aa r X i v : . [ m a t h . L O ] J u l DIFFERENTIAL EXPONENTIAL TOPOLOGICAL FIELDS
FRANC¸ OISE POINT ( † ) AND NATHALIE REGNAULT
Abstract.
We axiomatize a class of existentially closed exponential fields equipped withan E -derivation. We apply our results to the field of real numbers endowed with exp ( x )the classical exponential function defined by its power series expansion and to the field ofp-adic numbers endowed with the function exp ( px ) defined on the p -adic integers where p is a prime number strictly bigger than 2 (or with exp (4 x ) when p = 2). Introduction
The problem we address here is: given an elementary class of existentially closed expo-nential topological fields (where possibly the exponential function E is partially defined)whether the class of existentially closed differential expansions is an elementary class andhow it can be axiomatized. The model-complete theories of exponential fields we include inour analysis are the theory of the field of real numbers with the exponential function andthe field of p-adic numbers with the exponential function restricted to the subring of p-adicintegers. The derivations δ we consider are E -derivations, namely δ ( E ( x )) = δ ( x ) E ( x ), but δ is not assumed to be continuous.Independently, this question has also been considered by A. Fornasiero and E. Kaplanin the following setting. Given an o-minimal expansion K of an ordered field which ismodel-complete and expanded with a compatible derivation [12], they show that indeedthe class of existentially closed differential expansions is elementary and they provide anaxiomatization. A derivation δ is compatible with K if for any 0-definable C -function f : U → K , where U is an open subset of some cartesian product K n , we have δf (¯ u ) = P ni =1 ∂f∂x i (¯ u ) δ ( u i ), for any ¯ u ∈ U . In particular in case K expands an exponential field, suchderivation δ is an E -derivation. Their results apply to o-minimal fields K extending thefield of real numbers R and admitting an expansion to all restricted analytic functions. Inorder to show that the ”usual” derivations are compatible they have at their disposal thequantifier elimination result of J. Denef and L. van den Dries on the expansion R an of R with all these functions (with restricted division) and its extension by L. van den Dries, A.Macintyre and D. Marker for R an,exp , where exp is the exponential function given by theclassical power series [9].Our approach is different in several ways. On the one hand, we only work with L -structures where L is the language of exponential fields together with relation symbols (inorder to be able to define a basis of neighbourhoods of 0 in a quantifier-free way). Onthe other hand, we don’t restrict ourselves to the o-minimal context: as pointed above, wealso handle valued fields such as Q p the field of p-adics numbers or C p the completion ofthe algebraic closure of Q p , endowed with a partially defined exponential function (on the Date : July 29, 2020.1991
Mathematics Subject Classification.
Key words and phrases. exponential field, differential field, existentially closed.( † ) Research Director at the ”Fonds de la Recherche Scientifique (F.R.S.-F.N.R.S.)”. ( † ) AND NATHALIE REGNAULT valuation ring). More precisely, we work with a class of topological fields with a definabletopology, where an implicit function theorem holds (see Definition 3.12) and in the orderedcase with the lack of flat function property for certain definable functions (see Definition3.17). Note that both properties hold in o-minimal expansions of real-closed fields (or moregenerally in definably complete ordered fields) (for the implicit function theorem, see [8,page 113] and for lack of flat functions, see [26, Lemma 25]).Given an L -theory T of fields and a unary function symbol for a derivation δ , we denoteby T δ the L ∪ { δ } -theory consisting of T together with an axiom expressing that δ is an E -derivation. Our main result is: Theorem (later Theorem 4.3) Let T be a model-complete complete theory of topological L -fields whose topology is either induced by an ordering or a valuation. Assume in theordered case that the models of T satisfy an implicit function theorem (IFT) E and havethe lack of flat functions property (LFF) E and in the valued case that the models of T satisfy an analytic implicit function theorem (IFT) anE . Then the class of existentially closedmodels of T δ is elementary.Note that when we apply our result to ( R , exp ) where δ is now a compatible derivation,using the result of A. Wilkie on the model-completeness of ( R , exp ), by uniqueness of themodel-completion we get the same class of existentially closed exponential differential fields.However, it is unclear in an ordered exponential field model of the theory of ( R , exp ) whetherany E -derivation is compatible. (We cannot apply the argument used by A. Fornasieroand E. Kaplan since we don’t have quantifier-elimination in the language of ordered fieldstogether with the exponential function.)When we apply our result to the theories of respectively ( Q p , E p ) and ( C p , E p ), where E ( x ) := exp (4 x ) and E p ( x ) := exp ( px ), p = 2, we use a model-completeness result due toN. Mariaule [18], [19] (based on notes by A. Macintyre).The plan of the paper is as follows.In section 2, we first recall the notion of partial exponential fields and of the correspondingclosure operator, denoted by ecl-closure. It was introduced by A. Macintyre using the workof Khovanskii [17], then it plays a crucial role in the proof of A. Wilkie of the model-completeness of ( R , exp ). Later in a purely algebraic context, J. Kirby linked the ecl-closure with the cl-closure, defined through E -derivations. He showed that the two closureoperators coincide using a result of J. Ax on the Schanuel property in differential fieldsof characteristic 0. We recall those results and we slightly adapt J. Kirby’s results onextensions of E -derivations in order to be able to use them in the case of p-adically closedfields, where the exponential function is only defined on the valuation ring.Then in section 3, we recall the notion of E -varieties, generic points and torsors. Wealso recall the setting of topological fields [21]. We define the class of exponential fieldswe will be able to deal with, namely those satisfying the implicit function theorem and inthe ordered case the lack of flat functions. These hypotheses were used in the works of A.Wilkie and N. Mariaule recalled above.In section 4, we finally introduce a scheme of axioms (DL) E that will axiomatize a classof existentially closed differential exponential fields and show our main result. This schemeof axioms can be compared to the axiomatization of M. Singer of the closed ordered fields,called CODF. We also give a geometric interpretation of the scheme (DL) E . IFFERENTIAL EXPONENTIAL TOPOLOGICAL FIELDS 3
In the last section we show how to endow a topological exponential field of cardinality ℵ with a countable dense subfield with an E -derivation which satisfies this scheme of axioms.When the topology is induced by an ordering we point out that such ordered field can alsobe made a model of CODF. This kind of construction (for CODF) may be found in thework of M. Singer, and the theses of C. Michaux and Q. Brouette. Acknowledgments:
Part of these results appeared in the PhD thesis of Nathalie Reg-nault [24]. 2. E -derivations Preliminaries.
We will only consider commutative rings R of characteristic 0 with1 = 0. Let N ∗ := N \ { } , R ∗ := R \ { } . Denote by I ( R ) the subgroup of the invertibleelements of ( R ∗ , · , I, < ), denote I ≥ j := { i ∈ I : i ≥ j } (respectively( I >j ) := { i ∈ I : i > j } ).Let L rings := { + , · , − , , } be the language of rings; we will work in different expansions L of L rings such as L E := L rings ∪ { E } and L E,δ := L rings ∪ { E, δ } where E, δ are unaryfunctions. The L -formulas will be possibly with parameters and when we want to specifythem we will use L ( B ) with B a set of constants. Similarly L -definable sets will possiblybe definable with parameters. Our notation for tuples will be flexible: x (respectively a )will denote a tuple of variables (respectively a tuple of elements) but sometimes in orderto stress that we deal with tuples we will use ¯ x , respectively ¯ a , or bold letters e.g. x , a . Inthis section we will not make the distinction between an L -structure M and its domain M whereas from subsection 3.4 on, we will distinguish them. Definition 2.1. [6] An E -ring R is a ring equipped with a morphism E from the additivegroup ( R, + ,
0) to the multiplicative group I ( R ) satisfying E (0) = 1 and ∀ x ∀ y ( E ( x + y ) = E ( x ) · E ( y )). (So an E -ring can be endowed with an L E -structure.) An E -field is a fieldwhich is an E -ring.We will also consider partial E -fields, and so the corresponding language contains aunary predicate for the domain of the exponential function. We will first define partial E -domains. Definition 2.2.
Let F be an integral domain, namely a commutative ring with no non-zerozero-divisors. A partial E -domain is a two-sorted structure(( F, + F , · F , F , F ) , ( A, + A , A ) , E ) , where ( A, + A , A ) is a group and E : ( A, + A , A ) → I ( F ) is a group morphism. We identify( A, + A , A ) with an additive subgroup of ( F, + F , F ) and to stress it, we will denote it by A ( F ). When the domain of E is clear from the context, we will also simply use the notation( F, E ), even though E is only partially defined.A partial E -field F is a partial E -domain which is a field. A partial E -subfield F is apartial E -field which is a two-sorted substructure. We denote by F (¯ a ) E , where ¯ a ⊆ F , thesmallest partial E -subfield of F containing F and ¯ a and by F h ¯ a i E the smallest partial E -subring generated by F and ¯ a . When F = Q , we denote Q h ¯ a i E simply by h ¯ a i E . Tomake the distinction with the L rings -substructure, we denote by Q [¯ a ] the subring generatedby ¯ a .Note that in [15, Definition 2.2], one uses a stronger notion of partial E -fields, namely onerequires that A ( F ) is a Q -vector space, namely one endows A ( F ) with scalar multiplications FRANC¸ OISE POINT ( † ) AND NATHALIE REGNAULT ( · q ) q ∈ Q . Instead here, given two partial E -fields F ⊆ F , we replace that by the conditionthat A ( F ) is a pure subgroup of A ( F ). Notation 2.3.
Let F , F be two partial E -fields with F a substructure of F . Then thesubgroup A ( F ) is pure in A ( F ) iff for any a ∈ A ( F ) and n ∈ N ∗ , if na ∈ A ( F ), then a ∈ A ( F ). We use the notation A ( F ) ⊆ A ( F ).In addition, when the field F is endowed with a field topology and when lim n →∞ P ni ≥ x i i ! exists, we can consider the (partial) function x exp ( x ) := lim n →∞ P ni ≥ x i i ! . Then thedomain of exp ( x ) is a subgroup and a Q -vector space whenever F is closed under roots. Examples 2.1. (1) Let F be a partial E -field and consider the field of Laurent series F (( t )) (or moregenerally a Hahn field (see below)). Then, regardless of whether we put a topologyon F (( t )), we can always define exp ( x ) := P i ≥ x i i ! for x ∈ tF [[ t ]]. Indeed, byNeumann’s Lemma, the element exp ( x ) ∈ F [[ t ]] [10, chapter 8, section 5, Lemma].Then, we extend E on A ( F ) ⊕ tF [[ t ]] as follows. Write r ∈ A ( F ) ⊕ tF [[ t ]] as r + r where r ∈ A ( F ) and r ∈ t.F [[ t ]]. Define E on A ( F ) ⊕ tF [[ t ]] as follows: E ( r + r ) := E ( r ) .exp ( r ). So F (( t )) can be endowed with a structure of a partial E -field with A ( F (( t ))) := A ( F ) ⊕ tF [[ t ]].(2) More generally, under the same assumption on F , let ( G, + , − , , < ) be an abeliantotally ordered group, then the Hahn field F (( G )), can be endowed with a structureof a partial E -field defining E on the elements r ∈ A ( F ) ⊕ F (( G > )) similarly, where G > := { g ∈ G : g > } (respectively G ≥ := { g ∈ G : g ≥ } ). Namely decompose r as r + r with r ∈ A ( F ) and r ∈ F (( G > )). Then exp ( r ) ∈ F (( G ≥ )) againby Neumann’s Lemma and define E ( r ) := E ( r ) exp ( r ). So A ( F (( G ))) = A ( F ) ⊕ F (( G > )).(3) Let ¯ R := ( R , + , − , · , , , E ) where E ( x ) = exp ( x ) defined above.(4) Let ¯ C := ( C , + , − , · , , , E ) where E ( x ) = exp ( x ).(5) Let p be a prime number; when p = 2 set E p ( x ) := exp ( p x ) and when p >
2, set E p ( x ) = exp ( px ). Let C p be the completion of the algebraic closure of the fieldof p -adic numbers Q p (in C ). As examples of partial E -fields, we have the field of p -adic numbers ¯ Q p := ( Q p , + , − , · , , , E p ) or ¯ C p := ( C p , + , − , · , , , E p ). In thesetwo cases, E p is defined on the valuation ring Z p of Q p (respectively on the valuationring O p of C p ).We will investigate these examples further in section 5. Definition 2.4.
Let R be a (partial) E -ring. An E -derivation δ is a unary function on R satisfying:(1) δ ( a + b ) = δ ( a ) + δ ( b ),(2) the Leibnitz rule: δ ( a.b ) = δ ( a ) .b + a.δ ( b ),(3) ∀ a ∈ A ( δ ( E ( a )) = δ ( a ) .E ( a )).We will denote the differential expansion of R by R δ .For example, let F δ be a differential E -field ( δ can be the trivial derivation). We havealready seen how to extend E on F [[ t ]]. Then we extend δ on the field of Laurent series F (( t )) by setting δ ( t ) = 1 and by requiring it to be strongly additive. Then δ is again IFFERENTIAL EXPONENTIAL TOPOLOGICAL FIELDS 5 an E -derivation on F (( t )). Indeed, for x ∈ tF [[ t ]], we have δ ( exp ( x )) = P i ≥ δ ( x i i ! ) = δ ( x ) .exp ( x ) and for x ∈ F [[ t ]] with x = r + r where r ∈ A ( F ) and r ∈ t.F [[ t ]], wehave δ ( E ( r + r )) = E ( r ) .exp ( r ) .δ ( r ) + δ ( r ) .E ( r ) .exp ( r ) = δ ( x ) .E ( x ). This makes( F (( t )) , F [[ t ]] , exp, δ ) a differential (partial) E -field. Notation 2.5.
Let δ be an E -derivation on R . For m > a ∈ R , 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 )) ∈ R m +1 .Similarly, given an element a = ( a , . . . , a n ) ∈ R n , we write¯ δ m ( a ) := ( a , . . . , a n , . . . , δ m ( a ) , . . . , δ m ( a n )) ∈ R ( m +1) n . Denote by Q h a i E,δ the E -differential subring of R generated by a and Q .In section 2.3, we will consider in general the problem of extending E -derivations butfirst it is convenient to recall the notion of E -polynomials and differential E -polynomials.2.2. Free exponential rings.
The construction of free E -rings Z [ X ] E on finitely manyvariables X := ( X , . . . , X n ) (and more generally free E -rings R [ X ] E over ( R, E )) can befound in many places in the literature. It is initially due to B. Dahn. The elements of theserings are called E -polynomials in the indeterminates X . Here we will briefly recall theirconstruction, following [6] and [18]. When n = 1, we will use the variable X and since wewill also use differential E -polynomials, we will also allow X to denote a tuple of countablymany variables.Let R be an E -ring. Then the ring R [ X ] E is constructed by stages as follows: let R − := R , R := R [ X ] and A the ideal generated by X in R [ X ]. Then R = R ⊕ A . Let E − = E on R composed by the embedding of R − into R .For k ≥
0, set R k = R k − ⊕ A k and let t A k be a multiplicative copy of the additive group A k .For instance for k = 1, we get R = R [ t A ] and A is a direct summand of R in R .Then, put R k +1 := R k [ t A k ] and let A k +1 be the free R k − submodule generated by t a with a ∈ A k − { } . We have R k +1 = R k ⊕ A k +1 .By induction on k ≥
0, one shows the following isomorphism: R k +1 ∼ = R [ t A ⊕···⊕ A k ] , using the fact that R [ t A ⊕···⊕ A k ] ∼ = R [ t A ⊕···⊕ A k − ][ t A k ] [18, Lemma 2].We define the map E k : R k → R k +1 , k ≥
0, as follows: E k ( r ′ + a ) = E k − ( r ′ ) t a , where r ′ ∈ R k − and a ∈ A k .Finally let R [ X ] E := S k ≥ R k and extend E on R [ X ] E by setting E ( f ) := E k ( f ) for f ∈ R k . It is easy to check that it is well-defined. Let f ∈ R k +1 , then f = f k + g where f k ∈ R k and g ∈ A k +1 . So E ( f ) = E ( f k ) t g . By definition E ( f k ) = E k ( f k ) andso if f k = f k − + g k with f k − ∈ R k − and g k ∈ A k , we have E ( f k − ) = E k − ( f k − ) t g k .Unravelling f in this way, we get that E ( f ) = E ( f ) t g + g k + ··· + g with f = f + g + · · · + g k + g , f ∈ R, g ∈ A , · · · , g k ∈ A k , g ∈ A k +1 .Finally note that the above construction can be extended when R is a partial E -domain,the only change is that we only define E ( f ) for f as written above when f ∈ A ( R ).Using the construction of R [ X ] E as an increasing union of group rings, one can define onthe elements of R [ X ] E an analogue of the degree function for ordinary polynomials whichmeasures the complexity of the elements; it takes its values in the class On of ordinals and FRANC¸ OISE POINT ( † ) AND NATHALIE REGNAULT was described for instance in [6, 1.9] for exponential polynomials in one variable. Here wedeal with exponential polynomials in more than one variable and so we follow [17, section1.8].Let us denote by totdeg X ( p ) the total degree of p , namely the maximum of { P mj =1 i j :for each monomial X i · · · X i m m occurring (nontrivially) in p , m ∈ N ≥ } .Then one defines a height function h (with values in N ) which detects at which stage ofthe construction the (non-zero) element is introduced.Let p ( X ) ∈ R [ X ] E , then h ( p ( X )) = k , if p ∈ R k \ R k − , k > h ( p ( X )) = 0 if p ∈ R [ X ].Using the freeness of the construction, one defines a function rkrk : R [ X ] E → N :If p = 0, set rk ( p ) := 0,if p ∈ R [ X ] \ { } , set rk ( p ) := totdeg X ( p ) + 1 andif p ∈ R k , k >
0, let p = P di =1 r i .E ( a i ), where r i ∈ R k − , a i ∈ A k − \ { } . Set rk ( p ) := d .Finally, one defines the complexity function ordord : R [ X ] E → On as follows. Write p ∈ R k as p = p + p + · · · + p k with p i ∈ A i , 1 ≤ i ≤ k . Define ord ( p ) := P ki =0 ω i .rk ( p i ).Note that if p = 0, then there is q ∈ R [ X ] E such that ord ( E ( q ) .p ) < ord ( p ) (the proofis exactly the same as the one in [6, Lemma 1.10]).On R [ X ] E , we define n E -derivations ∂ X i as follows: ∂ X i ↾ R = 0 and ∂ X i X j = δ ij , where δ ij is the Kronecker symbol, 1 ≤ i, j ≤ n . Notation 2.6.
Assume that δ is an E -derivation on R . Let X := ( X , . . . , X n ), denoteby R { X } E the ring of differential E -polynomials over R in n differential indeterminates X , · · · , X n , namely it is the E -polynomial ring in indeterminates δ j ( X i ), 1 ≤ i ≤ n , j ∈ ω ,with by convention δ ( X i ) := X i . Let p ( X ) ∈ R { X } E . Let m ∈ N be the (differential)order of p (denoted by δ - ord ( p )) as classically defined in differential algebra [16, page 75] (if m = 0, then p is an ordinary E -polynomial). In particular we have that p can be written as p ∗ (¯ δ m ( X )) with ¯ δ m ( X ) = ( X , . . . , X n , δ ( X ) , . . . , δ ( X n ) , . . . , δ m ( X ) , . . . , δ m ( X n ) and p ∗ an ordinary E -polynomial. Lemma 2.7.
Let δ be an E -derivation on R . Let p ∈ R [ X ] E . Then there exists p δ ∈ R [ X ] E such that in the ring R { X } E , δ ( p ( X )) = P nj =1 δ ( X j ) ∂ X j p + p δ . Moreover there is a tuple ¯ e of elements of R such that p ∈ h ¯ e i E [ X ] E and p δ ∈ Q (¯ e, h δ (¯ e ) i E )[ X ] E . Furthermore whenever δ is trivial on R , p δ = 0 . Proof: Decompose p as: p = p + P ki =1 p i , with p ∈ R [ X ] and p i ∈ A i , i >
0. We proceedby induction on ord ( p ), namely we assume that for all q ∈ R [ X ] E with ord ( q ) < ord ( p ), wehave δ ( q ( X )) = P nj =1 δ ( X j ) ∂ X j q + q δ with q δ satisfying the conditions of the statement ofthe lemma..If ord ( p ) ∈ ω , namely p ∈ R [ X ], the statement of the lemma is well-known. Write p ( X ) = P a i , ··· ,i n .X i · · · X i n n , define p δ := P δ ( a i , ··· ,i n ) X i · · · X i n n . Then δ ( p ( X )) = P nj =1 δ ( X j ) ∂ X j p + p δ . Note that p δ ∈ δ ( R )[ X ] and ord ( p δ ) ≤ ord ( p ). If p is monic and n = 1, then ord ( p δ ) < ord ( p ).Now assume that ord ( p ) ≥ ω and that the induction hypothesis holds. IFFERENTIAL EXPONENTIAL TOPOLOGICAL FIELDS 7
Let k > p ∈ R k \ R k − . By additivity of the derivation, the way ord has been definedand the induction hypothesis, it suffices to prove it for p ∈ A k . So, write p = P di =1 r i E ( a i )with r i ∈ R k − and a i ∈ A k − \ { } ; so ord ( p ) = ω k d . We have that δ ( p ) = P di =1 ( δ ( r i ) + r i δ ( a i )) E ( a i ) . By induction hypothesis, δ ( r i ) = P nj =1 δ ( X j ) ∂ j r i + r iδ and δ ( a i ) = P nj =1 δ ( X j ) ∂ X j a i + a iδ . So we get that δ ( p ) = P nj =1 δ ( X j ) ∂ X j ( P di =1 E ( a i ) r i ) + P di =1 E ( a i )( r iδ + r i a iδ ). Put p δ := P di =1 E ( a i )( r iδ + r i a iδ ) ( † ).Let e i , c i be tuples of elements of R such that r i ∈ h e i i E [ X ] E , a i ∈ h c i i E [ X ] E . Then by in-duction hypothesis, r δi ∈ Q ( h e i i E , δ ( e i ))[ X ] E , a δi ∈ Q ( h c i i E , δ ( c i )[ X ] E . Let ¯ e := ( e , . . . , e d )and ¯ c := ( c , . . . , c d ). We have that p ∈ h ¯ e, ¯ c i E [ X ] E and by ( † ), p δ ∈ Q ( h ¯ e, ¯ c i E , δ (¯ e ) , δ (¯ c ))[ X ] E and if δ is trivial on R , then p δ = 0. (cid:3) Khovanskii systems.
Let F δ be an expansion of a partial E -field by an E -derivation δ (see Definition 2.4). Note that in [6], the condition of being an E -derivation was relaxedto: δ ( E ( x )) = rδ ( x ) E ( x ), for some r ∈ R ∗ . However if δ is an E-derivation, then rδ isalso an E -derivation, with r ∈ R . More generally, the set of E -derivations on R forms a R -module. Using E -derivations, J. Kirby defined a closure operator cl in E -rings and heshowed that cl induces a pregeometry on subsets of R [15, Lemma 4.4, Proposition 4.5]. Definition 2.8. [15, Definition 4.3] Let R be a partial E -ring and let A be a subset of R .Then, cl R ( A ) := { u ∈ R : δ ( u ) = 0 for any E − derivation δ vanishing on A } . If A ⊆ R , then cl( A ) is an E -subring and if R is field, it is an E -subfield.Note that in the algebraic case, when an element a is algebraic over a subfield endowedwith a trivial derivation δ , then δ ( a ) = 0 as well. Later, we will see an analog of thisproperty in the case of E -derivation working with a notion of E -algebraicity (see Lemma2.15). Notation 2.9.
Let R be an E -ring. In section 2.2, we recalled the construction of the ringof E -polynomials in X := ( X , · · · , X n ) over R . These E -polynomials induce functionsfrom R n to R and we will denote the corresponding ring of functions by R [ x ], where x := ( x , · · · , x n ) [6].Note that when R is a partial E -domain, we get the same ring of E -polynomials butwith an E -polynomial we can only associate a partially defined function on R (since E isonly defined on A ( R )).In [6, section 4], one can find a necessary condition on R under which the map sendingan E -polynomial p ( X ) to the corresponding function p ( x ) is injective. The condition is asfollows: there exist n E -derivations ∂ i on R [ x ] E , which are trivial on R and satisfy ∂ i ( x j ) = δ ij [6, Proposition 4.1]. Let f ∈ R [ x ] E , we denote by ∂ i f , the function corresponding tothe differential E -polynomial ∂ X i f . Notation 2.10.
Given f , · · · , f n ∈ R [ X ] E , ¯ f := ( f , · · · , f n ), we will denote by J ¯ f ( X ),the Jacobian matrix: ∂ X f · · · ∂ X n f ... . . . ... ∂ X f n · · · ∂ X n f n . FRANC¸ OISE POINT ( † ) AND NATHALIE REGNAULT
As usual, we denote by det ( J ¯ f ( X )) the determinant of the matrix J ¯ f ( X ); note that it isan E -polynomial. When we evaluate either J ¯ f ( X ) or its determinant at an n -tuple b ∈ R n ,we denote the corresponding values by J ¯ f ( b ), respectively det ( J ¯ f ( b )). Definition 2.11. [15, Definition 3.1] Let B ⊆ R be partial E-domains. We will adopt thefollowing convention. A Khovanskii system over B is a quantifier-free L E ( B )-formula infree variables x := ( x , · · · , x n ) of the form H ¯ f ( x ) := n ^ i =1 f i ( x ) = 0 ∧ det ( J ¯ f ( x )) = 0 , for some f , · · · , f n ∈ B [ X ] E . (We will sometimes omit the subscript ¯ f in the above formulaand possibly make explicit the coefficients ¯ c ∈ B of the E -polynomials ¯ f in which case, wewill use H ¯ c ( x ).)Let a ∈ R . Then a ∈ ecl R ( B ) if H ¯ f ( a , · · · , a n ) holds for some a , · · · , a n ∈ R with a = a , where H ¯ f is a Khovanskii system, f , · · · , f n ∈ B [ X ] E (assuming that a i ∈ A ( R ), 1 ≤ i ≤ n , if needed for the f ′ i s to be defined).The operator ecl is a well-behaved E -algebraic closure operator, satisfying the exchangeproperty [17], [14], [15, Lemma 3.3, Theorem 1.1]. A. Wilkie used it in his proof of themodel-completeness of the theory of ( ¯ R , exp ), where ¯ R denotes the ordered field of realnumbers. Then J. Kirby extracted ecl from this o-minimal setting and showed that itcoincides with the closure operator cl defined above [15, Propositions 4.7, 7.1]. Since theoperator cl F on subsets of an E -field F induces a pregeometry, we get a notion of dimensiondim F as follows: Definition 2.12.
Let F be a partial E -field, let x := ( x , . . . , x n ) and let C ⊆ A ( F ) with C = cl F ( C ), then for m ≤ n ,dim F ( x /C ) = m if there exist x i , . . . , x i m with 1 ≤ i < . . . < i m ≤ n such that x i j / ∈ cl( x i ℓ , C ; 1 ≤ ℓ = j ≤ m ) and x i ∈ cl F ( x i , . . . , x i m , C ) , ≤ i ≤ n. In order to show that cl ⊆ ecl, J. Kirby uses a result of J. Ax on the Schanuel propertyin differential fields of characteristic 0 [15, Theorem 5.1], in order to show the followinginequality: td( x , E ( x ) /C ) − ℓ dim Q ( x /C ) ≥ dim( x /C ) , ( † )where td( x , E ( x ) /C ) denotes the transcendence degree of the field extension Q ( x , E ( x ) , C )of Q ( C ) and ℓ dim Q ( x /C ) the dimension of the quotient h x , C i Q / h C i Q of the Q -vectorspaces: h x , C i Q generated by x and C by h C i Q generated by C . (When C = ∅ , ℓ dim Q ( x /C )is simply the linear dimension of the Q -vector-space generated by x .From now on we will also denote by dim F ( · /C ) the dimension induced by the closureoperator ecl F ( · /C ) and by ℓ dim the linear dimension of a vector-space. As usual we definethe dimension of a subset as the maximum of the dimension of finite tuples contained inthat subset (see Definition 3.3). Definition 2.13. [15, Definition 5.3] Let F be a partial E -field and F be a partial E -subfield of F . For any C ⊆ A ( F ), let d ( x /C ) := td( x , E ( x ) /C, E ( C )) − ℓ dim Q ( x /C ) . IFFERENTIAL EXPONENTIAL TOPOLOGICAL FIELDS 9
Then F ✁ F if for every tuple x in A ( F ), d ( x /F ) ≥ . Let M ⊆ M be two L -structures. Recall that the notation M ⊆ ec M means thatany existential formula with parameters in M satisfied in M is also satisfied in M . Letus note some straightforward properties of the ecl F relation (and how it depends on F ). Remark 2.14.
Let F ⊆ F be two partial E -fields. Suppose that F ⊆ ec F , then(i) A ( F ) ⊆ A ( F ) (see Notation 2.3),(ii) ecl F ( F ) = F , provided the number of solutions to a Khovanskii system is finite,and(iii) let ϕ ( x , . . . , x k , ¯ y ) be an existential formula, let a ∈ F , then if dim F ( ϕ ( F , a ) / h a i E ) ≥ k , then dim F ( ϕ ( F , a ) / h a i ) E ) ≥ k .Proof: Let us show (iii). Let b , · · · , b k ∈ F , k >
1, ecl F -independent over h a i E and suchthat ϕ ( b , . . . , b k , a ) holds. Let us show that b , . . . , b k remain ecl F -independent over h a i E .We proceed by contradiction assuming that b k ∈ ecl F ( b , . . . , b k − , h a i E ). So there are u = b k , u , . . . , u ℓ ∈ F and E -polynomials f , · · · , f ℓ with coefficients in h b , . . . , b k − , a i E such that H ¯ f ( b k , u , . . . , u ℓ ) holds. Since F ⊆ ec F , we would find witnesses u ′ , . . . , u ′ ℓ ∈ F such that H ¯ f ( b k , u ′ , . . . , u ′ ℓ ) holds, contradicting the ecl-independence of b , . . . , b k over h a i E . (cid:3) Lemma 2.15.
Let F ⊆ F , where F is a partial E -field and F is a partial E -domainendowed with an E -derivation δ . Then, given u ∈ ecl F ( F ) , we can extend δ to an E -derivation on u in a unique way. Proof: Let u ∈ ecl F ( F ), so for some n , there exist u = u, u , . . . , u n ∈ F such that H ( u , · · · , u n ) holds in F , for some Khovanskii system over F . Set u := ( u , · · · , u n ) and X := ( X , · · · , X n ). Let f , · · · , f n ∈ F [ X ] E be such that(1) n ^ i =1 f i ( u ) = 0 ∧ det ( J ¯ f ( u )) = 0 . Applying δ to f ( u ) , . . . , f n ( u ), and using the E -polynomials f δ , · · · , f δn obtained inLemma 2.7, we get(2) f δ ( u )... f δn ( u ) + J ¯ f ( u ) δ ( u )... δ ( u n ) = δ ( f ( u ))... δ ( f n ( u )) = 0So,(3) δ ( u )... δ ( u n ) = − J ¯ f ( u ) − · f δ ( u )... f δn ( u ) Note that J ¯ f ( X ) − = J ∗ ¯ f ( X )( det ( J ¯ f ( X )) − , so J ¯ f ( X ) − is a matrix whose entries are ra-tional E -functions with denominator det ( J ¯ f ( X )) . Since ecl has finite character, we may assume that f i ∈ h Q ( e i ) i E [ X ] E for some tuple e i and f δi ∈ Q ( h e i i E , δ ( e i ))[ X ] E (see Lemma 2.7). Let ¯ e := ( e , . . . , e n ); so we can expresseach δ ( u i ) , ≤ i ≤ n , as an E -rational function t i, ¯ f ( u ) with coefficients in h Q (¯ e, δ (¯ e )) i E .Then we extend δ to the E -subfield generated by F , u , . . . , u n . Since ecl = cl there is onlyone such E -derivation extending δ on F . ( † ) AND NATHALIE REGNAULT
We can also express the successive derivatives δ ℓ ( u i ), 1 ≤ i ≤ n, ℓ ∈ N , ℓ ≥
2, as E -rational function t ℓi, ¯ f ( u ) with coefficients in Q h ¯ δ ℓ (¯ e )) i E . Note that the E -polynomialappearing in the denominator is a power of det ( J ¯ f ( X )). We set t i, ¯ f ( u ) = t i, ¯ f ( u ). (cid:3) For later use, we need to make explicit the form of the rational functions t ℓi, ¯ f ( u ) as afunction of u but also of the coefficients of ¯ f (see section 4.3). Notation 2.16.
By equation (3), we have δ ( y i ) := t i, ¯ f ( y ) where y := ( y , . . . , y n ) and t i, ¯ f ( y ) is obtained by multiplying the matrix − J ¯ f ( y ) − by the column vector f δ ( y )... f δn ( y ) . Now by Lemma 2.7, there are tuples x i ∈ F such that f i belongs to h x i i E [ X ] E and f δi ∈ Q ( h x i i E , δ ( x i ))[ X ] E . To f δi , we associate an E -rational function f δ, ∗ i by replacing δ ( x i ) by the tuple x i .Let ¯ x j := ( x , . . . , x n ) with 0 ≤ j . Then we re-write t i, ¯ f ( y ) as an E -rational functionwith coefficients in Q , namely as t ∗ i, ¯ f ( y ; ¯ x , ¯ x )), 1 ≤ i ≤ n . Set t , ∗ i, ¯ f := t ∗ i, ¯ f and t , ∗ ¯ f :=( t , ∗ , ¯ f , . . . , t , ∗ n, ¯ f ). Then we define t , ∗ i, ¯ f by applying δ and substituting t j, ¯ f ( y ) to δ ( y j ) , ≤ j ≤ n , and ¯ x j to δ (¯ x j − ), 2 ≥ j ≥
1. So we get an E -rational function t , ∗ i, ¯ f ( y ; ¯ x , ¯ x , ¯ x ),1 ≤ i ≤ n . We iterate this procedure, namely we apply δ to t ℓ, ∗ i, ¯ f , we substitute t , ∗ k, ¯ f ( y ) to δ ( y k ) , ≤ k ≤ n , and ¯ x j +1 to δ (¯ x j ), j ≥
0, to obtain t ℓ +1 , ∗ i, ¯ f ( y ; ¯ x , . . . , ¯ x ℓ +1 ) , ≤ i ≤ n .We denote t ℓ +1 , ∗ ¯ f := ( t ℓ +1 , ∗ , ¯ f , . . . , t ℓ +1 , ∗ n, ¯ f ). Proposition 2.17.
Let F ⊆ F be two partial exponential fields and assume that A ( F ) ispure in A ( F ) , that F is generated as a field by A ( F ) ∪ E ( A ( F )) and that F ✁ F , thenevery E -derivation on F extends to F . Proof: This is essentially [15, Theorem 6.3] but there the running assumption on partial E -fields is that A ( F ) is a Q -vector space. Therefore, in [15, Proposition 5.6], we assumeby induction that A ( F β ) is a pure subgroup of A ( F ). When defining F β +1 , we take thedivisible hull in A ( F ) of the subgroup generated by A ( F β ) and ¯ x , where r β belongs to ¯ x and d (¯ x/F β ) is minimal. Let us also denote by h A ( F β ) i Q the Q -vector space generated by A ( F β ) in F .Then in [15, Theorem 6.3], we assume that A ( F ) ⊆ A ( F ) and we choose a , · · · , a n ∈ A ( F ) \ A ( F ) maximal Z -independent over A ( F ) and generating A ( F ) over A ( F ) inthe following way: for any b ∈ A ( F ) there are z , · · · z n ∈ Z , u ∈ A ( F ) and n ∈ N ∗ such that nb = P ni =1 z i a i + u . Note that if P i z i a i ∈ h A ( F ) i Q , then for some n ∈ N ∗ , n P i z i a i ∈ A ( F ). Since P i z i a i ∈ A ( F ) and A ( F ) ⊆ A ( F ), then P i z i a i ∈ A ( F ). Sothe element b in [15, Fact 6.4], does not belong to h A ( F ) i Q either. The rest of the proof issimilar since it only involves the spaces of derivations over F . (cid:3) Note that if F is a subfield of F and if A ( F ) =domain(E), A ( F ) =domain(E) ∩ F , then A ( F ) is pure in domain(E). Indeed, let n ∈ N ∗ and assume that u ∈ A ( F ) and n.u ∈ A ( F ).So u ∈ F and so u ∈ A ( F ) = A ( F ) ∩ F . Proposition 2.18.
Let F ⊆ F be two partial exponential fields. Assume that we have an E -derivation on F , then it extends to F . IFFERENTIAL EXPONENTIAL TOPOLOGICAL FIELDS 11
Proof: Consider the subfield C := ecl F ( F ) of F ; we have shown already that any E -derivation on F extends to C (see Lemma 2.15). By [15, Propositions 4.7, 7.1], C = cl F ( F ).Let F be the subfield generated by ( A ( F ) ∩ C ) ∪ ( E ( A ( F ) ∩ C )). We will show that F ✁ F which will enable us to apply the result of J. Kirby recalled above.Note that F is a subfield of C ( C is a partial exponential subfield [15, Lemma 3.3]).Set A ( F ) := A ( F ) ∩ F , then A ( F ) ⊆ A ( F ). Note that if u ∈ A ( F ), then E ( u ) ∈ E ( A ( F )) ∩ C ⊆ F . In order to show that F ✁ F , we take a finite tuple a ∈ A ( F ) and we calculate d ( a /A ( F )) := td( a , E ( a ) /A ( F ) ∪ E ( A ( F ))) − ℓ dim( a /A ( F )), where:td( a , E ( a ) /A ( F ) ∪ E ( A ( F )) denotes the transcendence degree of the subfield of F ,generated by a , E ( a ) over the subfield generated by A ( F ) ∪ E ( A ( F )) and ℓ dim( a /A ( F )) is the dimension of the quotient of two Q -vector spaces, the first onegenerated by a and A ( F ) and the second one by A ( F ).By Ax’s theorem [15, Theorem 5.1, Corollary 5.2],td( a , E ( a ) /C ) ≥ ℓ dim( a /C )) + dim( a /C ) , Moreover we have td( a , E ( a ) /A ( F ) ∪ E ( A ( F ))) ≥ td( a , E ( a ) /C ).Now let us show that ℓ dim( a /C ) = ℓ dim( a /A ( F )) . Suppose we have a Q -linear combi-nation u of elements of a belonging to C . So for some nonzero natural number n ∈ N ∗ , wehave that nu also belongs to A ( F ) (since u ∈ A ( F )). So, we get that nu ∈ A ( F ) ∩ C and so nu ∈ F ∩ A ( F ) = A ( F ), namely u ∈ h A ( F ) i Q . Therefore, ℓ dim( a /C ) = ℓ dim( a /A ( F ))and d ( a /F ) ≥ (cid:3) Corollary 2.19.
Let F ⊆ F be two partial exponential fields and let δ be an E -derivationon F . Assume that we have ℓ elements c , · · · , c ℓ ∈ F ecl -independent over F and d , · · · , d ℓ ∈ F . Then there is an E -derivation ˜ δ on F , extending δ and such that ˜ δ ( c i ) = d i , ≤ i ≤ ℓ . Proof: Since c , · · · , c ℓ ∈ F are ecl-independent over F , there are ℓ E-derivations δ i on F which are zero on F and such that δ i ( c j ) = δ ij , 1 ≤ i, j ≤ ℓ . By the preceding proposition,we have a derivation D on F extending δ . Consider D + P ℓi =1 f i δ i with f i ∈ F . Since theset of E -derivations on F forms an F - module, this is an E -derivation which extends δ byconstruction. We define ˜ δ as D + P ℓi =1 ( d i − D ( c i )) δ i (setting in the above expression thecoefficients f i to be equal to d i − D ( c i ), 1 ≤ i ≤ ℓ ). (cid:3) E-varieties and topological exponential fields
E-varieties.
Let K be a (partial) exponential E -field. Let X := ( X , · · · , X n ), f ∈ K [ X ] E and a ∈ K n , denote by ∇ f := ( ∂ X f ( X ) , · · · , ∂ X n f ( X )). and ∇ f ( a ) :=( ∂ X f ( a ) , · · · , ∂ X n f ( a )) . Definition 3.1.
Let g , · · · , g m ∈ K [ X ] E and let V n ( g , · · · , g m ) := { a ∈ K n : m ^ i =1 g i ( a ) = 0 } . An E -variety will be a definable subset of some K n of the form V n (¯ g ) for some ¯ g ∈ K [ X ] E .Sometimes we will need to consider the elements of an E -variety in an extension of K ; inthis case we will say that it is defined over K . Let V be an E -variety, then a is a regularpoint of V if for some ¯ g , V = V n (¯ g ) and ∇ g ( a ) , · · · , ∇ g m ( a ) are linearly independent over K (note that this implies that m ≤ n ). ( † ) AND NATHALIE REGNAULT
In the following, we will make a partition of variables of the g i ’s and consider the regularzeroes with respect to a subset of the set of variables. Notation 3.2.
Let 0 < n ≤ n and let f ∈ K [ X ] E . Denote by(4) ∇ n f := ( ∂ X n − n f, · · · , ∂ X n f ) . Consider the following subset of V n (¯ g ), with m ≤ n :(5) V regn,n (¯ g ) := { b ∈ K n : m ^ i =1 g i ( b ) = 0 & ∇ n g ( b ) , · · · , ∇ n g m ( b ) are K − linearly independent } . In case n = n , we simply denote V regn,n (¯ g ) by V regn (¯ g ).Furthermore, we need the following variant. Let ¯ i := ( i , · · · , i n ) be a strictly increasingtuple of natural numbers between 1 and n (of length 1 ≤ n ≤ n ). Then for f ∈ K [ X ] E ,we denote by(6) ∇ ¯ i f := ( ∂ X i f, · · · , ∂ X in f ) . We consider the subset of V n (¯ g ):(7) V regn, ¯ i (¯ g ) := { b ∈ K n : m ^ i =1 g i ( b ) = 0 & ∇ ¯ i g ( b ) , · · · , ∇ ¯ i g m ( b ) are K − linearly independent } . Note that in order to be non-empty we need that m ≤ n = | ¯ i | .3.2. Generic points.
Let K ⊆ L be partial E -fields. In section 2.3, we have seen thatecl L is a closure operator which coincides with cl L to which we associated the dimensionfunction dim L ( · /K ) (see Definition 2.12). As usual one defines the dimension of a definablesubset B ⊆ L n and the notion of generic points in B (see for instance [14]). Definition 3.3.
Let B be a definable subset of L n defined over K . The dimension of B over K is defined as dim L ( B/K ) := sup { dim L ( b /K ) : b ∈ B } . Let b ∈ B , then b is ageneric point of B over K if dim L ( b /K ) = dim L ( B/K ).We will need the following notion of subtuples.
Notation 3.4.
Let a := ( a , . . . , a n ) be an n -tuple in K and let X := ( X , . . . , X n ).Let 0 < m < n and let { i , . . . , i m } ˙ ∪{ j , . . . , j n − m } be a partition of { , . . . , n } , with1 ≤ i < . . . < i m ≤ n and 1 ≤ j < . . . < j n − m ≤ n .A m -subtuple of a is a m -tuple denoted by a [ m ] of the form ( a i , . . . , a i m ) and we denoteby a [ n − m ] := ( a j , . . . , a j n − m ).Given an E -polynomial f ( X ) ∈ K [ X ] E , we denote either by f ( a [ n − m ] , X i , . . . , X i m ) orby f a [ n − m ] ( X i , . . . , X i m ) the E -polynomial obtained from f when substituting for X j i theelement a j i , 1 ≤ i ≤ n − m . We adopt the same convention for L E -terms. Remark 3.5.
Let ¯ f = ( f , · · · , f m ) ⊆ K [ X ] E , a := ( a , · · · , a n ) ∈ V regn ( ¯ f ) ⊆ L n , 1 ≤ m ≤ n . Then:(1) There is a m -subtuple a [ m ] of a and a Khovanskii system over K h a [ n − m ] i E suchthat H ¯ f a [ n − m ] ( a [ m ] ) holds.(2) In particular dim L ( a /K ) ≤ n − m and if V n ( ¯ f ) = V regn ( ¯ f ), then dim L ( V n ( ¯ f ) /K ) ≤ n − m . IFFERENTIAL EXPONENTIAL TOPOLOGICAL FIELDS 13 E -ideals and differentiation. Let R be a partial E -ring. Let X := ( X , . . . , X n )and X ˆ i be the tuple X where X i is removed, 1 ≤ i ≤ n . Similarly for a ∈ R n , we denote a ˆ i := ( a , · · · , a i − , a i +1 , · · · , a n ). Definition 3.6.
Let I ⊆ R be an ideal of R . Then I is an E - ideal if( r ∈ I → E ( r ) − ∈ I ) . A prime E -ideal is a prime ideal which is an E -ideal.In R [ X ] E , an example of a prime E -ideal is Ann R [ X ] E ( a ) := { f ∈ R [ X ] E : f ( a ) = 0 } .(When the context is clear we will omit the superscript R [ X ] E .)As usual the definition of E -ideal is set-up in such a way that if I ⊆ R is an E -ideal,then on the quotient R/I , we have a well-defined exponential function given by: E ( r + I ) := E ( r ) + I for r ∈ A ( R ). So ( R/I, E ) is a again a partial E -ring.We now recall results from A. Macintyre on E -ideals closed under partial derivation. Fact 3.7. [17, Theorem 15 and Corollary]
Let R be a partial E -domain. Let ≤ i ≤ n .Let I ⊆ R [ X ] E be an E -ideal closed under the E -derivation ∂ X i . Then either I = 0 or I contains a non-zero element of R [ X ˆ i ] E . In particular, if I = 0 is closed under all E -derivations ∂ X i , ≤ i ≤ n and R is a field, then I = R [ X ] E . Let K ⊆ L be partial E -fields. Fact 3.7 actually shows that ecl L -independent elementsover K do not satisfy any hidden exponential-algebraic relations over K . Corollary 3.8.
Let a := ( a , · · · , a n ) ∈ L n be such that a , · · · , a n are ecl L -independentover K . Then there is no g ∈ K [ X ] E \ { } such that g ( a ) = 0 .Proof. By the way of contradiction assume there is g ∈ K [ X ] E be such that g ( a ) = 0. Thenfor i = 1 , · · · , n , ∂ X i g ( a ) = 0 otherwise a i ∈ ecl L ( K ( a ˆ i )). (Indeed, letting h ( X ) := g ( a ˆ i , X ),we would have H h ( a i ).) Hence the ideal Ann ( a ) is an E -ideal, closed under all partial E -derivations ∂ X i , 1 ≤ i ≤ n . So by Fact 3.7, since Ann ( a ) = 0, it is equal to K [ X ] E , acontradiction. (cid:3) Let K δ be an expansion of the partial E -field K by an E -derivation δ and let ˜ K be an E -field extending K . Let A ⊆ ˜ K n . Let I ( A ) ⊆ K [ X ] E be the set of E -polynomials withcoefficients in K which vanish on A , namely I K ( A ) = T a ∈ A Ann K [ X ] E ( a ). Note that it isan E -ideal as an intersection of E -ideals. Definition 3.9.
For A ⊆ ˜ K n , let τ ( A ) ⊆ ˜ K n be the E -torsor of A (over K ), namely: τ ( A ) := { ( a , b ) ∈ ˜ K n : a ∈ A and n X i =1 ∂ X i f ( a ) .b i + f δ ( a ) = 0 for all f ( X ) ∈ I K ( A ) } . Note that if we can find f i ( X ) ∈ Ann K [ X ] E ( a ), a ∈ A , 1 ≤ i ≤ m ≤ n such that ∇ f ( a ) , . . . , ∇ f m ( a ) are K -linearly independent, then setting T a := { b ∈ ˜ K n : n X i =1 ∂ X i f ( a ) .b i = 0 for all f ( X ) ∈ Ann K [ X ] E ( a ) } , we have that ℓ dim( T a ) ≤ n − m . ( † ) AND NATHALIE REGNAULT
Lemma 3.10.
Let K ⊆ ˜ K be partial E -fields, and let δ be an E -derivation on K . Let ¯ f = ( f , · · · , f m ) ⊆ K [ X ] E . Suppose that there are ( a , b ) ∈ ˜ K n such that a ⊆ ˜ K n is ageneric point of V regn ( ¯ f ) and ( a , b ) ∈ τ ( V regn ( ¯ f )) . Then there is an E -derivation δ ∗ on ˜ K extending δ , uniquely determined on ecl ˜ K ( K ( a )) and such that δ ∗ ( a i ) = b i , for i = 1 , . . . , n .Proof. Since a ∈ V regn ( ¯ f ), we have that ∇ f ( a ) , . . . , ∇ f m ( a ) are ˜ K -linearly independent. Bypermuting the coordinates of a , assume ∇ m f ( a ) , . . . , ∇ m f m ( a ) are ˜ K -linearly independent.Set a [ n − m ] := ( a , . . . , a n − m ) and a [ m ] := ( a n − m +1 , . . . , a n ). Note that det ( J ¯ f a [ n − m ] ( a [ m ] )) =0. Since n − m = dim ˜ K ( a /K ), a , · · · , a n − m are ecl ˜ K -independent.By Corollary 2.19, there is an E -derivation ˜ δ on ˜ K extending δ on K and such that˜ δ ( a i ) = b i , 1 ≤ i ≤ n − m .By assumption ( a , b ) ∈ τ ( V regn ( ¯ f )). In particular V mi =1 P nj =1 ∂ X j f i ( a ) b j + f δi ( a ) = 0 ( † ).We break the sum P nj =1 ∂ X j f i ( a ) b j in two parts: P n − mj =1 ∂ X j f i ( a ) b j , P nj = n − m +1 ∂ X j f i ( a ) b j .By assumption det ( J ¯ f a [ n − m ] ( a [ m ] )) = 0, so the fact that b satisfies ( † ) is equivalent to thefact that the subtuple b [ m ] satisfies the equation (8) below:(8) b n − m +1 ... b n = − J ¯ f a [ n − m ] ( a [ m ] ) − · f δ ( a ) − P n − mj =1 ∂ X j f ( a ) b j ... f δm ( a ) − P n − mj =1 ∂ X j f m ( a ) b j So there is only one such E -derivation satisfying δ ∗ ( a i ) = b i for i = n − m + 1 , · · · , n onecl ˜ K ( K ( a )) by Lemma 2.15. Note that by using Lemma 2.7, we explicitly define a mapping δ ∗ on K ( a ) E as follows. Let p ( X ) ∈ K [ X ] E , define δ ∗ ( p ( a )) := P nj =1 δ ∗ ( a i ) ∂ X i p ( a ) + p δ ( a )(note that p δ ∈ K [ X ] E ). Furthermore, by Corollary 3.8, since a n − m +1 , · · · , a n are ecl ˜ K -independent, Ann ( a [ n − m +1] ) ∩ K [ X ] E = { } and so given q ( X ), q ( X ) in K [ X ] E \ { } , wecan define δ ∗ ( q ( a [ n − m ] ) /q ( a [ n − m ] )) := q ( a [ n − m ] ) δ ∗ ( q ( a [ n − m ] )) − q ( a [ n − m ] ) δ ∗ ( q ( a [ n − m ] )) q ( a [ n − m ] ) . (cid:3) Topological E -fields. From now on L will be a relational extension of L E and let K be an L -structure which is a relational expansion of a partial E -field K . Denote by¯ K := ( K, + , − , · , , V denote a basis of neighbourhoods of 0. Then ( K , V ) is atopological L -field if V induces an Hausdorff topology such that the functions of L E areinterpreted by C -functions, namely continuously differentiable functions and the inversefunction is continuous on K \ { } and that each relation and its complement is a union ofan open set and the zero-set of a finite system of E -polynomials. This notion of topological L -fields extends the one given in [13, section 2.1]. We will say that K is endowed with adefinable topology if there is an L -formula χ ( x, y ) such that a basis of neighbourhoods of0 in K is given by χ ( K, d ), where d ∈ K n , n = | y | . Note that if K is endowed with adefinable topology, then any field K elementary equivalent to K can be endowed with adefinable topology using the same formula χ ( x, y ). A priori the topology can be discrete,but we will assume it is non-discrete in our main results. As usual, the cartesian productsof K are endowed with the product topology. Let x be a m -tuple, we will denote by ¯ χ ( x , y )the formula V mi =1 χ ( x i , y ). IFFERENTIAL EXPONENTIAL TOPOLOGICAL FIELDS 15
Notation 3.11.
Let ( K , V ) ⊆ ( ˜ K , W ) be two topological L -fields with ( ˜ K , W ) be a topo-logical extension of ( K , V ) [13, Definition 2.3], namely K is an L -substructure of ˜ K and forany V ∈ V there exists W ∈ W such that V = W ∩ K . Let W K := { W ∈ W : W ∩ K ∈ V} .On elements a, b ∈ ˜ K we have the equivalence relation a ∼ W K b which means that a − b belongs to every element of W K . (We will also use the notation a ∼ K b .)We will say that a non zero element a ∈ ˜ K is K -small if a ∼ W K a ∼ K K is endowed with a definable topology with corresponding formula χ ( x, ¯ y )and ˜ K an elementary extension of K endowed with a topology induced by χ , then ˜ K is atopological extension of K .3.5. Implicit function theorem.
In this subsection and from now on, we will assumethat the topology on K is non-discrete and that K is endowed with a definable topologywith corresponding formula χ .We now introduce the following implicit function theorem hypothesis that we put onthe class of fields under consideration. The implicit function theorem for C -functions(continuously differentiable functions), or C ∞ -functions (infinitely differentiable functions),or analytic functions is classically proven in fields like R , Q p (or more generally complete(non-discrete) valued fields) [2, section 1.5]. A. Wilkie stated it for any field K elementaryequivalent to an expansion of the field of reals [28, section 4.3], T. Servi recasted the resultsof Wilkie in definably complete expansions of ordered fields [26]. Definition 3.12.
A topological L -field K satisfies (IFT) F where F is a class of C -functionsin K , if the following holds.Let f , · · · , f m ∈ F , f i : K ℓ + m → K , 1 ≤ i ≤ m , let ( a , b ) ∈ K ℓ + m . Consider ¯ f a ( y ) :=( f ( a , y ) , · · · , f m ( a , y )) as functions defined on K m , | y | = m . Assume that ¯ f ( a , b ) = 0 andthat det ( J ¯ f a ( b )) = 0 (see Notation 2.10).Then there are neighbourhoods O a (respectively O b ) of a (respectively of b ) and C -functions g ( x ) , · · · g m ( x ) : O a → O b , | x | = ℓ such that m ^ i =1 g i ( a ) = b i ∧ ∀ x ∈ O a ( m ^ i =1 f i ( x , g ( x ) , · · · , g m ( x )) = 0) ∧ (9) J ¯ g ( x ) = − ( ∇ m ¯ f ( x , ¯ g ( x )) − ∇ ℓ ¯ f ( x , ¯ g ( x )) . (10)Keeping the same notation, this allows us to define a map ˆ: h ˆ h sending a function h : K ℓ + m → K to a function ˆ h : K ℓ → K : x h ( x , g ( x ) , · · · , g m ( x )). It is convenient tointroduce an ( ℓ + m )-tuple (˜ g ) of functions defined as follows: ˜ g i ( x ) = x i for 1 ≤ i ≤ ℓ and˜ g ℓ + i := g i ( x ), 1 ≤ i ≤ m . With this notation ˆ h ( x ) = h (˜ g ( x )). Notation 3.13.
As noted in [28, 4.3], when the topology on K is definable, this impliesthat whenever the functions f i are definable (or C ∞ ), the g i ’s are definable (or C ∞ ), usingthe above equations (9), (10). When F consists of the class of functions represented by E -polynomials, we denote this scheme by (IFT) E and when it consists of the class of C -functions L E -definable, we denote this scheme by (IFT) E . When the field K is a valuedfield, we will need to make the additional assumption that whenever the functions ¯ f areanalytic in a neighbourhood of a , then the functions ¯ g are also analytic in a neighbourhoodof a . In this last case, we will use the notation (IFT) anE , respectively (IFT) an E to indicatewe make this additional assumption. ( † ) AND NATHALIE REGNAULT
Remark 3.14. [17, 2.4, Notes a)]
Let K be a topological L -field satisfying (IFT) E . Let f , · · · , f m ∈ K [ x , y ] E , and ( a , b ) ∈ K ℓ + m , with | x | = ℓ, | y | = m . Let a ∈ K ℓ , b ∈ K m .Assume that H ¯ f a ( b ) holds, namely ¯ f ( a , b ) = 0 and det ( J ¯ f a ( b )) = 0 (see Definition 2.11).Then, by the implicit function theorem, there is an open neighbourhood O of b such thatfor all y ∈ O , ¯ f a ( y ) = 0 , namely b is an isolated zero of the system ¯ f a ( y ) = 0 . Notation 3.15.
Recall that a germ of a function f at a ∈ K n is an equivalence class,identifying two functions if they are equal on a neighbourhood of a . We will denote suchequivalence class containing f by [ f ] a . For f ∈ F , we will denote the set of germs associ-ated with elements of F at a , by G n a ( F ). When K is a valued field, we will consider theintersection G an,n a ( F ) of G n a ( F ) with the set of functions analytic in a neighbourhood of a . Lemma 3.16.
Let K be a topological L -field satisfying (IFT) E . Let ( a , b ) ∈ K ℓ + m and let f , · · · , f m , h : K ℓ + m → K be C functions on an open neighbourhood of ( a , b ) . Assumethat ¯ f ( a , b ) = 0 and assume that det ( J ¯ f a ( b )) = 0 . Then, keeping the same notations as inDefinition 3.12, ∇ ¯ f ( a , b ) , ∇ h ( a , b ) are K -linearly independent iff ∇ ˆ h ( a ) = 0 . Proof: The proof is the same as the one of [28, Lemma 4.7] (and it was also used in [18](see [18, Lemma 5.1.3])). (cid:3)
Definition 3.17.
A topological L -field K satisfies (LFF) F ( lack of flat functions ) where F is a class of definable C -functions in K , if, in case the topology on K is induced by anordering, the following holds: given an open interval U of K and given a system of lineardifferential equations (of order 1) with coefficients in F , there is a unique C -functionsolution of that system on U . As before, when F consists of the class of C -functions L E -definable, we denote this scheme by (LFF) E . Remark 3.18.
In case of R or any field elementary equivalent to it, the hypothesis (LFF) E is satisfied; in fact it suffices to assume that K is an ordered definably complete field (for aproof see [26, Theorem 1.5.1]).In the next proposition, we work in a Noetherian differential subring of K [ x ] E containinga finite given number of elements of K [ x ] E . Using the complexity function ord definedin K [ X ] E , it is always possible to find such a ring. Indeed an exponential polynomialcorresponds to an L E -term and those are constructed by induction in finitely many steps.So we place ourselves in the ordinary polynomial ring generated by all the (finitely many)sub-terms appearing in the construction and this ring is closed under differentiation.The result was first observed for ( ¯ R , exp ) by A. Wilkie [28] and then without the as-sumption of noetherianity, for definably complete structures by G. Jones and A. Wilkie[14]. One can follow the proof in [18, Proposition 5.1.4] given for ( Q p , E p ). (It uses Lemma3.16.) Proposition 3.19. [28, Theorem 4.9]
Assume that K is a topological field whose topologyis either induced by an ordering or a valuation.Let R n be a Noetherian subring of G n r ( E ) closed under differentiation. In case K is anordered field, assume that it satisfies (IFT) E and (LFF) E . In case K is a valued field,assume that K satisfies (IFT) anE and that R n ⊆ G an,n r ( E ) .Let m ∈ N and let f , . . . , f m ∈ R n . Let r ∈ K n and assume r ∈ V regn ( ¯ f ) . Then, exactlyone of the following is true: (a) n = m ; or, IFFERENTIAL EXPONENTIAL TOPOLOGICAL FIELDS 17 (b) m < n and for all h ∈ R n with h ( r ) = 0 , h vanishes on U ∩ V regn ( f , . . . , f m ) forsome open neighbourhood U containing r , (c) m < n and for some h ∈ R n , r ∈ V reg ( f , . . . , f m , h ) . (cid:3) Lemma 3.20.
Let K be a topological L -field endowed with a definable topology satisfyingsatisfying (IFT) E . Let K be a | K | + -elementary extension of K . Then there is an element t ∈ K \ ecl K ( K ) with t ∼ K . More generally for every n ∈ N ∗ there are n elements t , . . . , t n ∈ K ecl -independent over K and K -small.Proof. Consider the partial type tp K ( x ) consisting of L ( K )-formulas expressing that x ∼ K x / ∈ ecl ( K ). The first property is expressed by the set of formulas χ ( x, ¯ a ), where ¯ a variesin K and the second property by ¬∃ ¯ y H ¯ f ( x, ¯ y ) where ¯ f varies in K [ X, ¯ Y ] E . By Remark3.14, this set of formulas is finitely satisfiable. So tp K ( x ) is realized in an | K | + -saturatedextension of K (see for instance [20, Theorem 4.3.12]).Then by induction on n , assume we found n elements t , . . . , t n ecl-independent over K and K -small. Let K := ecl K ( K ( t , . . . , t n )). Then by Remark 3.14, | K | = | K | .Consider the partial type tp K ( x ) consisting of L ( K )-formulas expressing that x ∼ K x / ∈ ecl( K ). Again by Remark 3.14, it is finitely satisfiable and so it is realized in K by an element t n +1 such that t , . . . , t n +1 satisfy the requirement of the lemma. (cid:3) Proposition 3.21.
Let K be a topological L -field endowed with a definable topology, sat-isfying (IFT) E . Let ¯ f = ( f , · · · , f m ) ⊆ K [ X ] E , | X | = n > m . Suppose that there is a ∈ V regn ( ¯ f ) ∩ K n .Then there is an elementary L -extension ˜ K of K and b ∈ V regn ( ¯ f ) ∩ ˜ K n with b − a ∼ K ¯0 and dim ˜ K ( b /K ) = n − m . In particular, b is a generic point of V regn ( ¯ f ) ∩ ˜ K n .Proof. Let a ∈ V regn ( ¯ f ), then ∇ f ( a ) , · · · , ∇ f m ( a ) are linearly independent over K . Bypermuting the variables X , . . . , X n , assume that ∇ m f ( a ) , . . . , ∇ m f m ( a ) are K -linearlyindependent (see Notation 3.2). So we have det ( J ¯ f a [ n − m ] ( a [ m ] )) = 0, with a := ( a [ n − m ] , a [ m ] )(see Notation 3.4). By (IFT) E , there are definable neighbourhoods O ⊆ K n − m of a [ n − m ] , O ′ ⊆ K m of a [ m ] and definable functions g , . . . , g m from O → O ′ such that a [ m ] = g ( a [ n − m ] )and such that for all x ∈ O , V mi =1 f i ( x , g ( x ) , . . . , g m ( x )) = 0. By Lemma 3.20, there is anelementary L E -extension ˜ K of K containing n − m K -small elements t , · · · , t n − m whichare ecl-independent over K .Let t [ n − m ] := ( t , · · · , t n − m ) and b := a [ n − m ] + t [ n − m ] ∈ K n − m . Then b ∈ ˜ K are ecl-independent over K , a − b ∼ K V ni =1 f i ( b , g ( b ) , . . . , g m ( b )) = 0. (cid:3) Topological differential exponential fields
Differential fields expansions.
Throughout this section, we will place ourselves inthe same setting as in subsection 3.5. The language L is a relational expansion of L E defined as in section 3.4. We will always assume that the topological L -field K is endowedwith a definable field topology with corresponding formula χ . We will also assume that thetopology is either induced by an ordering or a valuation. Let L δ be the expansion of L bytwo unary function symbols: − for the inverse (extended to 0 − = 0 by convention) and δ for an E -derivation. Given K , we denote by K δ its L δ -expansion, namely an expansionof K by an E -derivation δ . Given an L -theory of topological L -fields, we denote by T δ thetheory T together with the axioms of E -derivation (see Definition 2.4). In particular if K | = T , then K δ is a model of T δ . ( † ) AND NATHALIE REGNAULT
By assumption on the language L , any L δ -term t ( x ) in the field sort with x = ( x , . . . , x n ),is equivalent, modulo the theory of differential E -fields, to a L δ -term t ∗ (¯ δ m ( x ) , . . . , ¯ δ m n ( x n ))where t ∗ is a L -term, for some ( m , . . . , m n ) ∈ N n . By possibly adding tautological con-junctions like δ k ( x i ) = δ k ( x i ) if needed, we may assume that all the m i ’s are equal. Weuse the following notation ¯ δ m ( x ) := ( x , δ ( x ) , . . . , δ m ( x )), with δ i ( x ) := ( δ i ( x ) , . . . , δ i ( x n )),1 ≤ i ≤ m . Therefore, we may associate with any quantifier-free L δ -formula ϕ ( x ) anequivalent L δ -formula, modulo the theory of differential E -fields, of the form ϕ ∗ ,m (¯ δ m ( x )), m ∈ N , where ϕ ∗ ,m is a L -quantifier-free formula which arises by uniformly replacing everyoccurrence of δ m ( x i ) by a new variable x mi in ϕ with the following choice for the order ofvariables ϕ ∗ ,m ( x , . . . , x m ), where x i = ( x i , . . . , x in ), 0 ≤ i ≤ m So we get ϕ ( x ) ⇔ ϕ ∗ ,m (¯ δ m ( x )) . We will call the least such m , the order of the quantifier-free L δ -formula ϕ . We will call anatomic formula of the form t ( x ) = 0, where t ( x ) is a L δ -term, an L δ -equation. Usually wewill usually drop the superscript m in the formula ϕ ∗ ,m .4.2. Scheme (DL) E . Given a model-complete theory T of topological L -fields, we want toaxiomatize the existentially closed differential expansions of models of T . By a scheme offirst-order axioms, we will express that certain systems of differential exponential equationshave a solution. In order to determine which ones, we first transform, using the processexplained above, an exponential differential equation in an ordinary exponential equation,taking into account all the possible ecl-relations among the variables. Since the derivationextends in a unique way to the ecl-closure, we enumerate partitions of the variables intotwo subsets: a first one where we impose no conditions and the other one where we expressthat there are regular solutions of an E -variety over this first subset of variables. Definition 4.1.
Let K δ be a differential topological L -field. Let V δ ( x ) be an L δ ( K )-definable subset, defined by a conjunction ϕ ( x ) of non-trivial L δ ( K )-equations of order m .Denote by x i := ( x i , . . . , x in ) , ≤ i ≤ m with x = x = ( x , · · · , x n ).Let n ≥ ℓ ≥ ℓ ≥ . . . ≥ ℓ m − ≥ x i [ ℓ i ] := ( x i , . . . , x iℓ i ), 0 ≤ i ≤ m −
1. We are goingto enumerate all possible Khovanskii systems expressing that each element x ij , ℓ i +1 ≤ j ≤ n ,of the subtuple ( x iℓ i +1 , . . . , x in ) of x i is in the ecl K -closure of x ℓ ] , . . . , x i [ ℓ i ] . For 0 ≤ i ≤ m − ℓ i < j ≤ n , let ¯ f j,i , be a tuple of E -polynomials with coefficients in Q (¯ c, x i [ ℓ i ] , . . . , x ℓ ] ),¯ c ∈ K , and consider the Khovanskii systems H ¯ f j,i ( x ij , z j,i ) with z j,i a tuple of new variablesexpressing this ecl-dependence (see Definition 2.11). Let H ( x , . . . , x m − , ¯ z ) be the L -formula: m − ^ i =0 n ^ j = ℓ i +1 H ¯ f j,i ( x ij , z j,i )where the tuple ¯ z := ( z ( ℓ i +1) ,i , . . . , z n,i ) ≤ i ≤ m − . We call such L (¯ c )-formula H , a Khovanskiiformula.Recall that whenever H ¯ f j,i ( x ij , z j,i ) holds, ℓ i + 1 ≤ j ≤ n , it implies that δ ( x ij ) is uniquelydetermined. We take it into account in the following way. We have that δ ( x ij , z j,i ) = t , ∗ ¯ f j ( x ij , z j,i ), where t , ∗ ¯ f j is a tuple of E -rational functions with coefficients in Q (¯ δ (¯ c ) , x i [ ℓ i ] , . . . , x ℓ ] )(see Notation 2.16).Let ϕ ∗ H ( x , . . . , x m , ¯ z ) be the L (¯ c ) ∪ { − } -formula we get from ϕ ∗ ( x , . . . , x m ) by addingfor each 0 ≤ k ≤ m − IFFERENTIAL EXPONENTIAL TOPOLOGICAL FIELDS 19 • the atomic formula ( x k +1 j , z j,k +1 ) = t , ∗ ¯ f j ( x kj , z j,k ), where t , ∗ ¯ f j is a tuple of E -rationalfunctions with coefficients in Q (¯ δ (¯ c ) , x k [ ℓ k ] , . . . , x ℓ ] ), ℓ k + 1 ≤ j ≤ n , • a formula expressing that the determinants of the Jacobian matrices occurring in theseKhovanskii systems are non-zero.Furthermore we will assume that clearing denominators, we put ϕ ∗ H in the followingequivalent form: a conjunction of E -polynomials equations (that we will denote by V ϕ ∗ H )and an E -polynomial inequation.Varying over all possible ecl-dependence relations (with coefficients in ¯ c ⊆ K ) among thevariables in the tuple x , . . . , x m − , we get an infinite disjunction over the Khovanskii for-mula H := H ( x , . . . , x m − , ¯ z ) of the form: W H ∃ ¯ z H ( x , . . . , x m − , ¯ z ) ∧ ϕ ∗ H ( x , . . . , x m , ¯ z ).Note that in case we do have a non-trivial relation between the x i , 0 ≤ i ≤ m −
1, withcoefficients in Q (¯ c ), they cannot be all ecl-independent over Q (¯ c ) by Corollary 3.8, but itmight be the case that the x i , 0 ≤ i ≤ m − Q (¯ c ) but we stillhave a non-trivial relation between the x i , 0 ≤ i ≤ m .The scheme (DL) E has the following form: for each L δ (¯ c )-formula ϕ ( x ) which is a finiteconjunction of L δ (¯ c )-equations of order m , for each Khovanskii L -formula H ( x , . . . , x m − , ¯ z ),we have: ∀ ¯ d ∀ x . . . ∀ x m ( ∃ ¯ zH ( x , . . . , x m − , ¯ z ) ∧ ϕ ∗ H ( x , . . . , x m , ¯ z )) → ( ∃ α ϕ ( α ) ∧ χ (¯ δ m ( α ) − ( x , . . . , x m ) , ¯ d )) . Note that by quantifying over the coefficients ¯ c , this scheme is first-order. Remark 4.2.
In a model K δ | = T δ of the scheme (DL) E , the differential points are dense inall cartesian products of K . Let O ⊆ K m +1 and ( a , . . . , a m ) ∈ O . Consider the L δ -formula ϕ ( x ) := δ m ( x ) = a m . The formula ϕ ∗ ( x , . . . , x m ) := x m = a m . Let V m − i =0 H i ( x i ) := x i − a i = 0, we find a differential solution b such that δ m ( b ) = a m and ¯ δ m − ( b ) is close to( a , . . . , a m − ). This is analogous to [13, Lemma 3.12].The main result of this section is: Theorem 4.3.
Let T be a model-complete complete theory of topological L -fields whosetopology is either induced by an ordering or a valuation. Assume in the ordered case thatthe models of T satisfy the schemes (IFT) E and (LFF) E and in the valued case that themodels of T satisfy the schemes (IFT) anE . Then the class of existentially closed models of T δ is axiomatized by T δ ∪ (DL) E . The strategy of the proof is the following. First show that a model K δ | = T δ satisfying(IFT) E and LFF E in the ordered case and IFT anE in the valued case can be embedded in˜ K δ | = T δ satisfying this scheme (DL) E . Second show that if T is model-complete, thenwe may choose ˜ K | = T . Finally show that if T δ ∪ (DL) E is consistent, then it gives anaxiomatization of the existentially closed models of T δ . We begin by realizing one instanceof the scheme (DL) E in a differential extension of K δ . Lemma 4.4.
Let K δ | = T δ and suppose K satisfies (IFT) E and (LFF) E in the ordered caseand IFT anE in the valued case. Let M be a | K | + - saturated elementary L -extension of K .Let ϕ ( x ) be a finite conjunction of L δ ( K ) -equations of order m , let H ( x , . . . , x m − , ¯ z ) be aKhovanskii formula with | x i | = n , ≤ i ≤ m . Assuming that for some a := ( a . . . , a m ) ∈ K , | a i | = n , we have: K | = ∃ ¯ z ( H ( a , . . . , a m − , ¯ z ) ∧ ϕ ∗ H ( a , . . . , a m , ¯ z )) , ( † ) AND NATHALIE REGNAULT then for any ¯ d ∈ K , we can find α ∈ M and we can extend δ on M such that M δ | = ϕ ( α ) ∧ ¯ χ (¯ δ m ( α ) − a , ¯ d ) . Proof.
For sake of simplicity suppose that m = 1. Let a := ( a , a ). Let ¯ c be the parametersfrom K occurring in the Khovanskii formula H and in the formula ϕ ∗ H . Suppose H isof the form V n − ℓi =1 H ( a ℓ + i , z i ), 0 < ℓ < n . Let ¯ u := ( u , . . . , u n − ℓ ) ∈ K be such that K | = V n − ℓi =1 H ( a ℓ + i , u i ). Let N be the length of ( a , ¯ u ). By Lemma 3.20, we can find t , . . . , t ℓ ∈ M which are K -small and ecl-independent over K .Let V ϕ ∗ H be the system of E -polynomial equations in unknowns x ℓ ] := ( x , . . . , x ℓ ),with coefficients in Q h ¯ c i δ and with parameters x , ¯ z := ( z ℓ +1 , , . . . , z n, ) occurring in ϕ ∗ H .We denote the corresponding tuple of E -polynomials by ¯ f := ¯ f ( x ℓ ; x , ¯ z ) and we will alsouse the notation ¯ f x , ¯ z ( x ℓ ] ). (Recall that for ℓ + 1 ≤ j ≤ n , the variables x j have beenreplaced by rational functions t , ∗ j depending on x ℓ ] , x ℓ ] , x j , z j, . Moreover, we have cleareddenominators and added the corresponding E -inequations.)First assume that | ¯ f | = ℓ and a ℓ ] is a regular zero of V ℓ ( ¯ f a , ¯ u ). Then we apply directlythe implicit function theorem (IFT) E . Let O be a definable neighbourhood of ( a , ¯ u ) and O be a definable neighbourhood of a ℓ ] and C functions g i from O to O , 1 ≤ i ≤ ℓ , suchthat ℓ ^ i =1 g i ( a , ¯ u ) = a i ∧ ∀ ¯ y ∈ O ( ℓ ^ i =1 f ¯ y,i ( g (¯ y ) , · · · , g ℓ (¯ y )) = 0)Let N be the length of ( a , ¯ u ) and n i := | u ℓ + i | , 1 ≤ i ≤ n − ℓ . Recall that we put theproduct topology on M N . Let π be the projection sending a tuple ( a , u ) of M N to thesubtuple a [ ℓ ] ∈ M ℓ and π i the projection sending ( a , u ) to the subtuple ( a ℓ + i , u ℓ + i ) ∈ M n i +1 ,1 ≤ i ≤ n − ℓ .Let ( a ℓ + i , u ℓ + i ) be regular zeroes of each system H i ( x ℓ + i , z ℓ + i ), 1 ≤ i ≤ n − ℓ , over Q (¯ c, a [ ℓ ] ). We apply (IFT) E in M and find a neighbourhood O , of a [ ℓ ] with O , ⊆ π ( O )and a neighbourhood O ,ℓ + i of ( a ℓ + i , u ℓ + i ) with O ,ℓ + i ⊆ π i ( O ) and definable functions h i, , . . . , h i,n i from O , to O ,ℓ + i such that(11) n − ℓ ^ i =1 h i, ( a [ ℓ ] ) = a ℓ + i ∧ n i ^ j =1 h i,j ( a [ ℓ ] ) = u ℓ + i,j ∧ ∀ ¯ y ∈ O , ( n − ℓ ^ i =1 H i ( h i, (¯ y ) , · · · , h i,n i (¯ y ))) . Let ¯ h i := ( h i, (¯ y ) , · · · , h i,n i (¯ y )) with ¯ y = ( y , . . . , y ℓ ). Let t [ ℓ ] := ( t , . . . , t ℓ ). Applying¯ h i to ( a [ ℓ ] + t [ ℓ ] ), we get a solution to each system H i ( x ℓ + i , z ℓ + i ), close to ( a ℓ + i , u ℓ + i ),1 ≤ i ≤ n − ℓ . Denote this solution by ( a ′ ℓ + i , u ′ ℓ + i ), 1 ≤ i ≤ n − ℓ . Let( e a , e u ) := ( a [ ℓ ] + t [ ℓ ] , a ′ ℓ +1 , . . . , a ′ n , u ′ ℓ +1 , . . . , u ′ n ) . Since ( e a , e u ) belongs to O ( M ), we may apply the functions g , . . . , g ℓ in order to obtain( g ( e a , e u ) , . . . , g ℓ ( e a , e u )) ∈ V ( ¯ f e a , e u ). Set (˜ b , . . . , ˜ b ℓ ) := ( g ( e a , e u ) , . . . , g ℓ ( e a , e u )).Since now a + t , · · · , a ℓ + t ℓ are ecl K -independent, we may define(12) δ ( a + t ) := ˜ b , . . . , δ ( a ℓ + t ℓ ) = ˜ b ℓ . Note that the values of the successive derivatives of ˜ b , · · · , ˜ b ℓ are determined since we canexpress δ (˜ b ) , . . . , δ (˜ b ℓ ) using that (˜ b , · · · , ˜ b ℓ ) is a regular zero of V ( ¯ f e a , e u ). By equation(11), a ′ ℓ +1 , . . . , a ′ n ∈ ecl M (¯ c, a + t , · · · a ℓ + t ℓ ), we can also express their derivatives in IFFERENTIAL EXPONENTIAL TOPOLOGICAL FIELDS 21 terms of a + t , · · · , a ℓ + t ℓ , a ′ ℓ +1 , . . . , a ′ n , the witnesses u ′ ℓ +1 , . . . , u ′ n and the derivatives of a + t , · · · , a ℓ + t ℓ , namely ˜ b , · · · , ˜ b ℓ . So first we extend δ on ecl M ( K, a + t , · · · , a ℓ + t ℓ ) sending the tuple a [ ℓ ] + t [ ℓ ] to (˜ b , . . . , ˜ b ℓ ) and then by Corollary 2.19 to M . Thisextension is uniquely determined on the subfield of M generated by K , a + t , . . . , a ℓ + t ℓ , a ′ ℓ +1 , . . . , a ′ n , u ′ ℓ +1 , . . . , u ′ n and ˜ b , . . . , ˜ b ℓ .Now assume that either | ¯ f | < ℓ or that a ℓ ] is not a regular zero of V ( ¯ f a , ¯ u ). Assume that | w | = ℓ and let, with x = ( x , . . . , x n ), w = ( w , . . . , w ℓ ), S ( a ℓ ] , a , ¯ u ) := { ( w , x , ¯ z ) : w ∈ V ( ¯ f x , ¯ z ) ∩ ( a ℓ ] , a , ¯ u ) + ¯ χ ( M, ¯ d ) & n − ℓ ^ i =1 H i ( x ℓ + i , z ℓ + i ) } . The set S ( a ℓ ] , a , ¯ u ) is non-empty since it contains ( a ℓ ] , a , ¯ u ). Let R ℓ + N be a Noetheriansubring of K [ w , x , ¯ z ] E closed under partial derivation and containing the E -polynomials¯ f ( w , x , ¯ z ). For ( w , x , ¯ z ) ∈ S ( a ℓ ] , a , ¯ u ) , let I ( w , x , ¯ z ) := Ann R ℓ + N ( w , x , ¯ z ), namely I ( w , x , ¯ z ) = { q ∈ R ℓ + N : q ( w , x , ¯ z ) = 0 } . Since R ℓ + N is Noetherian, there is ( r , s , s ) ∈ S ( a ℓ ] , a , ¯ u ) with | r | = ℓ , | s | = n and | ( s , s ) | = N such that I ( r , s , s ) is maximal. Since R ℓ + N isNoetherian, we can find h ( w , x , ¯ z ) , · · · , h p ( w , x , ¯ z ) ∈ R ℓ + N with p maximal ( † ) such that V pi =1 h i ( r , s , s ) = 0 and ∇ h s , s ) ( r ) , . . . , ∇ h p ( s , s ) ( r ) are K -linearly independent.Note that p ≥ E -polynomials whose all partial derivatives are equal to 0 is itself0 and the map sending an E -polynomial to the corresponding function is injective in thiscase.If p = ℓ , we proceed as above, replacing the tuple ¯ f by the tuple ¯ h and we use Proposition3.19.If p < ℓ , first note since for ( r , s , s ) ∈ S ( a ℓ ] , a , ¯ u ) , r ∈ V ( ¯ f s , s ). Then let 1 ≤ i < · · · < i p ≤ ℓ be strictly increasing indices such that the determinant of the matrix( ∇ ¯ i h s , s ) ( r ) , . . . , ∇ ¯ i h p ( s , s ) ( r )) is nonzero, with ¯ i := ( i , · · · , i p ). Decompose r into twosubtuples: r [ p ] and r [ ℓ − p ] (see Notation 3.4). We will add r [ ℓ − p ] to the parameters ( s , s )and apply the implicit function theorem (IFT) E to the corresponding square system. So wecan find in a neighbourhood of r [ p ] a point satisfying that system with coefficients close to( r [ ℓ − p ] , s , s ) and still get that this point belongs to V ¯ f , using Proposition 3.19 (b), sincewe assumed p maximal ( † ). (This is where we use the hypothesis (LFF) E .)Assume now that m >
1. Then we replace in the above discussion a ℓ ] by a m [ ℓ m − ] and weproceed as before. (cid:3) We will need later the following corollary which can be easily deduced from the proof ofthe above lemma.
Corollary 4.5.
Let
M ⊆ K | = T , assume that M δ | = T δ and that K is | M | + -saturated.Suppose K satisfies (IFT) E and (LFF) E in the ordered case and (IFT) anE in the valued case.Let ϕ ( x ) be a finite conjunction of L δ ( M ) -equations of order m , let H ( x , . . . , x m − , ¯ z ) bea Khovanskii formula with | a i | = n , ≤ i ≤ m . Assume that for some a := ( a . . . , a m ) ∈ K , the formula ∃ ¯ z ( H ( a , . . . , a m − , ¯ z ) ∧ ϕ ∗ H ( a , . . . , a m , ¯ z )) , holds in K . Then given d ∈ K , we can find α ∈ K and we can extend δ on ecl K ( M, α ) such that ϕ ( α ) holds and ¯ χ (¯ δ m ( α ) − a , d ) . (cid:3) ( † ) AND NATHALIE REGNAULT
Theorem 4.6.
Let T be a model-complete theory of topological L -fields. Let K | = T andassume that K satisfies (IFT) E and (LFF) E in the ordered case and IFT anE in the valuedcase. Then the differential expansion K δ can be embedded in a model ˜ K δ of T δ ∪ (DL) E . Proof: We adapt [13, Lemma 3.7] and [13, Proposition 3.9] to this exponential setting.The differential extension ˜ K δ will be built as the union of a chain of differential extensionsof K δ which will be in addition L -elementary extensions of K . In particular, we get that˜ K is an L -elementary extension of K . We first construct such extension ˜ K δ where all theinstances of the scheme (DL) E with coefficients in K are satisfied using transfinite inductionand then we repeat the construction replacing in the previous argument K δ by ˜ K δ and wedo it ω times. The union of this chain of extensions will be a model of the scheme (DL) E and an elementary extension of K (since T is model-complete).It suffices to show that given an instance of the scheme (DL) E , we can find an L δ extension K of K δ where it is satisfied, with K (cid:22) K .Let x = ( x , . . . , x n ), let ϕ ( x ) be an L δ ( K )-formula which is a conjunction of L δ ( K )-equations of order m , and let H ( x , . . . , x m , ¯ z ) be a Khovanskii L -formula with x = x . Let¯ χ ( K, ¯ d ) be a definable neighbourhood of 0 (in K n ( m +1) ) with ¯ d ∈ K . Let a = ( a , . . . , a n ) ∈ K and ¯ a = ( a , . . . , a m ) ∈ K , where a := a be such that ∃ ¯ zH ( a , . . . , a m − , ¯ z ) ∧ ϕ ∗ H (¯ a, ¯ z ) , holds in K .In Lemma 4.4, we constructed a differential extension K of K containing an element α such that ϕ ( α ) holds and such that ¯ δ m ( α ) is close to ¯ a , with respect to a given neighbour-hood ¯ χ ( · , ¯ d ) of 0. (cid:3) Recall that L is a first-order language satisfying the assumptions of section 3.4. Theorem 4.7.
Let T be a model-complete theory of topological L -fields. Assume that K | = T and that the differential expansion K δ is a model of T δ ∪ (DL) E . Then K δ isexistentially closed in the class of models of T δ . In particular if the theory T δ ∪ (DL) E isconsistent, then it is model-complete. Proof: Let K δ | = T δ ∪ (DL) E and suppose that K δ ⊆ ˜ K δ with ˜ K δ | = T δ .Let x = ( x , . . . , x n ) and ξ ( x ) be a quantifier-free L δ ( K )-formula of order m and assumethat for some tuple a ∈ ˜ K , ˜ K | = ξ ( a ). Since T is model-complete and K | = T , we mayassume that we are in the case where m ≥
1. Furthermore we may assume that ξ ( x ) is ofthe form ϕ ( x ) ∧ ¯ δ m ( x ) ∈ O , where ϕ ( x ) is a conjunction of L δ ( K )-equations and O is an L ( K )-definable open subset of some cartesian product of ˜ K .If the formula is of the form ¯ δ m ( x ) ∈ O , then we may conclude using the density ofdifferential points (see Remark 4.2). So, from now on, assume that there is a non-trivial L δ ( K )-equation occurring in ϕ ( x ).We consider all the ecl K -relations that may occur within the tuple ¯ δ m − ( a ). Set a i := δ i ( a ), 0 ≤ i ≤ m . If all a i , 0 ≤ i ≤ m −
1, are ecl K -independent, then by the scheme (DL E ),we can find a differential solution in K close to a . So from now on let us assume this isnot the case. Let a ℓ ] = ( a , . . . , a ℓ ) be the longest sub-tuple of a = ( a , . . . , a n ) which isecl-independent over K (which we may assume by re-indexing to be an initial subtuple sinceecl has the exchange property). (If there is no such ℓ , then a , . . . , a n ∈ ecl ˜ K ( K ) and theirsuccessive derivatives can be expressed in terms of a i , ¯ u i for some tuples of elements of ˜ K ,1 ≤ i ≤ n , and elements from K . So we can transform the L δ -formula ϕ into an L -formula IFFERENTIAL EXPONENTIAL TOPOLOGICAL FIELDS 23 and use the fact that T is model-complete.) Then we consider the ecl-relations among a over K and a ℓ ] . Again we re-index in order that they do not occur among elements of theinitial subtuple of a and we rename the corresponding subtuple e a and possibly permutethe indices of e a to match the indices. Suppose we got e u i , 0 ≤ i < m −
1. We considerthe ecl-relations among a i +1 over K and e a , . . . e a i . Again we re-index in order that they donot occur among elements of an initial subtuple of a i +1 and we rename the correspondingsubtuple g a i +1 as well as possibly permuting the indices of e a , . . . , e a i to match the indices.We proceed in this way getting successively e a , . . . , ] a m − . Assume the length of e a i is equalto ℓ i , 0 ≤ i ≤ m − n ≥ ℓ = ℓ ≥ ℓ ≥ . . . ≥ ℓ m − ≥ m = 1. Let H ( a ℓ +1 , ¯ u ℓ +1 ) , · · · , H n − ℓ ( a n , ¯ u n )be n − ℓ Khovanskii systems over K ( a , · · · , a ℓ ), witnessing that a ℓ +1 , · · · , a n belong toecl L ( K ( a , · · · , a ℓ )).Note that by Lemma 2.15 (and its proof), this implies that we can express δ ( a ℓ + i ) , δ (¯ u ℓ + i )in terms of a, ¯ u ℓ + i , δ ( a ) , · · · , δ ( a ℓ ), 1 ≤ i ≤ n − ℓ and finitely many elements of K andtheir derivative occurring as coefficients of the E -polynomials appearing in the Khovanskiisystems. Let ¯ u := (¯ u ℓ +1 , · · · , ¯ u n ).Let ϕ ∗ H be the L -formula constructed from ϕ and these Khovanskii systems (see Definition4.1).Since T is model-complete, there exists ¯ γ ∈ O ( K ) and ¯ z ∈ K such that ϕ ∗ H (¯ γ, ¯ z ) holds.Then we apply the scheme (DL) E and get a differential solution ¯ δ m ( α ) ∈ K satisfying ϕ ∗ H and close to ¯ γ . So K δ | = ξ ( α ). (cid:3) Geometric version of the scheme (DL) E . In this section we translate in geometricterms the scheme (DL) E . It is similar in spirit to the differential lifting scheme introducedby Pierce and Pillay, which gave another axiomatization of the class of differentially closedfields of characteristic 0 [22].For n ≤ m ∈ N ∗ , let π mn : K m → K n be the projection onto the first n coordinates andlet π m ( n,n ) : K m × K m → K n × K n : ( x, y ) ( π mn ( x ) , π mn ( y )). Definition 4.8.
Let K δ | = T δ , then K δ satisfies the scheme (DLg) E if the following holds.Let ˜ K be a | K | + -saturated L -elementary extension of K . Let W := W ( ¯ f ) ⊆ ˜ K n be an E -variety defined over K and let ¯ χ ( K, d ) be a neighbourhood of in K n with d in K .Suppose that ≤ dim ˜ K ( π nn ( W ) /K ) = ℓ < n . Let a be a generic point of π nn ( W ) with a [ ℓ ] a subtuple of a of ecl -independent elements over K and let ( a , b ) be a generic pointof W . Let u i be tuples of elements in ˜ K , ≤ i ≤ n − ℓ , witnessing that each componentof a [ n − ℓ ] belongs to ecl( K, a [ ℓ ] ) . Set ¯ u := ( u ℓ +1 , . . . , u n ) ∈ ˜ K m and assume that ( a , b ) ∈ π n + m )( n,n ) ( τ ( Ann K [ X ] E ( a , ¯ u )) , | X | = m + n , then we can find a differential point ( α, δ ( α )) ∈ W ∩ K n with ¯ χ (( α, δ ( α )) − ( a , b ) , d ) . Similarly to the proof of Lemma 4.4, one can show that if the theory of K is model-complete and if K δ satisfies ( IF T A ) E together with ( LF F ) E in the ordered case and(LFF) anE in the valued case, then K δ has an elementary extension which can be endowedwith a derivation extending δ and which satisfies the scheme (DLg) E . The trivial case whendim( W ) = 0 is handled as before by Lemma 3.20. ( † ) AND NATHALIE REGNAULT
Then one shows that if K δ satisfies the scheme (DLg) E and if its L -theory is model-complete, then it is existentially closed in the class of models of T δ . The proof is similarto the proof of Theorem 4.7. Indeed, let M δ | = T δ extending K δ . Let ¯ f ( β, δ ( β )) = 0 be asystem of E -polynomials satisfied by ( β, δ ( β )) and let W be the corresponding E -variety(defined over K ). Extract from β a maximal subtuple of ecl-independent elements over K and let ¯ u be a tuple of witnesses in M . Then ( β, δ ( β )) ∈ τ ( Ann K [ X ] E ( β, ¯ u )).Then we may apply the scheme (DLg) E .The scheme (DLg) E as stated is not first-order. The first issue concerns expressing that atuple is generic and the second is that a priori we have to consider all the E -polynomials inan annihilator. Concerning the second one, keeping the same notations as in Definition 4.8,one only needs the E -polynomials in Ann K [ X ] E ( a , ¯ u ) occurring in the Khovanskii systemsused to express that each component of a [ n − ℓ ] belongs to ecl( K, a [ ℓ ] ).5. Model-complete theories of (partial) exponential fields
In this section, we apply our previous results to theories of topological fields K wherethe topology is either induced by an ordering < or by a valuation map v . In the case ofvalued field K := ( K, v ) we will replace the valuation map by a binary relation div definedas follows: v ( a ) ≤ v ( b ) iff a div b. Denote by O K be the valuation ring of K and M K the maximal ideal of O K . Let D be a binary function symbol for division in the valuation ring O K , defined as follows: D ( x, y ) := (cid:26) xy if v ( x ) ≥ v ( y ) and y = 0 , The real numbers.
A. Wilkie showed that the theory of ( ¯ R , exp ) where ¯ R is theordered field of real numbers is model-complete [28, Second MainTheorem]. So setting T := T h ( ¯ R , exp ), Theorem 4.3 holds for models of T since they also satisfy (IFT) E and(LFF) E .5.2. The p -adic numbers. Let ( Q p , v ) be the valued field of p -adic numbers. A. Macintyreshowed that the theory of Q p admits quantifier elimination in the language of fields togetherwith the binary relation symbol div and for each n ≥
2, the predicates P n defined by P n ( x ) iff ∃ y y n = x .Then J. Denef and L. van den Dries showed that the theory of the valuation ring Z p of Q p (or the theory of Q p ) enriched by all restricted power series with coefficients in Z p together with the predicates P n , n ≥ D : Z p → Z p for division in Z p , admits quantifier elimination [5, Theorem (1.1)]. N. Mariaule showed that the theory ofthe valuation ring Z p of Q p expanded by the exponential function E p ( x ) (see Examples 2.1(5)) together with for each n ≥ E p ( x ) is model-complete [18, Theorem 4.4.5]. We will recall below precisely what are these decompositionfunctions.From that one can easily deduce that the theory of the partial exponential valued field( Q p , E p ) is model-complete in the language of fields together with the predicates P n , n ≥ E p ( x ) and the decomposition functions.(Note that N. Mariaule proves strong model-completeness [7, section 2 (2.2)]). So againTheorem 4.3 holds for T = T h ( Q p , E p ). IFFERENTIAL EXPONENTIAL TOPOLOGICAL FIELDS 25
Now let us recall what are these decomposition functions. They are the analog of thefunctions sin and cos in the real case, but their definition is more complicated since Q p hasinfinitely many proper algebraic extensions.The field Q p is bounded, namely for each fixed d ≥ d . So one may define a chain of finite algebraic extensions K n of Q p with the following properties:(1) K n contains any extension of degree n of Q p ,(2) K n is the splitting field of an irreducible polynomial q n ∈ Q [ X ] of degree N n .One may further assume that q n ∈ Z p [ X ]. Let β n be the root of q n ; so K n = Q p ( β n ) and V n := O K n = Z p ( β n ). Then V n is a Z p -module with basis 1 , β n , · · · , β N n − n . Let y ∈ V n andwrite it as P N n − i =0 x i β in . Then E p ( y ) = Q N n − i =0 E p ( x i β in ), with x i ∈ Z p and one adds thedecomposition functions for each E p ( xβ in ), namely functions from Z p to Z p which allowsto express E p ( xβ in ) in V n . Namely, write E p ( xβ in ) = P N n − j =0 ˜ c i,j,n ( x ) β in . Conversely, onecan express ˜ c i,j,n ( x ) in terms of E p ( x ) on O K n , β σn where σ varies in Gal ( K n / Q p ) and theinverse of the determinant of the Vandermonde matrix V associated to the roots of q n .Finally the decomposition functions c i,j,n ( x ) are obtained by multiplying the ˜ c i,j,n ( x ) bysome coefficient (the norm N K n / Q p ( det ( V )) [18, page 66]. (The issue is that det ( V ) mightbe of strictly positive valuation). Let L pEC be the language L E together with the thepredicates P n , n >
1, and the decomposition functions c i,j,n , 0 ≤ j ≤ N n , i, n ∈ N ∗ . Thenthe L pEC -theory T of ( Q p , E p ) is model-complete [18, Theorem 4.4.5]. Since Q p satisfiesthe analytic version of the implicit function theorem, we may apply Theorem 4.3.5.3. The completion of the algebraic closure of the p -adic numbers. Let C p be thecompletion of the algebraic closure of the field Q p of p-adic numbers. As a valued field, C p is a model of the theory ACVF ,p of algebraically closed valued fields of characteristic 0 andresidue characteristic p . It admits quantifier elimination in the language { + , − , · , , , div } [ ? ]. (Note that A. Robinson only proved model-completeness of the theory but the quantifierelimination result is easily deduced.) N. Mariaule showed that the theory of the valuationring O p of C p endowed with the exponential function E p ( x ) is model-complete [18, Theorem6.2.11]. From that one can easily deduce that the theory T of the partial exponentialvalued field ( C p , div , E p ) is model-complete. Since C p also satisfies the analytic versionof the implicit function theorem, we may apply Theorem 4.3. (Note that in this casesince C p is algebraically closed, one does not need to add additional functions such as thedecomposition functions).5.4. Non-standard extensions of Q p . Let (
K, v ) be a valued field extending ( Q p , v ).Let O K be the valuation ring of K and let O K h ξ i be the ring of strictly convergent powerseries over O K in ξ := ( ξ , · · · , ξ m ). An element f ( ξ ) is given by P ν ∈ N n a ν ξ ν , where ξ ν = ξ ν · · · ξ ν n n and v ( a ν ) + ∞ , when | ν | = ν + · · · + ν n + ∞ . Such f defines afunction from O K n to O K defined by f ( u ) = (cid:26) P ν ∈ N n a ν u ν for u ∈ O nK , L an is the language of rings augmented by a n -ary function symbol foreach f ∈ O K h ξ i and n ≥
1. Let D be a binary function symbol for division restrictedto the valuation ring as defined above. Let L an, div := L an ∪ { div } ∪ { P n : n ≥ } . and L Dan, div := L an, div ∪ { D } . Let K denote the L an, div -structure with domain K and the aboveinterpretation of the symbols of the language. In view of the way the functions f areinterpreted in both Q p and K , we have that Q p is an L an, div -substructure of K . Then usingthe quantifier elimination theorem of J. Denef and L. van den Dries, that if K is a model ( † ) AND NATHALIE REGNAULT of T h L an, div ( Q p ), then K is an elementary L an, div -extension of Q p . Now if we restrict thelanguage L an, div to the language L pEC , we get that the theory T of K in this restrictedlanguage is also model-complete (and in fact equal to the theory of ( Q p , E p ). In order toapply Theorem 4.3 to K δ , we need to check that K satisfies IFT anE . A way to do this is to geta universal axiomatisation of T h L an, div ( Q p ). It will imply that any definable function from O nK to O K is piecewise given by L an, div -terms and so analytic functions. (This argumentwas used for R an in [9].)We express that K ∗ / ( K ∗ ) n ∼ = Q ∗ p / ( Q ∗ p ) n and that cosets representative of the subgroup of n th powers can be found in N , namely for every x ∈ K ∗ there exist λ, r ∈ N with 0 ≤ r < n ,0 ≤ λ < p β ( n ) and β ( n ) = 2 v ( n ) + 1 and P n ( xλp r ) [1, Lemma 4.2]. This can be expressedby a finite disjunction and translates the fact that v ( K ∗ ) is a Z -groupThen we express that K is henselian in the following way. Let p ( X ) ∈ O K [ X ] be anordinary polynomial of degree n . Then one defines a function h n : O n +1 K → O K sending( a , . . . , a n , b ) u with a n X n + . . . + a X + a = 0, v ( p ( b )) > v ( ∂ X p ( b )) = 0 and v ( u − b ) > T to which we may apply Theorem 4.3.6. Construction of models of the scheme (DL) E In this section we will place ourselves in the same setting as in section 4.1. We showhow to endow certain exponential topological fields K satisfying (IFT) E and (LFF) E in theordered case and IFT anE in the valued case, with a derivation in such a way they become amodel of the scheme (DL) E . One can follow a similar strategy as in [3], [24] to endow certain(ordered) fields with a derivation in such a way they become a model of the scheme (DL)introduced in [13], generalizing for certain differential topological fields the axiomatizationCODF of closed ordered differential fields given by M. Singer in [27]. Proposition 6.1.
Let K be a topological L -field satisfying (IFT) E and (LFF) E in the or-dered case and IFT anE in the valued case. Assume that K is of cardinality ℵ with a countabledense subfield. Then we can endow K with a derivation δ such that K δ is a model of thescheme (DL) E . Proof: Denote by K a countable dense subfield of K . By Lowenheim-Skolem theorem,we may assume that K is an elementary substructure of K . Let ¯ d i ∈ K , i ∈ ω , be suchthat W i := χ ( K, ¯ d i ) is a basis of neighbourhoods of 0 in K and chosen such that it forms astrictly decreasing sequence of neighbourhoods with in addition W i + W i ⊆ W i +1 . Express K as K ( B ) with B a subset of elements of K which are ecl-independent over K (so | B | = ℵ ). Set B := ( t α ) α< ℵ . Claim 6.2.
Let ( W i ) i ∈ ω be a strictly decreasing sequence of neighbourhoods of in K .Then for each W i , i ∈ ω , and each ℓ ∈ ω , there are elements s , · · · , s ℓ ∈ W i that are ecl -independent over K and with the property that s j − t j ∈ K , ≤ j ≤ ℓ .Proof of Claim: Fix W i a neighbourhood of 0 in K and choose t , . . . , t ℓ ∈ B , ℓ ∈ ω . Since K is densein K , there are for each 0 ≤ j ≤ ℓ , r ji ∈ K such that t j − r j ∈ W i . Set s j := t j − r ji ,0 ≤ j ≤ ℓ . The elements s , · · · , s ℓ ∈ K , are ecl-independent over K and belong to W i . (cid:3) We will express K as the union of an elementary chain of countable subfields K α endowedwith a derivation δ α , α < ℵ , starting by putting on K the trivial derivation δ . IFFERENTIAL EXPONENTIAL TOPOLOGICAL FIELDS 27
The subfields K α have the following property. Given a neighbourhood of zero W i and any quantifier-free L δ ( K α )-formula ϕ ( x ) of order m and any Khovanskii system H (with parameters in K α ) and associated L -formula ϕ ∗ H (see Definition 4.1) such that H ( a , . . . , a m − , ¯ b ) ∧ ϕ ∗ H (¯ a, ¯ b ) holds in K α with ¯ a := ( a , . . . , a m ) and ¯ b ∈ K α , we canfind β ∈ K α +1 such that ϕ ( β ) holds and ¯ δ mα +1 ( β ) − ¯ a ∈ W i .By induction on α , assume we have constructed K ⊆ K α (cid:22) K a countable elementarysubstructure of K and suppose K α is endowed with a derivation δ α . Let ¯ x := ( x , . . . , x m ), x := x , | x | = n and, keeping the notations of Definition 4.1, set F α := {∃ ¯ zH ( x , . . . , x m − , ¯ z ) ∧ ϕ ∗ H (¯ x, ¯ z ) : K | = ∃ ¯ x ∃ ¯ z ( H ( x , . . . , x m − , ¯ z ) ∧ ϕ ∗ H (¯ x, ¯ z ))with ϕ varying over all the L δ ( K α ) − formulas of order m ≥ } . We will construct a differential extension K α +1 of K α containing t α , satisfying the scheme(DL) E relative to F α .Let ϕ ( x ) be an L δ ( K α )-formula of order m ≥ ∃ ¯ zH ( x , . . . , x m − , ¯ z ) ∧ ϕ ∗ H (¯ x, ¯ z ) ∈ F α with | x | = n . Assume H is of the form V n − ℓi =1 H ( u ℓ + i , ¯ z i ), with ¯ z := (¯ z , . . . , ¯ z n − ℓ ). Let t , · · · , t ℓ ∈ B . Let u := ( u , · · · , u n ) , ¯ b ∈ K be such that ϕ ∗ H (¯ u, ¯ b ) holds, where ¯ u :=( u , . . . , u m ). Then by Claim 6.2 and Corollary 4.5, there is an elementary extension of K α inside K , an element β ∈ K and a derivation ˜ δ α extending δ on K α and β such that ϕ ( β )holds and ˜ δ mα ( β ) − ¯ u ∈ W i . Furthermore we may assume that this extension is countableand ecl K -closed.We consider ecl( K α (˜ δ mα ( β )). In case t α does not belong to this subfield and we define˜ δ α ( t α ) = 1. Then let K α, = ecl( K α ( t α , ˜ δ mα ( β )). We enumerate F α and the extension K α,i corresponds to where the i th formula in F α has a differential solution close to thealgebraic one in some fixed neighbourhood W i of zero. Set K (1) α := S i K α,i . Then weredo the construction with K (1) α in place of K α with a smaller neighbourhood of zero. Set K α +1 := S m K ( m ) α . Note that K α +1 is countable.So we described what happens at successor ordinals and at limit ordinals we simplytake the union of the subfields we have constructed so far. Finally we express K as theunion of a chain of differential subfields and given any L δ -formula ϕ ( x ) of order m ≥ H ( x , . . . , x m − , ¯ z ) and an associated L -formula ϕ ∗ H such that for some¯ u = ( u , . . . , u m ) , ¯ b ∈ K with u := ( u , · · · , u n ) ϕ ∗ H (¯ u, ¯ b ) holds in K , we find an element ofthe chain K α such that ϕ ∈ F α and ¯ u, ¯ b ∈ K α . Therefore given a neighbourhood of zero W i , we have β ∈ K α +1 such that ϕ ( β ) holds and ¯ δ m ( β ) − ¯ u ∈ W i . (cid:3) Denote by L − the language L where we take off the exponential function and denote by T − the theory of the L − -reducts of the models of T . Let us assume that T − admits quantifierelimination. Then in [13], we showed that the class of existentially closed models of T − ,δ was elementary, assuming that the models of T satisfied Hypothesis (I). That last propertyis an analog for topological fields of the property of being large, property introduced byF. Pop [23]). Let us first recall the following notation. Given a differential polynomial p ( X ) ∈ K { X } of order m >
0, with | X | = 1, the separant s p of p is defined as s p := ∂∂δ m ( x ) p ∈ K { X } . Definition 6.3. [13, Definition 3.5]
The scheme of axioms (DL) is the following: given amodel K of T − ,δ , K satisfies (DL) if for every differential polynomial p ( X ) ∈ K { X } with ( † ) AND NATHALIE REGNAULT | X | = 1 and ord X ( p ) = m > , for variables y = ( y , . . . , y m ) it holds in K that ∀ z (cid:0) ( ∃ y ( p ∗ ( y ) = 0 ∧ s ∗ p ( y ) = 0) → ∃ x (cid:0) p ( x ) = 0 ∧ s p ( x ) = 0 ∧ χ τ (¯ δ m ( x ) − y , z ) (cid:1)(cid:1) . By quantifying over coefficients, the axiom scheme (DL) can be expressed in the language L − ,δ . Corollary 6.4.
Let K be a topological L -field satisfying (IFT) E and (LFF) E in the orderedcase and IFT anE in the valued case. Assume that K is of cardinality ℵ with a countabledense subfield. Then we can endow K with a derivation δ such that K δ is a model of theschemes (DL) E ∪ (DL) .Proof. We modify the proof of proposition above by also considering the instances of thescheme (DL) and alternating between solving a formula from scheme (DL) E to solving aformula from scheme (DL). We observe that if t , . . . , t n are ecl-independent, then they arealso algebraically independent by Corollary 3.8. (cid:3) Corollary 6.5.
Let K be an ordered real-closed exponential field. Assume that K is ofcardinality ℵ with a countable dense subfield. Then we can endow K with a derivation δ such that K δ is a model of CODF together with the scheme (DL) E . (cid:3) Remark 6.6.
Now let us examine a few cases when the field K has a dense countablesubfield.First, suppose that ( K, v ) is an henselian perfect valued field with value group G and ofequicharacteristic. Denote by { t g ∈ K : g ∈ G & v ( t g ) = g } , a family of elements of K whose set of values is G . Then by a result of Kaplansky, the residue field k isomorphicallyembed in the valuation ring of K [11, Lemma 3.8]. Let k be a dense subfield of k . Thenthe subring of K generated by k and { t g : g ∈ G > } is dense in the valuation ring of K .So in case | k | = ℵ and | G | = ℵ , then we do have a dense countable subfield.Second, suppose now that ( K, < ) is an ordered field. Either K is archimedean and so itembeds into R (and so Q is dense in K ), or the archimedean valuation on K is non trivialand so the residue field with respect to this archimedean valuation embeds in R (and Q is a dense subfield of the residue field). In case the value group G is countable, we havea countable dense subring in the valuation ring, namely the subring generated by Q and { t g : g ∈ G ≥ } . References [1] B´elair L., Le th´eor`eme de Macintyre, un th´eor`eme de Chevallet p -adique, Ann. Sc. Math. Qu´ebec,14 (2), 1990, 109-120.[2] Bourbaki N., Vari´et´es diff´erentielles et analytiques, Fascicule de r´esultats, Springer Berlin Heidel-berg, 2007.[3] Brouette Q., Differential algebra, ordered fields and model theory, PhD thesis, UMons (Belgium),september 2015.[4] Cluckers R., Lipshitz L., Strictly convergent analytic structures, J. Eur. Math. Soc. 19, 107-149,2017.[5] Denef, J., van den Dries, L., p-adic and real subanalytic sets, Ann. of Math. 128(1988), 79-138.[6] van den Dries, L., Exponential rings, exponential polynomials and exponential functions, PacificJournal of Mathematics, vol. 113, No 1, 1984, 51-66.[7] van den Dries, L., On the elementary theory of restricted elementary functions, J. Symbolic Logic53 (1988), no. 3, 796–808.[8] van den Dries, L., Tame topology and o-minimal structures. London Mathematical Society LectureNote Series, 248. Cambridge University Press, Cambridge, 1998.[9] van den Dries, L., Macintyre, A., Marker, D.,The elementary theory of restricted analytic fieldswith exponentiation, Ann. of Math. (2) 140 (1994), no. 1, 183-205. IFFERENTIAL EXPONENTIAL TOPOLOGICAL FIELDS 29 [10] Fuchs, L. Partially ordered algebraic systems, Pergamon Press, 1963.[11] Fornasiero A., Embedding Henselian fields into power series, Journal of Algebra 304 (2006) 112-156.[12] Fornasiero A., Kaplan E., Generic derivations on o-minimal structures, ArXiv:1905298v1, 17 May2019.[13] Guzy N., Point F., Topological differential fields, Ann. Pure Appl. Logic 161 (2010), no. 4, 570-598.[14] Jones G., Wilkie A., Locally polynomially bounded structures, Bull. London Math. Soc. 40 (2008)239-248.[15] Kirby J., Exponential algebraicity in exponential fields, Bull. London Math. Soc. 42 (2010) 879-890.[16] Kolchin E., Differential algebra and algebraic groups. Pure and Applied Mathematics, Vol. 54.Academic Press, New York-London, 1973.[17] Macintyre A., Exponential algebra in
Logic and algebra (Pontignano, 1994) , 191–210, LectureNotes in Pure and Appl. Math., 180, Dekker, New York, 1996.[18] Mariaule N.,
On the decidability of the p -adic exponential ring , PhD thesis, The University ofManchester (UK), 2013.[19] Mariaule N., Effective model-completeness for p -adic analytic structures, arXiv:1408.0610.[20] Marker D., Model Theory: An Introduction, Graduate Texts in Mathematics 217, 2002, Springer-Verlag New-York, Inc.[21] Pillay A., First order topological structures and theories. J. Symbolic Logic 52 (1987), no. 3,763–778.[22] Pierce D., Pillay A., A note on the axioms for differentially closed fields of characteristic zero, J.Algebra 204 (1998), no. 1, 108–115.[23] Pop, F., Henselian implies large, Ann. of Math. (2) 172 (2010), no. 3, 2183-2195.[24] Regnault N., On differential topological exponential fields, PhD thesis, UMons, september 2019.[25] Servi T., On the first-order theory of real exponentiation, PhD thesis, Scuola Normale Superioredi Pisa, 2006.[26] Servi T., Noetherian varieties in definably complete structures, Log. Anal. 1 (2008), no. 3-4, 187-204.[27] Singer M., The model theory of ordered differential fields. J. Symbolic Logic 43, no. 1, 82-91 (1978).[28] Wilkie A. J., Model completeness results for expansions of the ordered field of real numbers byrestricted Pfaffian functions and the exponential function, J. Amer. Math. Soc. 9 (1996), no. 4,1051-1094. Department of Mathematics (De Vinci), UMons, 20, place du Parc 7000 Mons, Belgium
E-mail address : [email protected] Department of Mathematics (De Vinci), UMons, 20, place du Parc 7000 Mons, Belgium
E-mail address ::