Remarks about Asymptotic Couples - Observation and Participation | MATH 103A, Papers of Mathematics

Download Remarks about Asymptotic Couples - Observation and Participation | MATH 103A and more Papers Mathematics in PDF only on Docsity! Some Remarks About Asymptotic Couples Matthias Aschenbrenner Department of Mathematics University of Illinois at Urbana-Champaign Urbana, IL 61801, U.S.A. Current address: Department of Mathematics University of California at Berkeley Berkeley, CA 94720, U.S.A. maschenb@math.uiuc.edu Abstract. Asymptotic couples (of H-type) try to capture the structure induced by the derivation of a Hardy field K on the value group of the natural valuation on K. In this note we continue the study of algebraic and model-theoretic aspects of asymptotic couples undertaken in [1]. We give a short exposition of some basic facts about asymptotic couples, and address a few topics left out in that paper: the (non-) minimality of the “closure” of an asymptotic couple of H-type, the Vapnik-Chernovenkis property for sets definable in closed asymptotic couples of H-type, and the relation of asymptotic couples of H-type to the “contraction groups” of [6]. Introduction Let K be a Hardy field, that is (see [3], [17]), an ordered differential field of germs at +∞ of real-valued differentiable functions defined on intervals (a,+∞), with a ∈ R. (So two such functions determine the same element ofK if they coincide on an interval (b,+∞) on which they are both defined; we will use the same letter for a function and its germ.) Every element f of K is ultimately monotonic, so limx→∞ f(x) exists as an element of R ∪ {±∞}. The valuation v : K× = K \ {0} → V = v(K×) associated to the place f 7→ limx→∞ f(x) (where we identify +∞ and −∞) has the crucial property that v(f ′) only depends on v(f), for f ∈ K× with v(f) 6= 0. (This is a consequence of L’Hospital’s Rule, see [17].) So we have a well-defined map ψ : V ∗ = V \ {0} → V given by ψ ( v(f) ) := v(f ′/f) for any f ∈ K× such that v(f) 6= 0. 1991 Mathematics Subject Classification. Primary 03C10, 06F20; Secondary 26A12, 12H05. Key words and phrases. Asymptotic couples, Hardy fields. 1 2 Matthias Aschenbrenner The pair V = (V, ψ), called the asymptotic couple of K, is of key importance in understanding the interaction of the ordering and the derivation of K. It has the following fundamental properties: For all elements f, g ∈ K× with a = v(f) 6= 0, b = v(g) 6= 0, (A1) ψ(ra) = ψ(a) for all r ∈ Z, r 6= 0, (A2) ψ(a+ b) ≥ min { ψ(a), ψ(b) } , where ψ(0) := ∞ > V , (A3) ψ(a) < ψ(b) + |b|. (See [14], Theorem 4.) Property (A2) expresses the fact that ψ is a valuation on the ordered abelian group V (taking values in V itself). In particular, it follows that for a, b as above, ψ(a + b) = min { ψ(a), ψ(b) } if ψ(a) 6= ψ(b). Property (A3) may be seen as a valuation-theoretic formulation of L’Hospital’s Rule, see [15]. Moreover, the map ψ is decreasing on the set of positive elements of V : For all a, b ∈ V , (H) 0 < a ≤ b =⇒ ψ(a) ≥ ψ(b). (Hence by (A1), ψ is increasing on the set of negative elements of V .) Note that if the Hardy field K contains the germ x of the identity function on R, then ψ(1) = 1, where we put 1 := v(x−1) > 0. By an asymptotic couple, we mean a pair V = (V, ψ) consisting of an ordered abelian group V and a map ψ : V ∗ → V satisfying (A1)–(A3) above, for all a, b ∈ V ∗. As in [2], we say that an asymptotic couple V = (V, ψ) is of H-type if (H) holds for all a, b ∈ V . We will sometimes also say “V is an H-asymptotic couple” instead of “V is an asymptotic couple of H-type.” Rosenlicht, in a series of papers ([14], [15], [16], [18]) studied in detail the asymptotic couples (V, ψ) where the ordered abelian group V has finite rank. The paper [1] contains an investigation of the basic model-theoretic properties of H-asymptotic couples. In this note, we want to supplement it by considering a few issues left open in that paper. Our hope is that insight into algebraic and model-theoretic properties of asymptotic couples will ultimately become useful in the recently initiated project of understanding the model theory of Hardy fields and the field of LE-series. (See [2], [5].) In section 1, we first review some basic facts about asymptotic couples. We only give a few proofs, referring to [1] and [2] for a more detailed exposition. From results of [1], it follows that the theory of H-asymptotic couples has a model companion (in a natural language), the theory of “closedH-asymptotic couples.” (The definition of a closedH-asymptotic couple, as well as the statement of the main theorem from [1], can be found in section 2. See [12] for the notion “model companion”.) In particular, each H-asymptotic couple V can be embedded into a closed one. In section 3, we show that in general there is no closedH-asymptotic couple containing V minimally. Section 4 consists of a few remarks about another model-theoretic property of the class of closed H-asymptotic couples, called the independence property. Finally, in section 5 we discuss a connection to Kuhlmann’s “contraction groups” from [6]. In [1], we mainly worked in the setting of “H-couples”: these are H-asymptotic couples (V, ψ) of a certain kind, where V has additional structure as an ordered vector space over an ordered field. In Section 6 of that paper, we showed how to adapt the results about H-couples to the case of H-asymptotic couples. Here, we right away restrict our attention to H-asymptotic couples, for convenience. We don’t assume familiarity of the reader with [1]. In fact, sections 1–4 of the present note may serve as a quick overview of some of the results from that paper. However, we will freely use basic model-theoretic notions (see e.g. [12]). Some Remarks About Asymptotic Couples 5 Corollary 1.4 The set (id+ψ) ( V >0 ) is closed upward. The set (id+ψ) ( V <0 ) is closed downward, and (− id+ψ) ( V >0 ) = (id+ψ) ( V <0 ) = { a ∈ V : a < ψ(x) for some x ∈ V ∗ } . (1.2) There is at most one element v ∈ V such that Ψ < v < (id+ψ) ( V >0 ) . If Ψ has a largest element, then there is no v ∈ V with Ψ < v < (id+ψ) ( V >0 ) . Proof Let a > x + ψ(x) for some x > 0; we want to show a ∈ (id+ψ) ( V >0 ) . Passing from (V, ψ) to (V, ψ − a) if necessary, we reduce to the case a = 0. Then [x] ≤ [ ψ(x) ] , hence a = 0 ∈ (id+ψ) ( V >0 ) by the previous remark and Propo- sition 1.1, (3.). So (id +ψ) ( V >0 ) is closed upward, and similarly one shows that (id+ψ) ( V <0 ) is closed downward. The equalities in (1.2) are clear except for the inclusion “⊇” in the last equa- tion. For this, let a, x ∈ V , x < 0, with a < ψ(x); we want to show that a ∈ (id+ψ) ( V <0 ) . As above, we may assume that a = 0. If [x] ≤ [ ψ(x) ] , it follows as before that 0 ∈ (id+ψ) ( V <0 ) . If [ ψ(x) ] < [x], then 0 < x+ ψ(x), hence 0 ∈ (id+ψ) ( V <0 ) , since (id+ψ) ( V <0 ) is closed downward. If u, v ∈ V satisfy ψ(w) ≤ u < v < w + ψ(w) for all w ∈ V >0, then v < (v − u) + ψ(v − u) ≤ (v − u) + u = v, a contradiction. This shows the rest. As a consequence of the last corollary, V \(id+ψ)(V ∗) has at most one element, and (id+ψ)(V ∗) 6= V if and only if Ψ has a supremum in V , and in this case V \ (id+ψ)(V ∗) = {supΨ}. We refer the reader to [1], Figure 1, for a picture of the behavior of the maps ψ and id+ψ on V ∗. 2 Closed H-Asymptotic Couples A cut of an H-asymptotic couple (V, ψ) is a set P ⊆ V which is closed down- ward, contains Ψ, and is disjoint from (id +ψ) ( V >0 ) . (So P < (id+ψ) ( V >0 ) .) By Corollary 1.4, an H-asymptotic couple (V, ψ) has at most two cuts, and it has two cuts if and only if Ψ < v < (id+ψ) ( V >0 ) for some v ∈ V . If Ψ has a maximum, then (V, ψ) has exactly one cut P = { a ∈ V : a ≤ ψ(x) for some x ∈ V ∗ } . Definition 2.1 An H-asymptotic couple V = (V, ψ) is closed if 1. V is divisible (as an abelian group), 2. (id +ψ)(V ∗) = V , and 3. Ψ = (id +ψ) ( V <0 ) . (In this case, P = Ψ is the only cut of V.) Example 1 Let K be a Hardy field containing R and closed under exponentia- tion (that is, f ∈ K ⇒ exp f ∈ K) and integration (i.e. f ∈ K ⇒ ∃g ∈ K : g′ = f). Then the asymptotic couple of K (as defined in the introduction) is a closed H- asymptotic couple. In [1], Definition 6.2, we also introduced the following notion, under the some- what technical name “H0-triple”: Definition 2.2 An asymptotic triple of H-type, or H-asymptotic triple for short, is a triple (V, ψ, P ), where (V, ψ) is an H-asymptotic couple and P a cut of (V, ψ), such that 1. V is divisible, and 6 Matthias Aschenbrenner 2. there exists a positive element 1 of V with ψ(1) = 1. (Equivalently, 0 ∈ (id+ψ) ( V <0 ) .) By Proposition 1.1, (2.), the element 1 in (2.) is uniquely determined. If (V, ψ, P ) is an H-asymptotic triple such that (V, ψ) is a closed H-asymptotic couple, then P = Ψ, and (V, ψ,Ψ) is called a closed H-asymptotic triple. We can naturally consider asymptotic couples (V, ψ) as model-theoretic struc- tures (V∞, ψ) in the first-order language L = {0,+,−, ψ,∞}. The H-asymptotic couples are then the models of a universal theory in L. Similarly, when dealing with H-asymptotic triples (V, ψ, P ) as model-theoretic objects, we construe them as LP -structures (V∞, ψ, 1, P ), where LP is the extension of L by 1. a constant symbol 1 for the distinguished element 1 ∈ V >0 with ψ(1) = 1, 2. a unary predicate symbol for P , and 3. unary function symbols δn for each n > 0, to be interpreted on V as the scalar multiplication by 1/n (and δn(∞) := ∞). The H-asymptotic triples are models of a universal theory in LP . Let T be the theory of closed H-asymptotic couples, in the language L, and let TP be the theory of closed H-asymptotic triples, in the language LP . One of the main results from [1] (Corollary 6.2) is: Theorem 2.3 The theory TP is complete, decidable, and has elimination of quantifiers. It is the model completion of the theory of H-asymptotic triples. From this we get immediately: Corollary 2.4 The theory T is the model companion of the theory of H- asymptotic couples. Remark The division symbols δn are included in the language LP in order to guarantee quantifier elimination for TP . Here is an instructive example to show that if we omit them, then in the resulting smaller language the theory of closed H-asymptotic triples would not eliminate quantifiers. Let (W,ψ) be a closed H-asymptotic couple. Choose an element b /∈ W in an ordered vector space W ′ := W ⊕ Qb over Q extending W , such that Ψ < b2 < (id+ψ) ( W>0 ) . Then, by Lemma 4.5 in [1], [W ] = [W ′], hence ψ extends uniquely to a map ψ′ : (W ′)∗ → W such that W ′ = (W ′, ψ′) is an H-asymptotic couple (Corollary 1.2). Note that [W ] = [W ′] implies Ψ′ = Ψ < b 2 < (id+ψ′) ( (W ′)>0 ) . Hence (W ′, ψ′) has two cuts. Now consider the ordered abelian group V := W ⊕ Zb ⊆ W ′. Since Ψ′ = Ψ ⊆ W , (V, ψ′|V ∗) is an H-asymptotic couple with (V, ψ′|V ∗) ⊆ (W ′, ψ′). One checks easily that the two distinct cuts of (W ′, ψ′) have the same intersection with V , namely { v ∈ V : v ≤ ψ(w) for some w ∈W } . 3 Non-Minimality of Closures According to Theorem 2.3, every H-asymptotic triple can be embedded into a closed H-asymptotic triple. In fact, in the course of the proof of this theorem, we showed a more precise statement: Some Remarks About Asymptotic Couples 7 Proposition 3.1 Every H-asymptotic triple V = (V, ψ, P ) has a closure, that is, a closed H-asymptotic triple Vc = (V c, ψc, P c) extending V, such that any embedding V → V ′ into a closed H-asymptotic triple V ′ extends to an embedding Vc → V ′. Any two closures of V are isomorphic over V. (See [1], Corollaries 5.3 and 6.1.) A natural question is if the closure Vc of an H-asymptotic triple V is always minimal over V, i.e. there exists no closed H- asymptotic triple W ⊇ V strictly contained in Vc as an LP -substructure. This turns out to be false, in a very strong way: Proposition 3.2 Let V = (V, ψ, P ) be an H-asymptotic triple which is not closed. Then the closure Vc of V is not minimal over V. (This is similar, e.g., to the situation encountered with differential fields and differential closures, [13].) Before we give a proof of Proposition 3.2, we outline how Vc is constructed from V. One first shows the following embedding statements (see [1], Lemmas 3.5, 3.6 and 3.7 for a proof): Lemma 3.3 Let V = (V, ψ, P ) be an H-asymptotic triple. 1. Suppose P has a largest element, and let V ε := V ⊕ Qε be an extension of the Q-vector space V . Then there exists a unique linear ordering of V ε, a unique map ψε : (V ε)∗ → V ε, and a unique subset P ε of V ε such that (V ε, ψε, P ε) is an H-asymptotic triple extending (V, ψ, P ) with ε > 0 and maxP = −ε+ ψε(ε). 2. Suppose there exists b ∈ V with P < b < (id+ψ) ( V >0 ) . Let V ε := V ⊕Qε be an extension of the Q-vector space V . Then there exists a unique linear ordering of V ε, a unique map ψε : (V ε)∗ → V ε, and a unique subset P ε ⊆ V ε such that (V ε, ψε, P ε) is an H-asymptotic triple extending (V, ψ, P ) with ε > 0 and b = ε+ ψε(ε). 3. Suppose b ∈ P\Ψ. Let V a := V ⊕Qa be an extension of the Q-vector space V . There exists a unique linear ordering of V a, a unique map ψa : (V a)∗ → V a, and a unique P a ⊆ V a, such that (V a, ψa, P a) is an H-asymptotic triple extending (V, ψ, P ) with a > 0 and ψa(a) = b. Note that (V ε, ψε) as in (1.) or (2.) of the lemma has the property that ψε ( (V ε)∗ ) has a maximum. So part (1.) applies to (V ε, ψε, P ε) in place of (V, ψ, P ). Also, if (id +ψ)(V ∗) = V , then (id+ψa) ( (V a)∗ ) = V a. Therefore, iter- ating (1.)–(3.), if necessary transfinitely often, we can obtain an increasing chain of H-asymptotic triples extending (V, ψ, P ) whose union is a closure of (V, ψ, P ). Proof of Proposition 3.2. Let V = (V, ψ, P ) be an H-asymptotic triple which is not closed. We have to find a closed H-asymptotic triple W with V ⊆ W which is strictly contained (as a substructure) in a closure of V. Let us first consider a special case: Lemma 3.4 Suppose that P does not have a supremum in V , and P \Ψ con- tains a strictly increasing sequence (an)n∈N. Then the closure of V is not minimal over V. Proof Using Lemma 3.3, (3.), we construct a strictly increasing sequence of H-asymptotic triples (Vn)n∈N such that 1. V0 = V = (V, ψ, P ), 10 Matthias Aschenbrenner In fact, in [1] (Proposition 6.2) we showed something more: the theory TP is weakly o-minimal, that is, for every closed H-asymptotic triple V = (V, ψ, P ), every LP -formula ϕ(x, y) with x = (x1, . . . , xm) and a single variable y, and every v ∈ V m, the set ϕV(v, y) is a boolean combination of cuts in (V,<). This also implies the corollary above, by Proposition 7.3 in [10]. We want to remark that the argument indicated here also works in the two-sorted setting of “closedH-triples” as defined in [1], thus giving a natural example of a locally o-minimal (but not weakly o-minimal) theory without the independence property. (See [1], Proposition 5.1 for the definition of “locally o-minimal” and a proof of the local o-minimality of the theory of closed H-triples.) Unlike in the weakly o-minimal case, it seems not to be known whether every locally o-minimal theory extending the theory of dense linear orders does not have the independence property. 5 Relation to Contraction Groups Our couples resemble the contraction groups of Kuhlmann [6], [7], and there is indeed a formal connection as indicated below. (A difference is that contraction groups have nothing like our cut P .) Contraction groups arise as follows: let K be a Hardy field closed under taking logarithms (i.e. f ∈ K>0 ⇒ log f ∈ K), with its valuation v : K× → V = v(K×). The logarithm map then induces a so-called contraction map χ : V <0 → V <0 by χ ( v(f) ) := v(log f) for all f ∈ K>0 with v(f) < 0, which we extend to a map V → V by requiring χ(−y) = −χ(y). If K is also closed under exponentiation, then V is divisible, and χ is surjective (χ(V ) = V ). This means that the pair (V, χ) (ordered group with contraction map) is a divisible centripetal contraction group, as axiomatized in [6], where it was shown that the elementary theory of non-trivial divisible centripetal contraction groups is com- plete and has quantifier elimination in its natural language. (See the appendix of [8] for an exposition of these results.) In the example above, we have for f ∈ K>0, with y = v(f) < 0: ψ(y) = v ( (log f)′ ) = v ( (log f)′/ log f ) + v(log f) = ψ ( χ(y) ) + χ(y) Let now (V, ψ) be any closed H-asymptotic couple. For y < 0 in V , let χ(y) = z be the unique solution in V ∗ of the equation z + ψ(z) = ψ(y). (5.1) For y > 0, set χ(y) := −χ(−y), and χ(0) := 0. It is easily seen that then (V, χ) is a non-trivial divisible centripetal contraction group; clearly χ is definable (without parameters) in (V, ψ). Hence in particular, (V, χ) is weakly o-minimal, by Propo- sition 6.2 in [1]. We want to point out that the weak o-minimality of the theory of non-trivial divisible centripetal contraction groups (proved in [7]; see also [8], The- orem A.34) is a consequence of its completeness and the preceding observation: any model of this theory can be elementarily embedded into one of the form (V, χ) with χ definable in a closed H-asymptotic couple (V, ψ) (by choosing (V, ψ) sufficiently saturated), and hence is weakly o-minimal. (As the theory of closed H-asymptotic couples, the theory of non-trivial divisible centripetal contraction groups does not have the Steinitz exchange property for the definable closure operation.) However, we cannot definably reconstruct ψ in (V, χ): Some Remarks About Asymptotic Couples 11 Proposition 5.1 In no divisible centripetal contraction group (V, χ) can one define, even allowing parameters, a function ψ : V ∗ → V such that (V, ψ) is a closed H-asymptotic couple and χ+ ψ ◦ χ = ψ on V <0. Before we can prove this we need some preparations. We let (V, ψ) denote a closed H-asymptotic couple. We also assume that 0 ∈ Ψ, so there exists 1 ∈ V ∗ such that ψ(1) = 1 > 0. Iterates of ψ. For n > 0, let ψn : V∞ → V∞ be the n-fold functional compo- sition ψ ◦ ψ ◦ · · · ◦ ψ. Put Dn := { v ∈ V : ψn(v) 6= ∞ } . For example D1 = V ∗, D2 = V ∗ \ ψ−1(0), etc. By induction on n one shows easily that ψn(Dn) = Ψ. Lemma 5.2 Let v ∈ V ∗ and n > 0 such that ψn(v) < 0. Then ψi(v) < 0 for all i = 1, . . . , n, and[ ψn(v) ] < [ ψn−1(v) ] < · · · < [ ψ(v) ] < [v]. Proof For n = 1, note that [v] ≤ [ ψ(v) ] and (1.1) imply ψ(v) ≥ ψ ( ψ(v) ) , hence −ψ(v) + ψ ( −ψ(v) ) ≤ 0 < (id+ψ) ( V >0 ) . Thus ψ(v) > 0, a contradiction. Assume inductively that the lemma holds for a certain n > 0. Let v ∈ Dn+1 with ψn+1(v) < 0. Applying the case n = 1 to ψn(v) instead of v gives [ ψn+1(v) ] <[ ψn(v) ] . By the inductive assumption the remaining inequalities will follow from ψn(v) < 0. Suppose ψn(v) ≥ 0. Then ψn(v) ∈ Ψ>0, thus [ ψn(v) ] ≤ [1] by (A3). Hence 0 > ψn+1(v) ≥ ψ(1) = 1 by (1.1), a contradiction. Let D∞ := ⋂ n>0Dn and Vinf := { v ∈ D∞ : ψn(v) < 0 for all n > 0 } , Vfin := V \ Vinf. Note that [v0] < [v] for all v ∈ Vinf, and that Vinf ∩ V >0 is closed upward and Vinf ∩ V <0 is closed downward. Remark The previous lemma, together with ψn(Dn) = Ψ, implies that for all n > 0, we can find an element v ∈ Dn such that all iterates ψ(v), ψ2(v), . . . , ψn(v) are negative. Hence if (V, ψ) is ℵ0-saturated, then Vinf 6= ∅. The proof of the next lemma is easy and left to the reader. Lemma 5.3 Vfin is a convex subspace of V , and (Vfin, ψ|V ∗fin) is a closed H- asymptotic couple. Moreover, ψ(Vinf) = Vinf ∩ V <0. Let χ be the contraction map defined by ψ(v) = χ(v) + ψ ( χ(v) ) for all v < 0. Lemma 5.4 Let v ∈ V <0 and ψ3(v) < 0. Then χ(v) = ψ(v)− ψ2(v). Proof We have [v] > [ ψ(v) ] , so ψ(v)− ψ2(v) < 0. We compute:( ψ(v)− ψ2(v) ) + ψ ( ψ(v)− ψ2(v) ) = ( ψ(v)− ψ2(v) ) + ψ2(v) = ψ(v). By the defining equation (5.1) of χ, it follows that χ(v) = ψ(v)− ψ2(v). 12 Matthias Aschenbrenner Proof of Proposition 5.1. Suppose (V, ψ) is a closed H-asymptotic couple such that we can define ψ in (V, χ). We may assume that (V, ψ) is ℵ0-saturated. For ease of notation we shall also assume that ψ is actually defined without parameters in (V, χ). (In the general case the role of Vfin below is taken over by the convex hull in V of a closure inside (V, ψ) of the substructure of (V, ψ) generated by the finitely many parameters used to define ψ.) If 0 ∈ (id+ψ) ( V <0 ) , then (V, ψ,Ψ) is a closed H-asymptotic triple. Otherwise, we let 1 ∈ V >0 be the unique solution to the equation x+ψ(x) = 0, and pass from (V, ψ) to (V, ψ0), where ψ0 := ψ+1−ψ(1), so that ψ0(1) = 1 > 0. We see that we may in fact assume that (V, ψ,Ψ) is a closed H-asymptotic triple, with a distinguished positive element 1. We now modify ψ to a function ψ̃ : V ∗ → V by putting ψ̃(v) := { ψ(v), if v ∈ V ∗fin ψ(v) + 1, if v ∈ Vinf. Then (V, ψ̃) is still an H-asymptotic couple, and ψ̃(Vinf) = ψ(Vinf), as is easily checked. Thus Ψ = ψ̃(V ∗), so (V, ψ̃) is even a closed H-asymptotic couple. Let χ̃ be the contraction map associated to (V, ψ̃). By completeness of the theory of closed H-asymptotic triples, the same formula that defines ψ in (V, χ) will define ψ̃ in (V, χ̃). By Lemma 5.4, χ = χ̃, hence ψ = ψ̃, contradiction. References 1. M. Aschenbrenner, L. van den Dries, Closed asymptotic couples, J. Algebra 225 (2000), 309– 358. 2. , H-fields and their Liouville extensions, Math. Z. 242 (2002), 543–588. 3. N. 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