Coincidence of dimensions in closed ordered differential fields
Pantelis E. Eleftheriou, Omar Leon Sanchez, Nathalie Regnault
aa r X i v : . [ m a t h . L O ] O c t COINCIDENCE OF DIMENSIONS IN CLOSED ORDEREDDIFFERENTIAL FIELDS
PANTELIS E. ELEFTHERIOU, OMAR LE ´ON S ´ANCHEZ, AND NATHALIE REGNAULT
Abstract.
Let K = hR , δ i be a closed ordered differential field, in the senseof Singer [20], and C its field of constants. In this note, we prove that, forsets definable in the pair M = hR , C i , the δ -dimension from [5] and the largedimension from [11] coincide. As an application, we characterize the definablesets in K that are internal to C as those sets that are definable in M andhave δ -dimension 0. We further show that, for sets definable in K , having δ -dimension 0 does not generally imply co-analyzability in C (in contrast tothe case of transseries). We also point out that the coincidence of dimensionsalso holds in the context of differentially closed fields and in the context oftransseries. Introduction
Pairs of fields have been extensively studied in model theory and arise natu-rally in various ways. If K = hR , δ i is a differentially closed field of characteristiczero (DCF ), a closed ordered differential field (CODF) or the differential field oftransseries T constructed in [2], and C = ker( δ ) is the field of constants in eachcase, then the reduct M = hR , C i is a pair of algebraically closed fields ([14], [16]),a dense pair of real closed fields ([7], [17]) or a tame pair ([15], [10]), respectively. Inall three cases, there is a natural notion of dimension, the differential or δ -dimension for definable sets in K . While we postpone the definitions until §
2, we do recallthat in [5, Corollay 5.27] it is shown that, in the case of CODFs, the δ -dimensionon definable sets coincides with the one obtained from δ -cell decomposition.In the above three cases (DCF , CODF, and transseries), the following implica-tions hold for a set X definable in K : X is internal to C ⇒ X is co-analyzable in C ⇒ X has δ -dimension 0 . In the case of transseries, it is shown in [3] that the latter two properties areequivalent, and in [10] that, when restricted to definable sets in the pair M = hR , C i , all three properties are equivalent. In this note, we prove that in the caseof CODFs the latter two properties are different (see examples in § M all three properties are equivalent. Thelatter result is a consequence of Theorem 1.1 below, which states that in M , the Date : October 12, 2020.2010
Mathematics Subject Classification.
Key words and phrases.
Closed ordered differential fields, dense pairs of o-minimal structures,differential and large dimension.The first author was supported by a Research Grant from the German Research Foundation(DFG) and a Zukunftskolleg Research Fellowship. δ -dimension coincides with the large dimension from [11], which we briefly describenext (and provide further details in § h R , Q rc i ,where R is the real field and Q rc the subfield of real algebraic numbers. This pairwas studied by A. Robinson in his classical paper [17], where the decidability of itstheory was proven. A systematic study of dense pairs hB , Ai of o-minimal structures(that is, A is a dense proper elementary substructure of B ) occurred much later in[7] by van den Dries. In [11] (as well as [4] and [12]), the pregeometric dimensionlocalized at A was studied. Namely, letting A and B be the underlying sets of A and B , respectively, and dcl denote the usual definable closure in the o-minimalstructure B , we set, for D ⊆ B ,scl( D ) = dcl( D ∪ A ) . For X a definable set in hB , Ai , the scl-dimension of X is called large dimension and is denoted by ldim( X ). In [11], in a much broader setting that includes densepairs, the large dimension was given a topological description via a structure theo-rem, much alike the topological description of the usual dcl-dimension via the celldecomposition theorem in the o-minimal setting.The main result of this note is the following theorem. Theorem 1.1.
Let K = hR , δ i be a closed ordered differential field and C its fieldof constants. Let X be a set definable in the pair M = hR , C i . Then ldim( X ) = δ - dim( X ) . With a bit more work, we can characterize the notion of being internal to C forsets definable in K . Corollary 1.2.
Let K = hR , δ i be a closed ordered differential field and C its fieldof constants. Let X be a set definable in K . Then X is internal to C ⇔ δ - dim( X ) = 0 and X is definable in M = hR , C i . In Section 4, in contrast to the case of transseries [3, Proposition 6.2], we give sev-eral examples of sets definable in K with δ -dimension 0 which are not co-analyzablein C (and hence also not internal to C ). Finally, in Section 5, we prove the ana-logues of Theorem 1.1 in the realm of differentially closed fields of characteristiczero and in the realm of transseries. Acknowledgements.
We are grateful to Anand Pillay for several comments andsuggestions that led to a significant improvement of a previous draft.2.
Preliminaries
Throughout we fix a real closed field R = h R, <, + , ·i and a derivation δ on R .Also, for us, definability is always meant with parameters.2.1. Closed ordered differential fields.
The differential field K = hR , δ i is aCODF if it is existentially closed among ordered differential fields. The class ofCODFs is first-order axiomatisable in the language of ordered differential rings; forinstance, Singer’s axioms [20] state that K | = CODF if and only if
OINCIDENCE OF DIMENSIONS IN CLOSED ORDERED DIFFERENTIAL FIELDS 3 ( † ) for differential polynomials f, g , . . . , g m ∈ K { x } with ord ( f ) = n ≥ ord ( g i ),if there is ( a , a , . . . , a n ) ∈ R n +1 such that as algebraic polynomials f ( a , . . . , a n ) = 0 , ∂f∂δ n x ( a , . . . , a n ) = 0 , and g i ( a , . . . , a n ) > , then there is b ∈ K such that as differential polynomials f ( b ) = 0 and g ( b ) = 0 . The theory CODF admits quantifier elimination [20] and elimination of imag-inaries [18] (in the language of ordered differential rings). Furthermore, by [6, § K = hR , δ i and A ⊆ R the definable and algebraic closuredcl K ( A ) = acl K ( A ) equals the real closure of the differential field generated by A . Notation.
Until Section 5, we fix a CODF K = hR , δ i , with C its field of contants,and M = hR , C i . By [5], the reduct M is a dense pair of real closed fields.We now prove a preliminary result (Lemma 2.2) that will be used in §
3. Let N be any of K , M , R , and let X be an arbitrary (not necessarily definable) subsetof R m . We say that X is internal to C in N if there is n > f : R n → R m definable in N such that X ⊆ f ( C n ). In Lemma 2.2, we prove thatfor definable sets in K , internality to C in N is invariant under varying N among K , M and R . We will need the following result on definable functions in a CODF . Lemma 2.1. If f : R n → R is a definable function in K , then there is d ≥ anda function F : R n ( d +1) → R definable in R such that f ( a ) = F ( a, δa, . . . , δ d a ) for all a ∈ R n . Proof.
Since for any A ⊆ R the definable closure dcl K ( A ) equals the real closure ofthe differential field generated by A , it follows, by a standard compactness argu-ment, that there is a partition X , . . . , X r of R n into sets definable in K and, forsome d ≥
0, there are functions F i : R n ( d +1) → R definable in R , for i = 1 , . . . , r ,such that f ( a ) = F i ( a, δa, . . . , δ d a ) for all a ∈ X i . By quantifier elimination and after possibly increasing d , for each i , there is a set T i ⊆ M n ( d +1) definable in R such that X i = { a ∈ R n : ( a, δa, . . . , δ d a ) ∈ T i } . Now define F : R n ( d +1) → R as F ( b ) = F ( b ) if b ∈ T F ( b ) if b ∈ T \ T ... ... F r ( b ) if b ∈ T r \ ( T ∪ · · · ∪ T r − )0 otherwiseThis F is the desired function. (cid:3) Lemma 2.2.
Assume X ⊆ R m is definable in K . If X is internal to C in K , then X is definable in the pair M = hR , C i and internal to C in R . We thank Marcus Tressl for pointing out the argument in the proof.
PANTELIS E. ELEFTHERIOU, OMAR LE ´ON S ´ANCHEZ, AND NATHALIE REGNAULT
Proof.
Since X is internal to C in K , there is n > f : R n → R m definable in K such that X ⊆ f ( C n ). Since X is definable in K , after possiblymodifying f , we may assume X = f ( C n ). By Lemma 2.1, there is d ≥ F : R n ( d +1) → R definable in R such that f ( a ) = F ( a, δa, . . . , δ d a ) for all a ∈ R n . Hence, f ( a ) = F ( a, , . . . ,
0) for all a ∈ C n . This shows that X is defined by X = { y ∈ R m : ∃ x ∈ C n with y = F ( x, , . . . , } Thus, X is definable in M and internal to C in R . (cid:3) In view of Lemma 2.2, we say that a set definable in K is C -internal if it is soin any of K , M , R .We now discuss the notion of differential dimension (or δ -dimension). For A ⊆ R we let cl δ ( A ) denote the set of elements of R that are differentially algebraicover A ; namely, a ∈ cl δ ( A ) if and only if a is a solution of a nonzero differentialpolynomial in one variable with coefficients in the differential field generated by A .One can also work over a set of parameters B ⊆ R and set cl δB ( A ) = cl δ ( A ∪ B ).This closure operator defines a pregeometry (localised at B ) on R , called the cl δB -pregeometry of K . Hence, we can induce a cl δB -dimension on finite tuples; namely,given a = ( a , . . . , a n ) ∈ R n we set cl δB -dim( a ) to be the maximal cardinality of acl δB -independent subtuple of a . That is, cl δB -dim( a ) = k if and only if, after possiblyre-ordering the tuple, a i / ∈ cl δB ( a , . . . , a i − ) for 1 ≤ i ≤ k and a j ∈ cl δB ( a , . . . , a k )for k < j ≤ n . It follows that cl δB -dim( a ) equals the differential transcendencedegree over B of the differential field generated by a .For definable sets in K we define δ -dimension as follows. Definition 2.3.
Let K ∗ = hR ∗ , δ i be a | R | + -saturated elementary extension of K . For any nonempty set X definable in K , we set X ∗ = X ( R ∗ ) (that is, therealisations in K ∗ of any formula over R defining X in K ). The δ -dimension isdefined as δ - dim( X ) = max a ∈ X ∗ cl δR - dim( a ) , where cl δR -dim( a ) is taken with respect to the cl δR -pregeometry of K ∗ . We note thatthe definition is independent of the choice of the (suitably saturated) elementaryextension K ∗ of K . For matter of convenience one sets δ -dim( ∅ ) = −∞ .In [5, Corollay 5.27] it was shown that, for definable sets in K , this notion of δ -dimension coincides with the one obtained from δ -cell decomposition. Furthermore,in § δ -dimension is a dimension function in the sense of [8]. The following summarizes properties shared by all dimensionfunctions. Fact 2.4. [8, § Let X ⊆ R n and Y ⊆ R m be definable in K . (1) If X is a finite nonempty set, then δ - dim( X ) = 0 . (2) δ - dim( R n ) = n . (3) If n = m , then δ - dim( X ∪ Y ) = max { δ - dim( X ) , δ - dim( Y ) } . (4) δ - dim( X × Y ) = δ - dim( X ) + δ - dim( Y ) . (5) if δ - dim( X ) = k , then there is a coordinate projection π such that δ - dim π ( X ) = k . OINCIDENCE OF DIMENSIONS IN CLOSED ORDERED DIFFERENTIAL FIELDS 5 (6)
Let f : R n → R m be a function definable in K . Then δ - dim f ( X ) ≤ δ - dim( X ) . Furthermore, for ≤ i ≤ n , if we let X i = { r ∈ R m : δ - dim f − ( r ) = i } , then X i is definable in K and δ - dim X i + i = δ - dim f − ( X i ) . (7) δ -dim is completely determined by its effect on subsets of R definable in K .Remark . We note that the definition of dimension function only requires prop-erties (1), (2), (3), (6); and so properties (4), (5), (7) are consequences of these.This is all justified in § Notions from dense pairs.
In this subsection we recall the necessary ma-terial on large dimension from [11]. Recall that K = hR , δ i is a CODF, with C its field of constants, and M = hR , C i . Let dcl denote definable closure in theo-minimal structure R . For A ⊆ R , we setscl( A ) := dcl( A ∪ C ) . We can work over a parameter set B ⊆ R and letscl B ( A ) := scl( A ∪ B ) = dcl( A ∪ B ∪ C ) . This closure operator defines a pregeometry (localised at B ) on R , called the scl B -pregeometry of M . In the usual manner, this induces a scl B -dimension on finitetuples of R . For sets definable in M we can define a dimension, as follows: Definition 2.6.
Let M ∗ = hR ∗ , C ∗ i be a | R | + -saturated elementary extensionof M . For any nonempty set X definable in M , we set X ∗ = X ( R ∗ ) (i.e., therealisations in M ∗ of any formula over R defining X in M ). The large dimension ,denoted by ldim, is defined asldim( X ) = max a ∈ X ∗ scl R - dim( a ) , where scl R -dim( a ) is taken with respect to the scl R -pregeometry of M ∗ . This isindependent of the choice of the (suitably saturated) elementary extension M ∗ (seefor instance [1, § ∅ ) = −∞ .As mentioned in the introduction, in [11] the large dimension was characterisedtopologically via a structure theorem. Furthermore, in Lemma 6.11 of the samepaper, it was shown that large dimension is a dimension function. Therefore largedimension satisfies properties (1)-(7) of Fact 2.4 (stated there for δ -dim). This isone of the key ingredients in the proof of Theorem 1.1. The other key ingredient isa characterisation, obtained in [11], of definable sets in M of large dimension zero.This is given in terms of the following notion (essentially coming from [9]). Definition 2.7.
Let X ⊆ R n be a set definable in M . We call X large if there is m ≥ f : R nm → R definable in R such that f ( X m ) contains anopen interval in R . We call X small if it is not large. Fact 2.8. [11, Corollary 3.11 and Lemma 6.11(3)]
Let X be a set definable in M .Then, ldim( X ) = 0 ⇐⇒ X is small ⇐⇒ X is internal to C PANTELIS E. ELEFTHERIOU, OMAR LE ´ON S ´ANCHEZ, AND NATHALIE REGNAULT Proof of Theorem 1.1
We prove Theorem 1.1 through a series of lemmas.
Lemma 3.1. δ - dim C = 0 .Proof. Let K ∗ = hR ∗ , δ i be an | R | + -saturated elementary extension of K = hR , δ i and C ∗ = C ( R ∗ ), see Definition 2.3. Any point a in C ∗ is a solution of the nonzerodifferential polynomial δx . Thus the differential transcendence degree of a is zero. (cid:3) Lemma 3.2.
Let X ⊆ R n be a set definable in M . If ldim X = 0 , then δ - dim X = 0 .Proof. Suppose ldim X = 0. By Fact 2.8, X is internal to C ; namely, there is afunction f : R m → R n definable in M , such that X ⊆ f ( C m ). By Lemma 3.1 and(3), (4), (6) of Fact 2.4, δ -dim X ≤ δ -dim f ( C m ) ≤ δ -dim C m = 0 . (cid:3) The following lemma is a useful fact on CODFs.
Lemma 3.3.
Let p ( x ) be a nonzero differential polynomial over R . The solutionset { c ∈ R : p ( c ) = 0 } is co-dense in R with respect to the order topology.Proof. This fact follows from the axioms of CODFs stated in ( † ) in § a, b ) be a nonempty open interval, weprove that there is c ∈ ( a, b ) such that p ( c ) = 0. Let f ( x ) = δ m x with m > ord( p ( x )), g ( x ) = p ( x ), g ( x ) = x − a and g ( x ) = b − x . One can find ¯ c =( c , . . . , c m ) ∈ R m such that, as algebraic polynomials, f (¯ c ) = 0, g (¯ c ) = 0, g (¯ c ) > g (¯ c ) >
0. By the axioms ( † ), we can find c ∈ R such that, as differentialpolynomials, g ( c ) = 0, g ( c ) > g ( c ) >
0. This yields p ( c ) = 0 and c ∈ ( a, b ),as desired. (cid:3) Lemma 3.4.
Let X ⊆ R n be a set definable in M . If δ - dim X = 0 , then ldim X = 0 .Proof. Suppose towards a contradiction that ldim
X >
0. By Fact 2.8, X is large,and hence, by definition, there is a function f : R nm → R definable in R suchthat f ( X m ) contains an open interval in R . On the other hand, by (4) and (6) ofFact 2.4, we have δ -dim f ( X m ) ≤ δ -dim X m = 0 . By compactness, there must exist a nonzero differential polynomial p ( x ) over R that vanishes in all of f ( X m ). As the latter contains an interval, this contradictsLemma 3.3. (cid:3) We can now prove Theorem 1.1 and Corollary 1.2.
Proof of Theorem 1.1.
By property (7) in Fact 2.4 of dimension functions, δ -dimand ldim will coincide in all sets definable in M as long as they coincide on definablesubsets of R . By (2) in Fact 2.4, δ -dim( R ) = 1 = ldim( R ), and so any nonemptydefinable subset of R has δ -dim and ldim either 0 or 1. Hence, it suffices to showthat for definable X ⊆ R we have δ -dim( X ) = 0 ⇐⇒ ldim( X ) = 0 . But this follows by putting together Lemmas 3.2 and 3.4. (cid:3)
OINCIDENCE OF DIMENSIONS IN CLOSED ORDERED DIFFERENTIAL FIELDS 7
Proof of Corollary 1.2.
Left-to-right is by Lemma 2.2. For right-to-left, by Theo-rem 1.1, ldim( X ) = 0, and, by Fact 2.8, X is internal to C . (cid:3) Examples of zero dimensional not co-analyzable sets
In this section we exhibit examples of sets definable in K with δ -dimension 0which are not co-analyzable in C (and hence also not internal to C ). We recallthat in the case of transseries no such example exists [3, Proposition 6.2]. Ourexamples come from classical constructions of strongly minimal sets in DCF thatare orthogonal to the constants (over suitable parameter sets). We carry on ourterminology from previous sections. Furthermore, for convenience, here we work ina universal (sufficiently saturated) model K = hR , δ i of CODF with constants C .We begin by recalling the definition of co-analyzability in C for subsets of R n (see [13] for further details). A set X ⊆ R n definable in K is said to be co-analyzablein C in -steps if X is finite, and in ( r + 1) -steps if there is a set Y ⊆ C × R n definable in K such that(1) the canonical projection π : C × R n → R n maps Y onto X , and(2) for each c ∈ C the fibre Y c = { a ∈ R n : ( c, a ) ∈ Y } is co-analyzable in C in r -steps. Co-analyzable means co-analyzable in r -steps for some r ≥ to CODF. Let F = R ( i ) and U = h F, δ i ; namely the underlying set of U issimply the algebraic closure of R and the derivation δ is the unique extension from R to F . From [21], we know U is a DCF . We recall that a strongly minimal set Z defined in U is said to be orthogonal to the constants if for any parameter set A ⊂ F over which Z is defined and generic point a ∈ Z over A we have that, forany tuple c of constants from U ,acl U ( A, a ) ∩ acl U ( A, c ) = acl U ( A ) . Here we also recall that, for any subset B ⊂ F , the algebraic closure acl U ( B ) is justthe field-theoretic algebraic closure of the differential field generated by B in F . Proposition 4.1.
Let Z ⊆ F n be a set definable in U over parameters from R thatis strongly minimal. Let X = Z ( R ) (the points of X whose entries are in R ). If Z is orthogonal to the constants in U and X is infinite, then X is not co-analizablein C .Proof. We first show that X is not 1-step co-analysable in C . Towards a contradic-tion, suppose it is, via a definable in K subset Y ⊆ C × R n . Let L be a small (withrespect to saturation) differential subfield of R over which all of Z , X , and Y aredefined. As X is infinite, let a ∈ X be such that a / ∈ acl K ( L ). Recall that acl K ( L )is just the (relative) field-theoretic algebraic closure of L in R . Let c ∈ C be suchthat ( c, a ) ∈ Y . In particular, a ∈ acl K ( L, c ). As c is a constant, this means that a is in the field-theoretic algebraic closure of L ( c ) in R and not in the field-theoreticalgebraic closure of L in R . This is of course also true after replacing R for F .As a is a generic point of Z over L (in the sense of U by strong minimality of Z ),this contradicts orthogonality of Z to the constants (in U ). Thus, X is not 1-stepco-analyzable. PANTELIS E. ELEFTHERIOU, OMAR LE ´ON S ´ANCHEZ, AND NATHALIE REGNAULT
We now prove X is not co-analyzable in r + 1-steps. If it were we would havea definable in K subset Y ⊆ C × R n such that the projection π : C × R n → R n maps Y onto X and all fibres Y c are r -step co-analyzable in C . Now take any c ∈ C such that the fibre Y c is infinite. As Y c is contained in X , we can applyinduction on r and the same argument as above to contradict orthogonality of Z to the constants. (cid:3) Remark . We note that the proof of Proposition 4.1 relies essentially only onthe fact that for A ⊆ R the algebraic closure acl K ( A ) equals the real closure in R of the differential field generated by A .Proposition 4.1 yields plenty of examples of sets definable in K of δ -dim zero andnot co-analyzable in C . For instance, those of the form X = Z ( R ) where Z is theManin kernel of a simple abelian variety defined over R which do not descend to theconstants in U . For more explicit examples, one can use the so-called Rosenlichtextensions [19], as follows. Corollary 4.3.
Let f ( x ) be either x − x or xx − . Then, the subset X ⊆ R definedby δ ( x ) = f ( x ) is not co-analyzable in C .Proof. The set X is infinite; indeed, for any differential field L , we can define aderivation on L ( x ) that maps x f ( x ). Furthermore, by a classical result ofRosenlicht (see [19, Proposition 2]), the subset of F defined by δ ( x ) = f ( x ) isorthogonal to the constants in U . The result now follows from Proposition 4.1. (cid:3) We conclude this section with an application on the existence of a proper CODFextension with the same field of constants. We are not aware of any such examplein the literature (and no such examples exist in the case of transseries).
Corollary 4.4.
There is a proper extension R S of CODFs with the sameconstants.Proof. This follows from the existence of a non-co-analyzable in C definable set(given by Corollary 4.3, for instance) and [3, Proposition 6.1(iv)]. (cid:3) Analogues of Theorem 1.1 in DCF and transseries In this section we point out the necessary adaptations of the arguments in § andtransseries.5.1. The case of DCF . To state the analogue of Theorem 1.1 in differentiallyclosed fields of characteristic zero, we recall some properties of pairs of ACFs, forwhich we use [1]. We fix an algebraically closed field F = h F, + , ·i and a derivation δ on F such that K = hF , δ i is a model of DCF . We let C be the field of constantsof K . The structure M = hF , C i is an elementary pair of algebraically closed fields(of characteristic zero).As in § F induces a pregeometry on M ; namely, for B ⊆ F one setscl B ( A ) = acl( A ∪ B ∪ C ) OINCIDENCE OF DIMENSIONS IN CLOSED ORDERED DIFFERENTIAL FIELDS 9 and this defines a pregeometry (localised at B ) on F , called the cl B -pregeometryof M . This induces a cl B -dimension on finite tuples of F , and for a nonempty set X definable in M , we define:dim ( X ) = max a ∈ X ∗ cl F - dim( a )where X ∗ = X ( F ∗ ) and M ∗ = hF ∗ , C ∗ i is a | F | + -saturated elementary extensionof M . One sets dim ( ∅ ) = −∞ .In [1] the dimension dim was studied (in the general context of pairs of alge-braically closed fields). In the same paper they showed that this is a dimensionfunction, and hence satisfies properties of (1)-(7) of Fact 2.4. On the other hand,the differential dimension, δ -dim, can be defined in the same way as in Definition 2.3for definable sets in K . Furthermore, δ -dim in K is also a dimension function (aspointed out in [2]).The analogue of Theorem 1.1 in the DCF setting is Theorem 5.1.
Let K = hF , δ i be a differentially closed field of characteristic zeroand C its field of constants. Let X be a set definable in the pair M = hF , C i . Then dim ( X ) = δ - dim( X ) . The key ingredient in the proof is the following dichotomy result established in[1, Proposition 1.1]
Fact 5.2.
For X ⊆ F definable in M , we have dim ( X ) ≤ or dim ( F \ X ) ≤ . We now prove Theorem 5.1. As dim and δ -dim are dimension functions, itsuffices to show that they agree on subsets of F definable in M (see (7) of Fact 2.4).As dim ( F ) = 1 = δ -dim( F ), it suffices to show that for definable X ⊆ F we havedim ( X ) = 0 ⇐⇒ δ -dim( X ) = 0 . By the dichotomy theorem (Fact 5.2), it actually suffices to show left-to-right in theabove equivalence. So assume dim ( X ) = 0. Let K ∗ = hF ∗ , δ i be a | F | + -saturatedelementary extension of K . We must show that for any a ∈ X ∗ = X ( F ∗ ) we havecl δF -dim( a ) = 0 (where the latter is taken with respect to the cl δF -pregeometry of K ∗ ). Let C ∗ be the constants of K ∗ . Since M ∗ := hF ∗ , C ∗ i is a | F | + -saturatedelementary extension of M and dim ( X ) = 0, we get cl F -dim( a ) = 0 (where thelatter is taken with respect to the cl F -pregeometry of M ∗ ). We thus have a ∈ cl F ( ∅ ) = acl F ∗ ( F, C ∗ ) = F ( C ∗ ) alg ⊂ F ∗ . As all elements in F ( C ∗ ) alg are differentially algebraic over F , we get that a ∈ cl δF ( ∅ ). Thus, cl δF -dim( a ) = 0, and so δ -dim( X ) = 0.5.2. The case of transseries.
Let K = h T , δ i be the differential field of transseriesas constructed in [2]. The field of constants in this case equals R , and the reduct M = h T , R i is a tame pair in the sense of [10]. The δ -dimension for definable setsin K was studied in [3] and was shown to be a dimension function. On the otherhand, in [1] the h T , R i analogue of the dim dimension was studied in [1] and wasshown to be a dimension function. Furthermore, putting together [3, Proposition4.1] and [1, Theorem 1.4], we have: Fact 5.3.
Let X be a definable set in M . Then dim ( X ) = 0 ⇐⇒ X is discrete ⇐⇒ δ - dim( X ) = 0 . The analogue of Theorem 1.1 is
Theorem 5.4.
Let K = h T , δ i be the differential field of transseries. Let X be aset definable in the pair M = h T , R i . Then dim ( X ) = δ - dim( X ) . Proof.
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OINCIDENCE OF DIMENSIONS IN CLOSED ORDERED DIFFERENTIAL FIELDS 11
Pantelis E. Eleftheriou, Department of Mathematics and Statistics, University ofKonstanz, Box 216, 78457 Konstanz, Germany
Email address : [email protected] Omar Le´on S´anchez, University of Manchester, Department of Mathematics, OxfordRoad, Manchester, M13 9PL.
Email address : [email protected] Nathalie Regnault, Department of Mathematics (De Vinci), UMons, 20, place duParc 7000 Mons, Belgium
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