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THE VALUATION DIFFERENCE RANK OF AN ORDERED DIFFERENCE FIELD 3 1 SALMAKUHLMANN,MICKAE¨LMATUSINSKI,ANDFRANÇOISEPOINT 0 2 n a Abstract. There are several equivalent characterizations of the valuation J rank of an ordered field (endowed with its natural valuation). In this paper, 1 we extend the theory to the case of an ordered difference field and introduce 1 thenotionofdifference rank. Wecharacterize thedifferencerankasthequo- tientmodulotheequivalencerelationnaturallyinducedbytheautomorphism ] (whichencodesitsgrowthrate). Inanalogytothetheoryofconvexvaluations, O weprovethatanylinearlyorderedsetcanberealizedasthedifferencerankof L anordereddifferencefield. . h 1. Introduction t a m The theory of realplaces andconvex valuations is a specialchapter in valuation theory; it is a basic tool in real algebraic geometry. Surveys can be found in [5], [ [6] and [7]. An important isomorphism invariant of an ordered field is its rank 1 as a valued field, which has several equivalent characterizations: via the ideals of v the valuation ring, the value group, or the residue field . This can be extended to 1 ordered fields with extra structure, giving a characterizationcompletely analogous 1 6 to the above, but taking into account the corresponding induced structure on the 2 ideals, value group, or residue field (see [2] for ordered exponential fields). 1. In this paper, we push this analogy to the case of an (ordered) difference field. 0 The leading idea is to identify the difference rank of a non-archimedean ordered 3 field as the quotient by the equivalence relationthat the automorphisminduces on 1 the set of infinitely large field elements. In Section 2 we start by a key remark : v regarding equivalence relations defined by monotone maps on chains, and briefly i review the theory of convex valuations and rank. In Section 3 we represent this X invariant via equivalence relations induced by addition and multiplication on the r a field. This approach allows us to develop in Section 4 the notion of difference compatible valuations, introduce the difference rank, and consider in particular isometries, weak isometries and ω-increasing automorphisms. The main results of the paper are in the last Section 5: Theorem 5.2 and its Corollaries 5.4, 5.3 and 5.5. Finally, we note that some of our notions and results generalize to the context of an arbitrary (not necessarily ordered) valued field, see [3]. 2. Preliminaries on the rank and principal rank of an ordered field We begin by the following key observation: 1991 Mathematics Subject Classification. Primary03C60, 06A05, 12J15: Secondary 12L12, 26A12. SupportedbyaResearchinParisgrantfromInstitutHenriPoincaré. 1 2 SALMAKUHLMANN,MICKAE¨LMATUSINSKI,ANDFRANÇOISEPOINT Remark 2.1. Let ϕ be a map from a totally ordered set S into itself, and assume that ϕ is order preserving, i. e. a≤a′ implies ϕ(a) ≤ϕ(a′), for all a, a′ ∈S. We assume further that ϕ has an orientation, i. e. ϕ(a) ≥ a for all a ∈ S (ϕ is a right shift) or ϕ(a) ≤ a for all a ∈ S (ϕ is a left shift). We set ϕ0(a) := a and ϕn+1(a):=ϕ(ϕn(a)) for n∈N :=N∪{0}. 0 It is then straightforwardthat the following defines an equivalence relation on S: (i) If ϕ is a right shift, set a ∼ a′ if and only if there is some n ∈ N such that ϕ 0 ϕn(a)≥a′ and ϕn(a′)≥a, (ii) If ϕ is a left shift, set a ∼ a′ if and only if there is some n ∈ N such that ϕ 0 ϕn(a)≤a′ and ϕn(a′)≤a. The equivalence classes [a] of ∼ are convex and closed under application of ϕ. ϕ ϕ By the convexity, the order of S induces an order on S/∼ such that [a] <[b] if ϕ ϕ ϕ and only if a′ <b′ for all a′ ∈[a] and b′ ∈[b] . ϕ ϕ OnthenegativeconeG<0ofanorderedabeliangroupG,thearchimedeanequiv- alence relation∼ is obtainedby setting ϕ(a):=2a, andv is the map a7→[a] . ϕ G ϕ The order on Γ:=G<0/∼ is the one induced by the order of G<0 as above. We ϕ call v (G<0) := Γ the value set of G. By convention we also write v (G) := Γ G G extendingthearchimedeanequivalencerelationtothepositiveconeofGbysetting v (g):=v (−g). The natural valuation v on G satisfies the ultrametric triangle G G G inequality, and in particular we have: v (x+y)=min{v (x),v (y)} if sign(x) = G G G sign(y). Wegathersomebasicfactsaboutvaluationscompatiblewiththeorderofanordered field. Throughoutthispaper,K willbeanon-archimedeanorderedfield,andv will denote its non-trivial naturalvaluation, that is, its valuation ring R is the convex v hull of Q in K. We set: P := K>0 \R , G = v(K) and Γ = v (G<0). The K v G naturalvaluationonK satisfiesv(x+y)=min{v(x),v(y)}ifsign(x)=sign(y)and for all a,b∈K :a≥b>0 =⇒ v(a)≤v(b). Letw beavaluationofK,withvaluationringR ,valuationidealI ,valuegroup w w w(K) and residue field Kw. Then w is called compatible with the order if and only if it satisfies, for all a,b∈Ka≥b>0 =⇒ w(a)≤w(b). Forthefollowingcharacterizationsofcompatiblevaluations,see[4]Proposition5.1, or [5] Theorem 2.3 and Proposition 2.9, or [7] Lemma 3.2.1, or [8] Lemma 7.2: Lemma 2.2. The following assertions are equivalent: 1) w is a valuation compatible with the order of (K,<), 2) R is a convex subset of (K,<), w 3) I is a convex subset of (K,<), w 4) I <1, w 5) the image of the positive cone P of (K,<) under the residue map K ∋a7→aw ∈Kw is a positive cone Pw in Kw. Avaluationcompatible withthe orderisthus alsosaidto be aconvex valuation. Foreveryconvexvaluationw,wedenotebyU>0 :={a∈K |w(a)=0∧a>0}the w groupofpositiveunitsofR . Itisaconvexsubgroupoftheorderedmultiplicative w group (K>0,·,1,<) of positive elements of K. THE VALUATION DIFFERENCE RANK OF AN ORDERED DIFFERENCE FIELD 3 Let w and w′ be valuations on K. We say that w′ is finer than w, or that w is coarser than w′ if Rw′ ⊂6= Rw. This is equivalent to Iw ⊂6= Iw′. By Lemma 2.2 4), if w′ is convex, then w also is convex. Conversely, a convex subring containing 1 is a valuation ring, see [1] Section 2.2.2. The natural valuation v of K is the finest convex valuation (and is characterized by the fact that its residue field is archimedean). ThesetRofallvaluationringsR ofconvexvaluationsw 6=v (i. e. w allcorseningsofv) istotallyorderedbyinclusion. Itsordertypeis calledthe rank of (K,+,·,0,1,<). For convenience, we will identify it with R. For example, the rank of an archimedeanordered field is empty since its natural valuation is trivial. The rank of the rational function field K = R(t) with any order is a singleton: R={K}. Recall that the set of all convex subgroups G 6= {0} of the value group G is w totally ordered by inclusion. Its order type is called the rank of G. Note that the rank of an ordered field (respectively of an ordered group) is an isomorphism invariant. To every convex valuation ring R , we associate a convex subgroup w G := {v(a)|a∈K ∧ w(a)=0} = v(U>0). We call G the convex subgroup w w w associated to w. Note that G = {0}. Conversely, given a convex subgroup G v w of v(K), we define w : K → v(K)/G by w(a) = v(a)+G . Then w is a convex w w valuation with v(U>0) = G (and v is finer than w if and only if G 6= {0}). We w w w call w the convex valuation associated to G . We summarize: w Lemma 2.3. The correspondence R 7→G is an order preserving bijection, thus w w R is (isomorphic to) the rank of G. We want to analyze the rank of G further and relate it to the value set Γ of G. For G 6= {0} a convex subgroup, we associate Γ := v (G<0) a non-empty w w G w final segment of Γ. Conversely, if Γ is a non-empty final segment of Γ, then w G = {g | g ∈ G,v (g) ∈ Γ }∪{0} is a convex subgroup, with Γ = v (G ). w G w w G w Let us denote by Γfs the set of non-empty final segments of Γ, totally ordered by inclusion. We summarize: Lemma 2.4. The correspondence G 7→Γ is an order preserving bijection, thus w w the rank of G is (isomorphic to) Γfs. Theorem2.5. ThecorrespondenceR 7→Γ isanorder preservingbijection, thus w w R is (isomorphic to) Γfs. A final segment which has a smallest element is a principal final segment. Let Γ∗ denotethesetΓwithitsreversedordering,thenΓ∗ is(isomorphicto)thetotally ordered set of principal final segments: Lemma 2.6. The map from Γ to Γfs defined by γ 7→ {γ′ | γ′ ∈ Γ,γ′ ≥ γ} is an order reversing embedding. Its image is the set of principal final segments. RecallthataconvexsubgroupG ofGiscalledprincipalgenerated byg,g ∈G, w if G is the minimal convex subgroup containing g. The principal rank of G is w the subset of the rank of G consisting of all principal G 6={0} . w Lemma 2.7. Let G 6= {0} be a convex subgroup. Then G is principal if and w w only if v (G )=Γ is a principal final segment. G w w 4 SALMAKUHLMANN,MICKAE¨LMATUSINSKI,ANDFRANÇOISEPOINT Lemma 2.8. The map G 7→ minv (G ) is an order reversing bijection. Thus w G w the principal rank of G is (isomorphic to) Γ∗. A convex subring R 6= R is a principal convex subring generated by a w v fora∈P ifR is thesmallestconvexsubringcontaininga. Theprincipal rank K w of K is the subset Rpr of R consisting of all principal R ∈R. w Theorem 2.9. The correspondence R 7→ Γ is an order preserving bijection w w between Rpr and the principal rank of G, thus Rpr is (isomorphic to) Γ∗. 3. The rank and principal rank via equivalence relations In this section, we exploit Remark 2.1 to give an interpretation of the rank and principal rank as quotients via an appropriate equivalence relation, thereby providing alternative proofs for Theorem 2.5 and Theorem 2.9. It is precisely this approachthat we will generalize to the difference rank in the next sections. Consider the following commutative diagram: ϕ - P P K K with ϕ(a) := a2 for all a∈P , K v /// v ϕ (v(a)) := v(ϕ(a)) for all a∈P , ? ? G K ϕ G<0 G- G<0 that is ϕG(g)=2g for all g ∈G<0, and ϕ (v (g)) := v (ϕ (g)) for all g ∈G<0, Γ G G G vG /// vG that is ϕ (γ) = γ for all γ ∈ Γ, so that ϕ is ? ? Γ Γ just the identity map. ϕ Γ- v (G) v (G) G G Weconsidertheequivalencerelationsassociatedtothemonotonemaps ϕ, ϕ and G ϕ as in Remark 2.1, and note that ∼ is just multiplicative equivalence ∼· on Γ ϕ P , ∼ justarchimedeanequivalenceonGand∼ justequalityonΓ. Suppose K ϕG ϕΓ that ∼ and ∼ are two equivalence relations defined on the same set. Recall that 1 2 ∼ is said to be coarser than ∼ if ∼ -equivalence implies ∼ -equivalence. The 1 2 2 1 following straightforward observation will be useful for the proof of Theorem 3.2 below: Lemma 3.1. The equivalence relation ∼ is coarser than the archimedean equiv- ϕ alence relation with respect to addition on P . The equivalence classes of ∼ are K ϕ closed under addition and multiplication. We further note that ϕn(v(a)) = v(ϕn(a)) and ϕn(v (g)) = v (ϕn(g)) G Γ G G G thus a∼ a′ if and only if v(a)∼ v(a′) if and only if v (v(a))∼ v (v(a′)). ϕ ϕG G ϕΓ G THE VALUATION DIFFERENCE RANK OF AN ORDERED DIFFERENCE FIELD 5 ThuswehaveanorderreversingbijectionfromP /∼ ontoΓ/∼ =Γ. Thusthe K ϕ ϕΓ chain[P /∼ ]is of initial segments of P /∼ orderedby inclusion is isomorphic K ϕ K ϕ to Γfs. Theorems 2.5 and 2.9 will therefore immediately follow from the following result Theorem 3.2. The rank R is isomorphic to [P / ∼ ]is and the principal rank K ϕ Rpr is isomorphic to the subset of [P /∼ ]is of initial segments which have a last K ϕ element. Proof. First we note that if R is a convex valuation ring, then clearly R>0\R>0 w w v is an initial segment of P , and moreover [a] is the last class in case R is K ∼ϕ w principal generated by a. Furthermore, if R intersects an equivalence class [a] w ∼ϕ thenitmustcontainit,sincethesequencean;n∈N iscofinalin[a] andR isa 0 ∼ϕ w convexsubring. Weconcludethat(R>0\R>0)/∼ isaninitialsegmentofP /∼ . w v ϕ K ϕ Conversely set I = {[a] |a∈R>0\R>0}. Given I ∈[P /∼ ]is, we show that w ϕ w v K ϕ there is a convex valuation ring R such that I = I. Given I, let (SI) denote w w thesettheoreticunionofthe elementsofI and−(SI)thesetofadditiveinverses. Set R = −(SI)∪R ∪(SI) . We claim that R is the required ring. Clearly, w v w I =I. Further R is convex (by its construction), and strictly contains R . We w w v leaveit to the reader,using Lemma 3.1, to verify that R is a ring,andthat R is w w principal generated by a if [a] is the last element of I. (cid:3) ∼ϕ Note that the principal rank determines the rank, that is if ordered fields have (isomorphic) principal ranks, then they have (isomorphic) ranks. 4. The difference analogue of the rank In this section, we develop a difference analogue of what has been reviewed above. Thatis, wedevelopa theoryofdifference compatiblevaluations,inanalogy to the theory of convex valuations. The automorphism will play the role that multiplication plays in the previous case. Let σ be an order preserving automorphism of K. We assume σ 6= identity, i.e. (K, < ,σ) is a non-trivial ordered difference field. Note that σ satisfies for all a,b ∈ K : v(a) ≤ v(b) if and only if v(σ(a)) ≤ v(σ(b)) and thus induces an order preserving automorphism σ and σ such that the following diagram commutes: G Γ σ - P P K K v /// v with σ (v(a)) := v(σ(a)) for all a∈P , G K ? ? σ G<0 G- G<0 and σ (v (g)) := v (σ (g)) for all g ∈G<0. vG /// vG Γ G G G ? ? σ Γ- v (G) v (G) G G 6 SALMAKUHLMANN,MICKAE¨LMATUSINSKI,ANDFRANÇOISEPOINT Nowletw be a convexvaluationonK. Sayw is σ-compatibleifforalla,b∈K : w(a)≤w(b) if and only if w(σ(a))≤w(σ(b)). The subset R := {R ∈ R; R is σ- compatible } is the σ-rank of (K,<,σ). σ w w Similarly, the subset of all convex subgroups G 6= {0} such that σ (G ) = G w G w w (σ -invariant) is the σ-rank of G. Finally, we denote by σ -Γfs the subset of final G Γ segments Γ such that σ (Γ )=Γ . w Γ w w The following analoguesof Lemmas 2.2, 2.3 and2.4 are verifiedby straightforward computations, using basic properties of valuations rings on the one hand and of automorphisms on the other (e.g. σ(A) ⊆ B if and only if A ⊆ σ−1(B) and σ(A)⊆B if and only if σ(−A)⊆−B): Lemma 4.1. The following assertions are equivalent for a convex valuation w: 1) w is σ–compatible 2) w is σ−1–compatible 3) σ(R ) = R w w 4) σ(I ) = I w w 5) σ(R>0\R>0) = R>0\R>0 w v w v 6) the map Kw → Kw defined by aw 7→ σ(a)w is well-defined and is an order preserving automorphism of Kw (with the ordering induced by Pw). We shall call R σ-compatible if any of the above equivalent conditions holds. w Lemma 4.2. The correspondence R 7→G is an order preserving bijection from w w R onto the σ -rank of G. σ G Lemma 4.3. The correspondence G 7→Γ is an order preserving bijection from w w the σ -rank of G onto σ -Γfs. G Γ We deduce fromLemmas 4.2 and 4.3 the following information. An automorphism σ is an isometry if v(σ(a)) = v(a) for all a ∈ K, equivalently σ is the identity G automorphism, and a weak isometry if σ is the identity automorphism. Every Γ isometry is a weak isometry. Note that if Γ is a rigid chain, then σ is necessarily a weak isometry. If σ is a weak isometry, then σ (v (g)) = v (σ (g)) = v (g), Γ G G G G thus g is archimedean equivalent to σ (g) for all g, and so every convex subgroup G is σ -invariant. G Corollary 4.4. If σ is a weak isometry, then R = R. σ Corollary 4.5. The correspondence R 7→minΓ is an order (reversing) isomor- w w phism from R ∩Rpr onto the chain {γ; σ (γ)=γ} of fixed points of σ . σ Γ Γ At the other extreme σ is said to be ω-increasing if σ(a)>an for all n∈N and 0 all a∈P . K Remark 4.6. Note that σ is ω-increasing if and only if σ is a strict left shift, Γ thatis, σ (γ)<γ forallγ ∈Γ.Thusifσ ω-increasingthenσ hasnofixedpoints. Γ Γ Corollary 4.7. If σ is ω-increasing then R ∩Rpr is empty. σ Recall that the Hahn group over the chain γ and components R, denoted H R, Γ is the totally orderedabelian groupwhose elements are formal sums g :=Pg 1 , γ γ THE VALUATION DIFFERENCE RANK OF AN ORDERED DIFFERENCE FIELD 7 with well-ordered support g := {γ ; g 6= 0}. Addition is pointwise and the or- γ der lexicographic. Similarly, the the field of generalized power series over the ordered abelian group G (or Hahn field over G), denoted R((G)), is the totally ordered field whose elements are formal series s := Ps tg , with well-ordered g support s := {g; s 6= 0}. Addition is pointwise, multiplication is given by the g usual convolution formula, and the order is lexicographic. Lemma 4.8. Any order preserving automorphism σ of the chain Γ lifts to an Γ order preserving automorphism σ of the Hahn group G over Γ, and σ lifts in G G turn to an order preserving automorphism σ of the Hahn field over G. Proof. Set σG(Pgγ1γ):=Pgγ1σΓ(γ) and σ(Psgtg):=PsγtσG(g). (cid:3) Corollary 4.9. Given any order type τ there exists a maximally valued non-trivial ordered difference field (K,<,σ), such that the order type of R ∩Rpr is τ. σ Proof. Set µ := τ∗, and consider e.g. the linear ordering Γ := P Q≥0, that is, µ the concatenationof µ copies of the non-negative rationals. Fix a non-trivialorder automorphismη ofQ>0. Define σ tobe the uniquelydefinedorderautomorphism Γ ofΓfixingevery0∈Q≥0ineverycopyandextendingηotherwiseoneverycopy. Set e.g. G:=H R. Then σ lifts canonically to σ on G. Now set e.g. K :=R((G)). Γ Γ G Again σ lifts canonically to an order automorphism of K, our required σ. (cid:3) G In the next section, we will exploit appropriate equivalence relations to define the principaldifferencerankandconstructdifferencefieldsofarbitrarydifferencerank. 5. The principal σ-rank Our aim now is to state and prove the analogues to Theorems 3.2, 2.5 and 2.9. However, scrutinizing the proof of Theorem 3.2 we quickly realize that in order to obtain an analogue of Lemma 3.1 (which is essential for the arguments), we need further condition on σ. Thus from now on we will assume that σ(a) ≥ a2 for all a ∈ P . It follows by induction that σn(a) ≥ a2n. Thus given n ∈ N , there K 0 exists l ∈ N such that σl(a) ≥ an. Note that our condition on σ is fulfilled for 0 ω-increasing automorphism. A convex subring R 6= R is σ-principal generated by a for a ∈ P if R is w v K w the smallest convex σ-compatible subring containing a. The σ-principal rank of K is the subset Rpr of R consisting of all σ-principal R ∈R. σ σ w The maps σ, σ and σ are order preserving and we can define the corresponding G Γ equivalence relations ∼ , ∼ and ∼ . As before we have σ σG σΓ a∼ a′ if and only if v(a)∼ v(a′) if and only if v (v(a))∼ v (v(a′)). σ σG G σΓ G Thus we have an order reversing bijection from P / ∼ onto Γ/ ∼ . Thus the K σ σΓ chain [P /∼ ]is of initial segments of P /∼ ordered by inclusion is isomorphic K σ K σ to [Γ/∼ ]fs. σΓ Lemma5.1. (i)Theequivalencerelation∼ iscoarserthanthearchimedeanequiv- σ alence relations with respect to addition and multiplication on P . K (ii) The equivalence classes of ∼ are thus closed under addition, multiplication σ and σ. 8 SALMAKUHLMANN,MICKAE¨LMATUSINSKI,ANDFRANÇOISEPOINT Proof. If a is archimedean equivalent to b then v(a) = v(b) so v(a) ∼ v(b) σG certainly and therefore a ∼ b. If a is multiplicatively equivalent to b so that σ an ≥ b and bn ≥ a for some n ∈ N , then choose l large enough so that σl(a) ≥ 0 an and σl(b) ≥ bn. Clearly, the condition on σ implies that a ∼ σ(a). Recall σ that the natural valuation on K satisfies v(x+y) = min{v(x),v(y)} if sign(x) = sign(y). One easily deduces from this fact and (i) that the equivalence classes of σ are closed under addition. Similarly, the natural valuation v on G satisfies G v (x+y)=min{v (x),v (y)} ifsign(x)=sign(y). Againoneeasilydeduces from G G G thisfactand(i)thattheequivalenceclassesofσareclosedundermultiplication. (cid:3) We can now prove: Theorem 5.2. The σ-rank R is isomorphic to [P / ∼ ]is and the principal σ- σ K σ rank Rpr is isomorphic to the subset of [P / ∼ ]is of initial segments which have σ K σ a last element. Proof. FirstwenotethatifR isaconvexσ-compatiblevaluationring,thenclearly w R>0\R>0isaninitialsegmentofP . Furthermore,ifR intersectsaσ-equivalence w v K w class [a] then it must contain it, since the sequence σ(a)n;n ∈ N is cofinal in ∼σ 0 [a] and R is a convex subring. We conclude that (R>0\R>0)/∼ is an initial ∼σ w w v σ segment of P /∼ and moreover [a] is the last class in case R is σ- principal K σ ∼σ w generatedbya. ConverselysetI = {[a] |a∈R>0\R>0}.GivenI ∈[P /∼ ]is, w σ w v K σ we show that there is a σ-compatible convex valuation ring R such that I =I. w w Given I, let (SI) denote the set theoretic union of the elements of I and −(SI) the set of additive inverses. Set R =−(SI)∪R ∪(SI) . We claim that R is w v w therequiredring. Clearly,I =I. FurtherR isconvex(byits construction),and w w strictlycontainsR . We leaveit tothe reader,using Lemma5.1, to verifythat R v w is a σ-compatible subring,andthat R is σ-principalgeneratedby a if [a] is the w ∼σ last element of I. (cid:3) Corollary 5.3. R is (isomorphic to) (Γ/∼ )fs. σ σΓ Corollary 5.4. Rpr is (isomorphic to) (Γ/∼ )∗. σ σΓ We now can ω-increasing automorphisms of arbitrary principal difference rank: Corollary 5.5. Given any order type τ there exists a maximally valued ordered field endowed with an ω-increasing automorphism of principal difference rank τ. Proof. Set µ := τ∗, and consider e.g. the linear ordering Γ := P Q, that is, the µ concatenationofµcopiesofthenon-negative. Letℓbee.g. translationby−1onQ. Defineσ tobetheuniquelydefinedorderautomorphismofΓextendingℓonevery Γ copy. It is a left shift. Set e.g. G := H R. Then σ lifts canonically to σ on G. Γ Γ G Now set e.g. K := R((G)). Again σ lifts canonically to an order automorphism G of K, our required σ. (cid:3) References [1] Engler,A.J.andPrestel,A.: Valued Fields,SpringerMonographsinMath.(2005) [2] Kuhlmann,S.: Ordered exponential fields,FieldsInstituteMonographs 12(2000) [3] Kuhlmann, S. 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