γ either Z -group or a divisible group.Then, the theory T ∗ δ has L -open core.Proof. Let K be a model of T ∗ δ and Γ be its value group. In all cases, the L -theory T satisfies either ( † ) or ( †† ). Note that by Ax-Kochen/Erˇsov, if Γ is a Z -group, then k (( t Γ )) hasan elementary substructure of the form k (( t Z )). Moreover, in each case, Γ ∞ is L -definablycomplete. The result follows by Theorem 6.0.7. (cid:3) Theorem ([29, Theorem 0.2]) . The theory
CODF has L -open core.Proof. Since RCF is definably complete, the result follows by Theorem 6.0.7. Alternatively,this follows from Theorem 6.0.8. Indeed, every model of CODF embeds into a model ofRCVF ∗ δ , and hence, the L -open core of RCVF ∗ δ implies the result for CODF. (cid:3) The proof given in [29] uses the fact that CODF is L δ -definable complete together withthe following criterion due to A. Dolich, C. Miller and C. Steinhorn [10]: any expansion of adensely ordered abelian group has “o-minimal open core” if it eliminates the quantifier ∃ ∞ and is definably complete. Since CODF eliminates the quantifier ∃ ∞ (Theorem 3.5.5), theresult follows.6.1. Continuous L δ -definable functions and correspondences. In this section we proveTheorems 6.0.1 and 6.0.2. We will also need the following two lemmas showing that L δ -definable correspondences with an L -definable domain are essentially compositions of L -definable correspondences with the derivation.6.1.1. Proposition.
Let K be a model of T ∗ δ . Let X ⊆ K n be an L -definable set and f : X ⇒ K be an L δ -definable ℓ -correspondence with ℓ > . Then, there are d ∈ N , an L -definable set Y ⊂ K n ( d +1) and an L -definable ℓ -correspondence F : Y ⇒ K , such that forevery x ∈ X f ( x ) = F (¯ δ d ( x )) . Proof.
Let ˜ X := graph( f ). By Lemma 3.5.10, there is a natural number d and an L -definablesubset Z ⊂ K ( n +1)( d +1) such that given any ( a, b ) ∈ K n × K , (¯ δ d ( a ) , ¯ δ d ( b )) ∈ Z if and only if( a, b ) ∈ graph( f ).Moreover if ˜ X a is finite, then | ˜ X a | = | Z ¯ δ d ( a ) | . Let π be the projection from K ( n +1)( d +1) to K n ( d +1) and let Y := π ( Z ) ∩ { ¯ x ∈ K n ( d +1) : ∃ = ℓ ¯ y (¯ x, ¯ y ) ∈ Z ¯ x } . Define F : Y → K by F (¯ e ) := π ( Z ¯ e ), where π is the projection on the first coordinate. Let a ∈ K n , then F (¯ δ d ( a )) = π ( Z ¯ δ d ( a ) ) = X a . (cid:3) The following is an analogous result in the valued context.
Proposition (Valued) . Let K be a model of T ∗ δ . Let X ⊆ K n be an L -definable set and f : X ⇒ Γ ∞ be an L δ -definable correspondence. Then there are m ∈ N and an L -definable ℓ -correspondence F : Y ⇒ Γ ∞ , Y ⊂ K n ( m +1) , such that for every x ∈ Xf ( x ) = F (¯ δ m ( x )) . Proof.
For x = ( x , . . . , x n ) and ξ a single variable (varying in Γ ∞ ), let ϕ ( x, ξ ) be an L δ ( K )-formula defining f . Let ϕ ∗ (¯ x, ξ ) be the corresponding L ( K )-formula with ¯ x = (¯ x , . . . , ¯ x n ), ℓ (¯ x i ) = m + 1. Consider the L -definable set: Y := { ¯ u ∈ K n ( m +1) : ( u , . . . , u n ) ∈ X ∧ ∃ = ℓ ξϕ ∗ (¯ u, ξ ) } . and the ℓ -correspondence F (¯ u, ξ ) defined by the L ( K )-formula ψ ℓ (¯ u, ξ ) := ϕ ∗ (¯ u, ξ ) ∧ ¯ x ∈ Y. For x ∈ X and ξ ∈ f ( x ), we have that ϕ ( x, ξ ) holds, so ϕ ∗ (¯ δ m ( x ) , ξ ) too. Since | f ( x ) | = ℓ , weget that ψ (¯ δ m ( x ) , ξ ) holds. (cid:3) We have all tools to show Theorems 6.0.1 and 6.0.2.
Proof of Theorem 6.0.1:
We proceed by induction on dim( X ), the case dim( X ) = 0 beingclear. By cell decomposition (Theorem 2.7.1) and the induction hypothesis (possibly changing n and ℓ ), we may suppose that X is open in K n .By Proposition (6.1.1), let F : Y ⊆ K n ( m +1) ⇒ K be an L -definable ℓ -correspondence suchthat for all x ∈ X and y ∈ K y ∈ f ( x ) ⇔ y ∈ F (¯ δ m ( x )) . If m = 0 there is nothing to show, so suppose m > π : K n ( m +1) → K n be the projection sending each block of ( m + 1) tuples to its firstelement. Without loss of generality, we may suppose that π ( Y ) = X .Given x = ( x , . . . , x n ) ∈ K n and z = ( z , . . . , z n ) ∈ K nm , we let ( x, z ) π denote the element( x , z , . . . , x n , z n ) ∈ K n ( m +1) . In particular, π (( x, z ) π ) = x for all x ∈ K n .6.1.3. Claim.
We have Y = π − ( X ) . Since ∇ m ( X ) ⊆ Y ⊆ π − ( X ), it suffices to show that ∇ m ( X ) is dense in π − ( X ). Let( x, z ) π ∈ π − ( X ) with z = ( z , . . . , z n ) ∈ K nm . Let U an open neighbourhood of 0 such that( x, z ) π + U n ( m +1) ⊆ π − ( X ). Since X is open, by Lemma 3.4.1, for each i ∈ { , . . . , n } thereis u i ∈ x i + U such that ¯ δ m ( u i ) ∈ ( x i , z i ) + U m +1 . Letting u = ( u , . . . , u n ), we have that¯ δ m ( u ) ∈ ( x, z ) π + U n ( m +1) . This shows the claim.By the claim, dim( Y ) = n ( m + 1). Indeeddim( Y ) = dim( Y ) = dim( π − ( X )) = n ( m + 1) , where the last equality holds since X is open. Define˜ Y := ® ¯ x ∈ Y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) there is an open set V ⊆ Y of ¯ x such that F | V is continuous ´ . By Proposition 2.6.10, it holds thatdim Ä Y \ ˜ Y ä < dim( Y ) . OPOLOGICAL FIELDS WITH A GENERIC DERIVATION 37
Claim.
The following holds dim Ä X \ π ( ˜ Y ) ä < dim( X ) . Suppose for a contradiction this is not the case. Therefore there is an L -definable open set U ⊆ X such that U ∩ π ( ˜ Y ) = ∅ . This implies that π − ( U ) ⊆ Y \ ˜ Y , and therefore that n ( m + 1) = dim( π − ( U )) dim Ä Y \ ˜ Y ä < dim( Y ) = n ( m + 1) , a contradiction. This shows the claim.By Claim 6.1.4 and the induction hypothesis, we may suppose π ( ˜ Y ) = X . The theoremfollows directly from the following final claim.6.1.5. Claim.
For x ∈ X and all y such that ( x, y ) π ∈ ˜ Y , F (( x, y ) π ) = f ( x ) . Suppose for a contradiction this is not the case and let y be such that ( x, y ) π ∈ ˜ Y but F (( x, y ) π ) = f ( x ). Therefore, there is z ∈ F (( x, y ) π ) \ f ( x ) (since both are ℓ -correspondences).Let U be an open neighbourhood of 0 such that z + U is disjoint from f ( x ) + U . By thedefinition of ˜ Y , let V ⊆ U be an open neighbourhood of 0 such that ( x, y ) π + V n ( m +1) ⊆ ˜ Y and F | ( x,y ) π + V n ( m +1) is continuous. By Lemma 2.6.2, we may assume (possibly shrinking V ) that graph( F | V ) is the disjoint union of the graphs of ℓ continuous definable functions g , . . . , g ℓ from ( x, y ) π + V n ( m +1) to K . Suppose without loss of generality that (( x, y ) π , z ) ∈ graph( g ). By the continuity of f , let U ⊆ V be an open neighbourhood of 0 such that f ( x + U n ) ⊆ f ( x ) + U . Let V ⊆ U be such that, g | ( x, y ) π + V n ( m +1)0 ⊂ z + U . By Lemma3.4.1, there is w ∈ X such that ¯ δ m ( w ) ∈ ( x, y ) π + V n ( m +1)0 . Since F (¯ δ m ( w )) = f ( w ), there is z ′ ∈ f ( w ) such that z ′ ∈ z + U , which contradicts that z + U and f ( x ) + U are disjoint. (cid:3) Proof of Theorem 6.0.2.
The proof is an immediate analogue of the proof of Theorem 6.0.1,replacing Proposition 6.1.1 by 6.1.2, Proposition 2.6.10 by 2.6.11 and noting that a strongerversion of Lemma 2.6.2 holds in this context since the graph of a definable Γ ∞ -valued ℓ -correspondence is the disjoint union of the graphs of ℓ definable Γ ∞ -valued functions (evenglobally). (cid:3) Appendix A. Classical transfers
Through this section we let T be an open L -theory of topological fields. Let U be a monstermodel of T ∗ δ and A be some small subset. We let h A i be A together with the differential closureof the elements of A in the field sort.A.0.1. Lemma.
Let x be a tuple of variables of field sort and let z be a tuple of variables ofother sorts. Then for a ∈ S x ( U ) and b ∈ S z ( U ) , the L δ -type tp δ ( a, e/A ) is determined by theinfinite sequence of L -types { tp (¯ δ m ( a ) , e/ h A i ) : m ∈ N } .Proof. This follows by relative quantifier elimination (Theorem 3.4.2) and the fact that forevery field sort quantifier free L δ -formula ϕ ( x, z ) over A , the formula ϕ ∗ as defined in Notation3.2.1 is an L -formula over h A i . (cid:3) A.0.2.
Corollary.
Let x and z be as in the previous lemma. Let ( a i , e i ) i ∈ I be a sequence where a i ∈ S x ( U ) and e i ∈ S z ( U ) . Then the sequence is L δ -indiscernible sequence over A if and onlyif for each m ∈ N , the sequence (¯ δ m ( a i ) , e i ) i ∈ I is L -indiscernible over h A i . (cid:3) A.0.3.
Theorem. T is NIP if and only if T ∗ δ is NIP.Proof. Suppose T ∗ δ is not NIP. Let ϕ ( x, z ; y, w ) be a partitioned L δ -formula with IP where x, y are tuples of field sort and z, w are tuples of other sorts. By Theorem 3.4.2, we may assumethat ϕ has no field sort quantifiers. Then, since ϕ has IP so does the L -formula ϕ ∗ in T . Theconverse is clear, since being NIP is preserved by reducts (see Remark 3.3.7). (cid:3) To show the transfer of distality, a dividing line introduced by P. Simon in [36], we willuse the following equivalent definition of distality which appears in [17]. In the followingdefinition we let L be any first order language, T be a complete L -theory and U be a monstermodel of T .A.0.4. Definition ([17, Definition 1.3]) . Let ϕ ( x , . . . , x n ; y ) be a partitioned L -formula, where x i , 1 i n is a p -tuple of variables and y is a q -tuple of variables, p, q >
0. Then ϕ is distal(in T ) if for every b ∈ U q , and every indiscernible sequence ( a i ) i ∈ I in U p such that(1) I = I + c + I , where both I , I are (countable) infinite dense linear orders withoutend points and c is a single element with I < c < I ,(2) the sequence ( a i ) i ∈ I + I in U p is indiscernible over b ,then U | = ϕ ( a i , . . . , a i n ; b ) ↔ ϕ ( a j , . . . , a j n ; b ) with i < . . . < i n , j < . . . < j n in I .A theory T is distal if every formula is distal in T .The transfer of distality from T to T ∗ δ is an unpublished result of A. Chernikov. Theconverse has not been, to our knowledge, observed before. Note that since distality is notpreserved under reducts, the converse implication is not straightforward as in Theorem A.0.3.Examples of distal open L -theories of topological fields include RCF, p CF d and RCVF. Incontrast, the theory ACVF ,p is not distal (see [36]).A.0.5. Theorem. T is distal if and only if T ∗ δ is distal.Proof. Let us check that in T ∗ δ every formula is distal. Let ϕ ( x , z , . . . , x n , z n ; y, w ) be apartitioned L δ -formula where each x i is a p -tuple of field sort variables, each z i is a q -tupleof variables of fixed sorts S , . . . , S q (none being the field sort), y is a tuple of field sortvariables and w is a tuple of other sorts. Let U be a monster model of T ∗ δ . By Theorem3.4.2, we may assume ϕ has no field sort quantifiers. Let m be the order of ϕ Take an L δ -indiscernible sequence ( a i , e i ) i ∈ I in U where ( a i , e i ) ∈ S x ( U ) × S z ( U ) and I = I + c + I with I , I infinite dense linear orders without end points. Let ( b, d ) be a tuple in S y ( U ) × S w ( U ),and assume that ( a i , e i ) i ∈ I + I is L δ -indiscernible over ( b, d ). Then, by Corollary A.0.2, thesequence (¯ δ m ( a i ) , e i ) i ∈ I (resp. (¯ δ m ( a i ) , e i ) i ∈ I + I ) is L -indiscernible (resp. L -indiscernibleover B where B = { ¯ δ m ( b ) : m ∈ N } ∪ { d } )) for every m ∈ N . Since T is distal, the partitioned L -formula ϕ ∗ (¯ x , z , . . . , ¯ x n , z n ; ¯ y, w )is distal, which easily implies the distality of ϕ .For the converse, suppose ϕ ( x , z , . . . , x n , z n ; y, w ) is an L -formula which is not distal in T . Consider the L δ -formula ψϕ ( x , z , . . . , x n , z n ; y, w ) ∧ n ^ j =1 δ ( x i ) = 0 ∧ δ ( y ) = 0 . OPOLOGICAL FIELDS WITH A GENERIC DERIVATION 39
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TU Dresden, Fachrichtung Mathematik, Institut f¨ur Algebra, 01062 Dresden.
E-mail address : pablo.cubides [email protected] Department of Mathematics (De Vinci), UMons, 20, place du Parc 7000 Mons, Belgium
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