Definable completeness of P -minimal fields and applications
DDEFINABLE COMPLETENESS OF P -MINIMAL FIELDS ANDAPPLICATIONS PABLO CUBIDES KOVACSICS AND FRANC¸ OISE DELON
Abstract.
We show that every definable nested family of closed and bounded subsets of a P -minimal field K has non-empty intersection. As an application we answer a question ofDarni`ere and Halupczok showing that P -minimal fields satisfy the “extreme value property”:for every closed and bounded subset U ⊆ K and every interpretable continuous function f : U → Γ K (where Γ K denotes the value group), f ( U ) admits a maximal value. Two furthercorollaries are obtained as a consequence of their work. The first one shows that everyinterpretable subset of K × Γ nK is already interpretable in the language of rings, answeringa question of Cluckers and Halupczok. This implies in particular that every P -minimal fieldis polynomially bounded. The second one characterizes those P -minimal fields satisfying aclassical cell preparation theorem as those having definable Skolem functions, generalizing aresult of Mourgues. A celebrated result of Miller [12] shows that every o-minimal expansion of the real fieldis either polynomially bounded or the exponential function is definable in it. In contrast, itfollows from the work of Darni`ere and Halupczok [9] that every P -minimal expansion of Q p ispolynomially bounded. In fact, they showed more generally that every P -minimal expansion of Q p is relatively P -minimal , that is, every interpretable subset of Q p × Z n (where Z stands herefor the value group) is already interpretable in the language of rings. However, the questionwhether every P -minimal field is relatively P -minimal remained open. We settle this questionas a consequence of the following strong form of definable completeness for P -minimal fields,which yields in particular that all P -minimal fields are polynomially bounded. Theorem (A).
Let K be a P -minimal field. Every definable nested family of closed andbounded subsets of K has non-empty intersection. Let us start by putting the previous theorem in context. Recall that a valued field (
K, v ) isspherically complete if every nested family of balls has non-empty intersection. It is completeif the same condition holds for nested families of balls for which the set of radii is cofinal inthe value group of (
K, v ). Examples of spherically complete fields include all locally compactvalued fields and Hahn fields like K (( t R )) for any field K . The field C p is an example of acomplete but not spherically complete valued field.For first order expansions of a valued field ( K, v ), definable completeness and definablespherical completeness correspond to the analogous conditions restricted to definable nestedfamilies of balls. These are weaker conditions: without being spherically complete, C p isdefinably spherically complete in the language of valued fields L div = (+ , − , · , , , div), wherethe binary predicate div( x, y ) is interpreted by v ( x ) ≤ v ( y ). Respectively, any countableelementary substructure of C p in L div is definably complete but not complete.Since definable spherical completeness (and definable completeness) is first-order express-ible, it is not difficult to see that all p -adically closed and all algebraically closed valued fieldsare definably spherically complete in L div . It is therefore natural to ask whether these prop-erties are preserved in tame expansions of such fields. Concerning algebraically closed valuedfields, the second author showed in [10] that there are C -minimal expansions of algebraically a r X i v : . [ m a t h . L O ] J u l PABLO CUBIDES KOVASCICS AND FRANC¸ OISE DELON closed valued fields which are not even definably complete. As shows Theorem (A), this doesnot arise in P -minimal expansions of p -adically closed fields, which shows a strong differencebetween these two notions of minimality.The idea of considering definable nested families of closed and bounded sets (instead ofjust balls) can be traced back to the work of Miller [13] on definable completeness in orderedstructures. To briefly explain how Theorem (A) relates to other properties of P -minimal fieldsand how it is used to settle some open questions in this area, let us first give some informalbackground on cell decomposition and cell preparation. All formal definitions will be givenin Section 1.Let K be a P -minimal field, Γ K denote the value group of K and v : K → Γ K ∪ {∞} denote the valuation map. In [14], Mourgues characterized the class of P -minimal fields satis-fying a classical cell decomposition theorem as the class of P -minimal fields having definableSkolem functions (see later Theorem 1.1.4). Keeping the discussion informal, by classical celldecomposition we mean that every definable set can be decomposed into finitely many cellswhich are defined in the spirit of Denef’s classical definition in [11]. In his original result,Denef proved more than just a cell decomposition result as he also partitioned the domain ofa definable function into finitely many cells in which the function satisfies further properties.Although Denef did not use this terminology, we will make the distinction and call this secondand a priori stronger result about definable functions classical cell preparation .After [14], it remained open if the class of P -minimal fields having definable Skolem func-tions further satisfies a classical cell preparation theorem. In [9], Darni`ere and Halupczokcharacterized the class of P -minimal fields satisfying such a preparation theorem as the classof P -minimal fields having definable Skolem functions and satisfying the following additionalproperty (see later Theorem 1.1.7). Definition (Extreme value property) . For every closed and bounded subset U ⊆ K and everyinterpretable continuous function f : U → Γ K , f ( U ) admits a maximal value.Cell-preparation was obtained in [9] by first showing that P -minimal fields with the extremevalue property are relatively P -minimal, i.e., every interpretable subset of K × Γ nK is inter-pretable in L ring . Although the extreme value property can be easily verified for P -minimalexpansions of Q p , it remained unknown whether the extreme value property and/or relative P -minimality hold in every P -minimal field (or even in every P -minimal field with definableSkolem functions). We use Theorem (A) precisely to show that every P -minimal field has theextreme value property. Theorem (B).
Every P -minimal field has the extreme value property. As a consequence of the results in [9], we obtain thus the following.
Theorem (C).
Every P -minimal field is relatively P -minimal. As mentioned above, Theorem (C) yields that every P -minimal field is polynomially bounded(for a formal definition see the introduction of [6]). We would like to point out that it remainsopen to know whether every C -minimal expansion of an algebraically closed non-trivially val-ued field is polynomially bounded. Some partial results in this direction appear in [6], wherethe authors show that every C -minimal expansion of an algebraically closed field with valuegroup Q (e.g., C p , F p alg (( t Q ))) is polynomially bounded.The following is another corollary of Theorem (B) and the work of Darni`ere and Halupczok. Theorem (D).
Let ( K, L ) be a P -minimal field. Then the following are equivalent EFINABLE COMPLETENESS OF P -MINIMAL FIELDS AND APPLICATIONS 3 (1) ( K, L ) has definable Skolem functions(2) ( K, L ) has classical cell preparation. It is worthy to mention that P -minimal fields without definable Skolem functions do existby a result of the first author and Nguyen [8].The article will be structured as follows. In Section 1 we provide all needed backgroundon P -minimality including definitions of cells, cell decomposition and cell preparation. Wewill follow the terminology from [1] and make essential use of the clustered cell decompositiontheorem proven there. Definable nested families are introduced in Section 2, where we proveTheorem (A) and its consequences. 1. Preliminaries
Throughout this article we let K denote a p -adically closed field, that is, a field elementarilyequivalent to a finite extension of Q p in the language of rings L ring . Note that div is L ring -definable in such a field. We let Γ K denote the value group, v : K → Γ K ∪ {∞} the valuationmap, O K the valuation ring and k K the residue field. For a subset Y ⊆ Γ K and γ ∈ Γ K , wedefine Y >γ := { γ (cid:48) ∈ Y : γ < γ (cid:48) } . Concerning balls, B γ ( a ) denotes the ball around a withradius γ : B γ ( a ) := { x ∈ K : v ( x − a ) (cid:62) γ } . The topological closure of a set X ⊆ K n is denoted by cl ( X ). Let (cid:36) K be a uniformizerin K . For a positive integer m >
0, write ac m : K × → ( O K /(cid:36) mK O K ) × for the m th angularcomponent map , the unique group homomorphism such that ac m ( (cid:36) K ) = 1 and ac m ( u ) ≡ u mod (cid:36) mK for any unit u ∈ O K . Existence, uniqueness and definability of such maps was shownin Lemma 1.3 of [4]. We extend them to K by setting ac m (0) = 0. For positive integers n, m ,let Q n,m be the set Q n,m := { x ∈ K × : v ( x ) ≡ n ) and ac m ( x ) = 1 } . Note that for λ ∈ K × and x ∈ λQ n,m , λ encodes the values of v ( x )(mod n ) and ac m ( x ).For L a language extending L ring , the structure ( K, L ) is P -minimal if for every structure( K (cid:48) , L ) elementarily equivalent to ( K, L ), every L -definable subset X ⊆ K (cid:48) is L ring -definable.Hereafter, L -definable means definable with parameters in the language L . For our purposes, itwill be sometimes convenient to work in a two sorted language L where we include the valuegroup as a new sort in the language of Presburger arithmetic L Pres := (+ , − , <, ( ≡ n ) n ∈ N ∗ )(for details see [7, Section 2]). We write ( K, L ) to indicate that we work in the two-sortedlanguage. The following result of Cluckers shows in particular that L -definable subsets ofΓ K are L Pres -definable.
Theorem 1.0.1 (Cluckers [3] Lemma 2 and Theorem 6) . Let ( K, L ) be a P -minimal field.The value group is stably embedded and its induced structure is that of a pure Z -group. Inaddition, if Y ⊆ Γ mK is definable, v − ( Y ) is L ring -definable. (cid:3) Remark . As a consequence of the previous theorem, every L -definable bounded set Y ⊆ Γ K has a maximal element. Equivalently, if Y has no maximal element, it must becofinal in Γ K . This shows in particular that for P -minimal fields, the notions of definablecompleteness and definable spherical completeness are equivalent. PABLO CUBIDES KOVASCICS AND FRANC¸ OISE DELON
Cells, cell decomposition and cell preparation.
From now on we work in a P -minimal field ( K, L ). By definable we mean L -definable. We will use ‘and’ for logicalconjunction since the symbol ‘ ∧ ’ will be reserved for something else (see later Section 1.2).Let S denote a definable parameter set (i.e. a definable subset of some product of sorts whichwill play the role of parameters). A Γ K -cell condition over S is a formula of the form(E1) C ( s, γ ) := s ∈ S and α ( s ) (cid:3) γ (cid:3) β ( s ) and γ ≡ k (mod n ) , where α, β are definable functions S → Γ K , squares (cid:3) , (cid:3) may denote either < or ∅ (i.e. ‘nocondition’), γ is a variable ranging over Γ K and 0 (cid:54) k < n are two integers. If S = ∅ , then α, β simply denote elements of Γ K . A Γ K -cell over S is simply the set of elements satisfyinga Γ K -cell condition over S .Let D ⊆ Γ K be a Γ K -cell defined by a cell condition C as in (E1) over S = ∅ (hence fixing k and n ). A function g : D → Γ K is said to be linear if g ( γ ) = a ( γ − k ) n + δ, where a ∈ Z and δ ∈ Γ K . Using Theorem 1.0.1, the following is a special case of [3, Theorem1]: Theorem 1.1.1 (Cluckers) . Let ( K, L ) be a P -minimal field. Let g : Y ⊆ Γ K → Γ K be adefinable function. Then there is a finite partition of Y into Γ K -cells Y , . . . , Y n such that g | Y i is linear. (cid:3) Let us now define K -cells. A K -cell condition C over S is a formula of the form(E2) C ( s, c, t ) := s ∈ S and α ( s ) (cid:3) v ( t − c ) (cid:3) β ( s ) and t − c ∈ λQ n,m , where t and c are variables over K , α, β are definable functions S → Γ K , squares (cid:3) , (cid:3) maydenote either < or ∅ , λ ∈ K and n, m ∈ N \{ } . The variable c is called the center of C . A K -cell condition C is called a condition, resp. a 1 -cell condition if λ = 0, resp. λ (cid:54) = 0.Again, if S = ∅ then α, β denote elements of Γ K .To define K -cells we need the following additional notion. Let C be a K -cell condition over S . Given a function σ : S → K , we let C σ denote the set C σ := { ( s, t ) ∈ S × K : C ( s, σ ( s ) , t ) } . For Σ ⊆ S × K , we let C Σ denote the set C Σ := { ( s, t ) ∈ S × K : ( ∃ c )( c ∈ Σ s and C ( s, c, t )) } . A definable set Σ ⊆ S × K is called a multi-ball over S , if for every s ∈ S the fibre Σ s is theunion of finitely many balls with the same radius. For an integer (cid:96) >
0, we say a multi-ballΣ over S is of order (cid:96) , if for every s ∈ S the fibre Σ s is a union of (cid:96) disjoint balls (with thesame radius). Definition 1.1.2. A classical K -cell over S is a set of the form C σ with C a K -cell conditionover S and σ : S → K a definable function. A clustered K -cell over S is a set of the form C Σ where Σ is a multi-ball over S of order (cid:96) for some (cid:96) >
0. A K -cell over S is either a classicalor a clustered K -cell over S .It is worthy to mention that the definition of clustered K -cell given in [1] contains furtherproperties which we omitted in Definition 1.1.2 as we will not need them in our arguments(see [1, Definition 3.4] for more details). We will only need two additional properties whichwe gather in the following remark. EFINABLE COMPLETENESS OF P -MINIMAL FIELDS AND APPLICATIONS 5 Remark . Let X ⊆ S × K be a definable set and let X , . . . , X d be a cell decompositionof X over S .(1) We may suppose that every classical K -cell X i over S is defined by a cell condition C ( s, c ( s ) , t ) as in (E2) such that (cid:3) = ∅ . Indeed, when (cid:3) is < , we can view X i as aclustered K -cell given by C Σ where Σ isΣ := { ( s, y ) ∈ S × K : ( ∀ t )( C ( s, c ( s ) , t ) ↔ C ( s, y, t )) . which is a multi-ball of order 1.(2) If X i is a clustered cell C Σ where Σ is a multi-ball of order (cid:96) over S and C is a cellcondition as in (E2), we may suppose that the function β ( s ) is bounded by the radiusof some (any) ball in Σ s (see also the explanation given [1] after Definition 1.4).We can now rephrase Mourgues’ main result in [14], which shows in particular that in theabsence of definable Skolem functions, classical cells are not enough to describe definable sets.We say that a (one sorted) P -minimal field ( K, L ) has classical cell decomposition , if for everyinteger n (cid:62)
1, every definable set X ⊆ K n can be decomposed into finitely many classical K -cells. Recall that a structure M has definable Skolem functions if for every definable set X ⊆ M n +1 there is a definable function g : π ( X ) → M such that ( x, g ( x )) ∈ X for all x ∈ π ( X ), where π denotes the projection of M n +1 onto the first n coordinates. Theorem 1.1.4 (Mourgues) . Let ( K, L ) be a P -minimal field. Then the following are equiv-alent.(1) ( K, L ) has definable Skolem functions;(2) ( K, L ) has classical cell decomposition. (cid:3) The main theorem of [1] shows that clustered cells are enough to describe definable subsetsof P -minimal fields without assuming the existence of Skolem functions. Theorem 1.1.5 (Clustered cell decomposition) . Let ( K, L ) be a P -minimal field and X ⊆ S × T be a definable set where T is either K or Γ K . Then X can be decomposed into finitelymany T -cells over S . (cid:3) Let us now define what classical cell preparation is. Let C σ be a classical K -cell over S and f : C σ → K be a definable function. Suppose C is a K -cell condition over S as given by theformula in (E2). We say that f is prepared if there are an integer k and a definable function δ : S → K such that for each ( s, t ) ∈ C σ v ( f ( s, t )) = v ( δ ( s )) + kv ( t − σ ( s )) + v ( λ − k ) n . When S = ∅ , δ is assumed to be a single element of K and if λ = 0, we use as a conventionthat k = 0 and 0 = 1. Definition 1.1.6.
The structure ( K, L ) has classical cell preparation if given definable func-tions f j : X ⊆ K n → K for j = 1 , . . . , r , there exists a finite partition of X into classical K -cells C over K n − such that each function f j | C is prepared and continuous for each K -cell C .Any structure ( K, L ) having classical cell preparation also has classical cell decomposition.Classical cell preparation for p -adically closed fields ( K, L ring ) was proved by Denef in hisfoundational article [11]. It was later extended by Cluckers for the sub-analytic language( K, L an ) in [5] (see [5] or [8] for a definition). His result is slightly stronger as he shows PABLO CUBIDES KOVASCICS AND FRANC¸ OISE DELON moreover that prepared functions may be chosen to be not only continuous but even analytic(and analogously for centers).We can now formally state the result of Darni`ere and Halupczok from [9] quoted in theintroduction as follows (see more precisely [9, Theorems 1.3 and 5.3 ]).
Theorem 1.1.7 (Darni`ere-Halupczok) . Let ( K, L ) be a P -minimal field. The following areequivalent:(1) K has definable Skolem functions and satisfies the extreme value property;(2) K has classical cell preparation. (cid:3) We will further need the following result, which corresponds to [2, Lemma 3.2].
Lemma 1.1.8.
Let K be a P -minimal field and f : Γ K → K be a definable function. Then f has finite image. (cid:3) The meet-semi lattice tree of closed balls.
Let T ( K ) denote the set of closed ballsof K with radius in Γ K ∪ {∞} . Ordered by inclusion, T ( K ) is a meet semi-lattice tree. Welet x ∧ y denote the meet of two elements x, y ∈ T ( K ). Given x ∈ T ( K ), we let B ( x ) be theset of elements of K in the closed ball associated with x . We let rad : T ( K ) → Γ K ∪ {∞} denote the radius function, namely, the function sending a point x ∈ T ( K ) corresponding tothe closed ball B γ ( a ) to γ . We will often identify points of K with leaves of T ( K ) (i.e., those x ∈ T ( K ) such that rad( x ) = ∞ ).For a ∈ K , the branch of a in T ( K ), in symbols Br( a ), is the set of x ∈ T ( K ) such that a ∈ B ( x ). Every branch of T ( K ) with cofinal radii (i.e., a linearly ordered subset H of T ( K ),maximal with respect to inclusion and such that { rad( x ) : x ∈ H } is cofinal in Γ K ) can beidentified with (the branch of) an element b in the completion (cid:98) K of K . We thus extend thenotation and write Br( b ) for the branch in T ( K ) of b ∈ (cid:98) K .Note that T ( K ), ∧ and rad are interpretable (without parameters) in any valued field.Abusing of terminology, we will speak about definable subsets of T ( K ) instead of saying“interpretable subsets”.We finish this section with two slightly technical lemmas. Lemma 1.2.1.
Let I ⊆ Γ K be a cofinal subset which is in addition well-ordered. Let ( x γ ) γ ∈ I be a sequence of elements in T ( K ) such that for every c ∈ K , there is ε c ∈ Γ K such that thefunction f c : I >ε c → Γ K given by γ (cid:55)→ rad( c ∧ x γ ) is the trace on I >ε c of a definable functionon Γ K . Then, there is a cofinal subset I (cid:48) ⊆ I such that one of the following holds:(1) ( x γ ) γ ∈ I (cid:48) is constant;(2) f | I (cid:48) is strictly decreasing;(3) the set J := { γ ∈ I : ( ∀ δ ∈ I >γ )( ∃ γ (cid:48) ∈ I >δ )( ∃ γ (cid:48)(cid:48) ∈ I >γ (cid:48) )( x γ ∧ x γ (cid:48) = x γ ∧ x γ (cid:48)(cid:48) < x γ (cid:48) ∧ x γ (cid:48)(cid:48) ) } is cofinal in Γ K .Proof. For c ∈ K , since f c is the trace of a definable function, and I >ε c is cofinal in Γ K , byTheorem 1.1.1, there is a cofinal subset I (cid:48) of I such that f c restricted to I (cid:48) is linear and henceeither strictly increasing, strictly decreasing or constant. If f c | I (cid:48) is strictly increasing, then (3)would hold. If f c | I (cid:48) is strictly decreasing, then for large enough γ we have that f c ( γ ) = f ( γ ),and (2) would hold. Therefore, possibly taking a larger ε c , we may assume that the function f c is constant on I >ε c for every c ∈ K . Assuming (3) does not hold, let γ ∈ I be such that I (cid:62) γ ∩ J = ∅ . Thus, there is δ ∈ I >γ such that for every γ (cid:48) , γ (cid:48)(cid:48) ∈ I with δ < γ (cid:48) < γ (cid:48)(cid:48) either x γ ∧ x γ (cid:48) (cid:54) = x γ ∧ x γ (cid:48)(cid:48) or x γ ∧ x γ (cid:48)(cid:48) (cid:62) x γ (cid:48) ∧ x γ (cid:48)(cid:48) . EFINABLE COMPLETENESS OF P -MINIMAL FIELDS AND APPLICATIONS 7 Pick any c ∈ B ( x γ ). Since f c is constant on I >ε c , given γ (cid:48) , γ (cid:48)(cid:48) ∈ I >ε c we must have that x γ ∧ x γ (cid:48) = x γ ∧ x γ (cid:48)(cid:48) . Therefore, if m := max { δ, ε c } < γ (cid:48) < γ (cid:48)(cid:48) , then x γ ∧ x γ (cid:48) = x γ ∧ x γ (cid:48)(cid:48) = x γ (cid:48) ∧ x γ (cid:48)(cid:48) . Since the residue field is finite, this can only occur if (1) holds for I (cid:48) = I >m . (cid:3) Lemma 1.2.2.
Let A ⊆ Γ K × ( T ( K ) \ K ) be a definable set and let Y be its projection to the Γ K -coordinate. Assume that(1) Y is bounded below and cofinal in Γ K ;(2) there is a positive integer (cid:96) such that A γ has cardinality (cid:96) for each γ ∈ Y ;(3) given γ ∈ Y , rad( x ) = rad( y ) and rad( x ∧
0) = rad( y ∧ for all x, y ∈ A γ ;(4) the function g : Y → Γ K given by γ (cid:55)→ rad( x ) for some (any) x ∈ A γ is monotoneincreasing.Then, the image of the function h : Y → Γ K given by γ (cid:55)→ rad( x ∧ for some (any) x ∈ A γ ,is bounded below.Proof. Suppose for a contradiction that h ( Y ) is unbounded below. By Theorem 1.1.1, possiblyreplacing Y by a cofinal subset, we may suppose that h is linear and strictly decreasing.Consider the definable subset of K W := (cid:91) γ ∈ Y (cid:91) x ∈ A γ B ( x ) . By assumption, W contains elements of arbitrarily small valuation. Suppose D , . . . , D k forma cell decomposition of W (over ∅ ) with D i := { x ∈ K : α i (cid:3) ,i v ( x − a i ) (cid:3) ,i β i and x − a i ∈ λ i Q n i ,m i } . For x ∈ K such that v ( x ) ∈ h ( Y ) and v ( x ) < min i { v ( a i ) , α i , β i } , we have that x ∈ W if and only if for some i ∈ { , . . . , k } , (cid:3) ,i = ∅ and x ∈ λ i Q n i ,m i . For m := max { m i } , there is γ ∈ Y such that, for all γ ∈ Y >γ and all i ∈ { , . . . , k } h ( γ ) + m < min i { v ( a i ) , α i , β i , g ( γ ) } . But then, W ∩ ( B h ( γ ) (0) \ B h ( γ )+1 (0)) is the union of (cid:96) balls of radius strictly bigger than h ( γ ) + m (since g is increasing) which shows that W ∩ ( B h ( γ ) (0) \ B h ( γ )+1 (0)) (cid:54) = (cid:91) i D i ∩ ( B h ( γ ) (0) \ B h ( γ )+1 (0))for sufficiently small values of h ( γ ), which contradicts that W = (cid:83) i D i . (cid:3) Nested families and definable completeness
Definable nested families.
Although the most natural acronym for definable nestedfamilies was ‘denef’, avoiding temptation, we will use the shorter ‘ dnf ’. Definition 2.1.1.
Let X ⊆ Γ K × K be a definable set and π denote the projection onto thefirst coordinate. We say that X is a definable nested family , in short dnf , if(1) for every γ ∈ π ( X ), the fibre X γ is non-empty and(2) X γ (cid:48) ⊆ X γ for every γ, γ (cid:48) ∈ π ( X ) such that γ < γ (cid:48) .A dnf X is said to be a strict dnf if moreover(2’) X γ (cid:48) (cid:40) X γ for every γ, γ (cid:48) ∈ π ( X ) such that γ < γ (cid:48) . PABLO CUBIDES KOVASCICS AND FRANC¸ OISE DELON
Convention 2.1.2.
Let X be a dnf . For π and π the projections onto the first and secondcoordinates, we set Y := π ( X ) Z := π ( X ) . For a subset Y (cid:48) ⊆ Y , we define the subfamily X | Y (cid:48) as X | Y (cid:48) := { ( γ, x ) ∈ X : γ ∈ Y (cid:48) } . We saythat X has non-empty intersection if (cid:84) γ ∈ Y X γ (cid:54) = ∅ . We let η : Z → Y ∪ { + ∞} be the definable function given by η ( x ) : (cid:40) γ if x ∈ X γ and ( ∀ γ (cid:48) ∈ Y >γ )( x / ∈ X γ (cid:48) )+ ∞ otherwise , picking the biggest γ ∈ Y such that x ∈ X γ if existing, and + ∞ if x lies in the intersectionof all X γ . Finally, since Y is L P res -definable, there is a definable successor function on Y defined by γ (cid:55)→ γ + := min { γ (cid:48) ∈ Y : γ (cid:48) > γ } . In view of condition (1) in Definition 2.1.1, if Y has a maximal element then X has non-empty intersection. On the other hand, if Y has no maximal element, by Remark 1.0.2 Y iscofinal in Γ K . Lemma 2.1.3.
Let X be a dnf with empty intersection. Then there is a cofinal definablesubset Y (cid:48) ⊆ Y such that X | Y (cid:48) is a strict dnf .Proof. Consider the definable function µ : Y → Y defined by µ ( γ ) := min { γ (cid:48) ∈ Y : γ (cid:48) (cid:62) γ and X γ (cid:48) + (cid:40) X γ } . Since X has empty intersection, µ is well-defined. Note moreover that µ is monotone in-creasing. We show that Y (cid:48) := µ ( Y ) satisfies the desired property. Since µ ( γ ) (cid:62) γ and µ ismonotone, Y (cid:48) is cofinal. To show that X | Y (cid:48) is strict, pick µ ( γ ) , µ ( δ ) ∈ Y (cid:48) such that µ ( γ ) < µ ( δ )for γ, δ ∈ Y . This implies that µ ( γ ) < δ (indeed, arguing by the contrapositive, if δ (cid:54) µ ( γ )holds, then µ ( δ ) (cid:54) µ ( µ ( γ )) = µ ( γ )). Therefore, X µ ( δ ) ⊆ X δ ⊆ X µ ( γ ) + (cid:40) X µ ( γ ) , which showswhat we wanted. (cid:3) By cell decomposition in Γ K we obtain as a corollary Corollary 2.1.4.
Let X be a dnf with empty intersection. Then, there are integers k, n (cid:62) and α ∈ Y such that the Γ K -cell (E3) C := (cid:26) γ ∈ Γ K (cid:12)(cid:12)(cid:12)(cid:12) α < γγ ≡ k (mod n ) (cid:27) , is a subset of Y . Moreover, we may assume n (cid:62) by replacing it by n . (cid:3) The next step towards Theorem (A) is to prove the special case in which all fibres are balls,that is, to show that P -minimal fields are definably complete. Proposition 2.1.5.
Every P -minimal field is definably complete, that is, every dnf of ballshas non-empty intersection.Proof. Suppose not and let X be a dnf which is a counterexample. By Lemma 2.1.3 we mayassume that X is a strict dnf . Let δ : Y → Γ K be the definable function sending γ to theradius of the ball X γ . Replacing Y by δ ( Y ), we may assume that X γ is a ball of radius γ forall γ ∈ Y . By Corollary 2.1.4, we may furthermore assume that Y is a Γ K -cell defined as in EFINABLE COMPLETENESS OF P -MINIMAL FIELDS AND APPLICATIONS 9 (E3) for an integer n (cid:62)
2. Moreover, our assumptions imply that η ( Z ) ⊆ Y , so no element in Z has + ∞ as its image. Let γ be the minimal element in Y . Consider the definable set W := { x ∈ X γ : ( ∀ y ∈ X η ( x ) + )( η ( x ) = v ( x − y ) and ac ( x − y ) = 1) } . Let us first give a geometrical description of the set W . For q equal to the cardinality of theresidue field k K , each ball X γ is the disjoint union of exactly q subballs of radius γ + 1. Foreach γ ∈ Y , the set W contains exactly one of these subballs. Figure 1 shows a picture of W . Figure 1.
The set W corresponds to the union of the grey balls. X η ( x ) + X η ( x ) ++ B γ +1 ( y ) x yX η ( x ) = B γ ( y ) By P -minimality, the set W is L ring -definable, and thus, by Denef’s classical cell decom-position, there is a finite set of classical K -cells D such that W is the disjoint union of all D ∈ D , where D := { x ∈ K : α D (cid:3) D, v ( x − σ D ) (cid:3) D, β D and x − σ D ∈ λ D Q n D ,m D } , with α D , β D ∈ Γ K , σ D , λ D ∈ K and n D , m D ∈ N ∗ . Claim 2.1.6.
For no γ ∈ Y and no cell D ∈ D we have that X γ ⊆ D . Suppose X γ ⊆ D for some γ ∈ Y and some cell D ∈ D . This implies that X γ ⊆ W . Let x ∈ X γ be such that η ( x ) = γ and let y ∈ X γ be such that y ∈ X η ( x ) + . Then, the ball B γ +1 ( y ) ⊆ W . By our choice of Y (i.e., n (cid:62) γ + 1 / ∈ Y , which implies that W ∩ B γ +1 ( y ) (cid:54) = B γ +1 ( y ), a contradiction. This shows the claim.Fix D ∈ D and let γ D ∈ Y be such that σ D / ∈ X γ D (which exists since otherwise σ D witnesses already that X has non-empty intersection). Since Y is cofinal in Γ K we mayfurther suppose, possibly replacing γ D by a bigger value in Y , that for all x, y ∈ X γ D (E4) v ( σ D − x ) + m D = v ( σ D − y ) + m D < v ( x − y ) . Suppose that X γ D ∩ D (cid:54) = ∅ . In this case, equation (E4) implies that X γ D ⊆ D contradictingthe claim, so X σ D ∩ D = ∅ for every D ∈ D . To conclude, take γ ∈ Y such that γ > γ D forall D ∈ D , which exists since Y is cofinal in Γ K . By construction X γ has empty intersection with every cell D , which contradicts that D is a decomposition of W as W ∩ X γ (cid:54) = ∅ for every γ ∈ Y . (cid:3) We are ready to show Theorem (A) which we now rephrase:
Theorem (A).
Let X ⊆ Γ K × K be a dnf of closed and bounded sets. Then (cid:84) γ ∈ Y X γ (cid:54) = ∅ .Proof. By Lemma 2.1.3 we may assume X is a strict dnf and Y is cofinal in Γ K (otherwisethe result follows directly). Moreover, we may also suppose that Z is bounded. By clusteredcell decomposition (Theorem 1.1.5), X is equal to a finite disjoint union of K -cells X , . . . , X d over Γ K for some positive integer d . Possibly replacing Y by a cofinal subset, we may furtherassume none of these cells has empty fibres. We obtain the result by a series of cases andreductions. Step 1:
We may assume each X i is a clustered K -cell. For suppose X i is a classical K -cell.By Remark 1.1.3 and since Z is bounded, we may assume that( γ, x ) ∈ X i ⇔ γ ∈ Y and α ( γ ) < v ( t − c i ( γ )) and t − c i ( γ ) ∈ λQ n i ,m i , where c i : Y → K is a definable function. By Lemma 1.1.8, c i has finite image, so by takinga cofinal subset of Y , we may suppose c i is constant, say with value a ∈ K . Since each X γ is closed, a belongs to the intersection of X . Step 2:
By Step 1, suppose X is a clustered K -cell with associated multi-ball Σ over Γ K of smallest order (cid:96) (cid:62) X , . . . , X d , and let r (cid:62) (cid:96) . Call ( (cid:96), r ) the couple associated to the partition of X into cells X , . . . , X d . Note that if (1 ,
1) is the couple associated to some partition of X , then X is a dnf of balls and has non-empty intersection by Proposition 2.1.5. By induction on associatedcouples (in the lexicographic order), we may further suppose that any other dnf of closed andbounded sets admitting a cell decomposition into clustered K -cells with non-empty fibres andsmaller associated couple than ( (cid:96), r ) has non-empty intersection. Let A ⊆ Y × ( T ( K ) \ K )be the definable set such that for every γ ∈ Y , the fibre A γ consists precisely of the setof the (cid:96) closed balls of Σ γ . Each fibre A γ is thus a finite antichain in T ( K ) \ K such thatrad( x ) = rad( y ) for all x, y ∈ A γ (by definition of multi-ball). In particular, the definablefunction g : Y → Γ K , γ (cid:55)→ rad( x ) for some (any) x ∈ A γ is well-defined. By Theorem 1.1.1 and Corollary 2.1.4, we may further assume that g is linearand hence either constant or strictly increasing. Note that g cannot be strictly decreasingsince Z is bounded. Given c ∈ K , consider the following definable functions h min c : Y → Γ K , γ (cid:55)→ min { rad( c ∧ x ) : x ∈ A γ } h max c : Y → Γ K , γ (cid:55)→ max { rad( c ∧ x ) : x ∈ A γ } h dif c : Y → Γ K , γ (cid:55)→ h max c ( γ ) − h min c ( γ ) . Step 3:
We may suppose that for each c ∈ K there is ε c such that h dif c ( γ ) = 0 for all γ ∈ Y >ε c . Indeed, if (cid:96) = 1, then h max c = h min c and the result is trivial. So suppose (cid:96) > c ∈ K such that the set Y (cid:48) = { γ ∈ Y : h dif c ( γ ) > } is cofinal in Y . Replacing Y by Y (cid:48) , we may then suppose h dif c ( γ ) > γ ∈ Y . But then we can express Σ as a EFINABLE COMPLETENESS OF P -MINIMAL FIELDS AND APPLICATIONS 11 disjoint union Σ = Σ ∪ Σ whereΣ := { ( γ, t ) ∈ Y × K : t ∈ B ( x ) , x ∈ A γ , rad( x ∧ c ) = h min c ( γ ) } Σ := Σ \ Σ . Both Σ and Σ are multi-balls. Possibly replacing Y by a cofinal subset, we may supposethey are multi-balls of fixed orders (cid:96) , (cid:96) < (cid:96) . This shows that we can express X as a disjointunion of two clustered K -cells with multi-balls of order smaller than (cid:96) , and the result followsby induction on associated couples. This shows the claim of this step. Simplifying notation,for each c ∈ K , we let h c : Y >ε c → Γ denote the definable function h c ( γ ) = h min c ( γ ) = h max c ( γ ). Step 4:
Let I ⊆ Y be a cofinal well-ordered subset and ( x γ ) γ ∈ I be a sequence such that x γ ∈ A γ for each γ ∈ I . By Step 3, the hypotheses of Lemma 1.2.1 are satisfied. Indeed, forevery c ∈ K , function f c : I >ε c → Γ K given by γ (cid:55)→ rad( c ∧ x γ ) is the trace of the definablefunction h c above defined. Therefore, by Lemma 1.2.1, there is a cofinal subset I (cid:48) ⊆ I suchthat one of the following holds:(1) ( x γ ) γ ∈ I (cid:48) is constant;(2) f | I (cid:48) is strictly decreasing;(3) the set J := { γ ∈ I : ( ∀ δ ∈ I >γ )( ∃ γ (cid:48) ∈ I >δ )( ∃ γ (cid:48)(cid:48) ∈ I >γ (cid:48) )( x γ ∧ x γ (cid:48) = x γ ∧ x γ (cid:48)(cid:48) < x γ (cid:48) ∧ x γ (cid:48)(cid:48) ) } is cofinal in Γ K .In the remaining steps we deal with each of these cases. Step 5:
Suppose (1) holds and let x denote the constant value of ( x γ ) γ ∈ I (cid:48) . Then, the set Y (cid:48) := { γ ∈ Y : x ∈ A γ } is definable and contains I (cid:48) (so in particular, it is cofinal). Thus,without loss of generality suppose Y (cid:48) = Y . Furthermore, we may suppose (cid:96) = 1. Indeed,if (cid:96) >
1, we could express Σ as a disjoint union Σ = Σ ∪ Σ where Σ = Y × B ( x ) andΣ = Σ \ Σ . Both Σ and Σ are multi-balls of smaller order than (cid:96) , and the result will followby induction on associated couples. When (cid:96) = 1, we have that Σ = Y × B ( x ) and hence, forall γ ∈ Y t ∈ X ,γ ⇔ ( ∀ c ∈ B ( x ))( α ( γ ) < v ( t − c ) < β ( γ ) and t − c ∈ λQ n,m ) , where α, β are definable functions and n, m are integers and λ ∈ K . By Theorem 1.1.1and possibly replacing I (cid:48) by a cofinal subset, we may assume that both α and β are linearfunctions. Now, β cannot be strictly decreasing since Z is bounded (and no cell has emptyfibres). It cannot be strictly increasing either since β ( γ ) < rad( x ) (see 1.1.3). Thus, β must be constant. Similarly, α cannot be strictly decreasing since Z is bounded, nor strictlyincreasing since α ( γ ) < β ( γ ) (again, as no cell has empty fibres). Therefore, both α and β must be constant functions. But this shows that X ,γ is the same set for all γ ∈ Y , whichyields that any element in X ,γ is in the intersection of X . Step 6:
Let us show (2) cannot hold. For suppose it does. Replacing Y with Y >ε and I (cid:48) with I (cid:48) >ε , we may suppose f is the trace of the definable function h : Y → Γ K . Since(2) holds, by Theorem 1.1.1 and Corollary 2.1.4, we may further assume that h is strictlydecreasing. In particular, h ( Y ) is coinitial in Γ K . This contradicts Lemma 1.2.2. Step 7:
Suppose (3) holds. Let us first show that g is strictly increasing. Consider thedefinable subset of YY (cid:48) := { γ ∈ Y : ( ∀ ε ∈ Y >γ )( ∃ δ ∈ Y >ε )( ∃ x ∈ A γ )( ∃ y ∈ A ε )( ∃ z ∈ A δ )( x ∧ y < y ∧ z ) } . By (3), Y (cid:48) is cofinal in Y . Consider the definable subset of Γ K given by G := { rad( x ∧ y ) : x ∈ A γ , y ∈ A δ , γ, δ ∈ Y (cid:48) } . The set G is definable and, by the choice of Y (cid:48) , it has no maximal element. Then, G is cofinalin Γ K , but this cannot be the case if the radius g is constant, as any element in G will bebounded by the constant value of g . This shows, g must be strictly increasing.Replacing Y by a definable cofinal subset of Y (cid:48) , we may suppose the following: for every γ ∈ Y and every x ∈ A γ , there is b ∈ (cid:98) K such that for every γ ∈ Γ K , there are ε, δ ∈ Y with γ < ε < δ , y ∈ A ε and z ∈ A δ such that x ∧ y < y ∧ z and γ < rad( y ∧ z ) and ( x ∧ y ) ∈ Br( b ) . Indeed, if this condition does not hold for all x ∈ A γ and all γ in a final segment of Y , one canagain express Σ as a disjoint union of two multi-balls of lower order, and the result followsby induction on associated couples. Let F be the set of all such elements b in (cid:98) K . We split intwo final cases. Case 1:
Suppose some b ∈ F is isolated. Then there is x ∈ T ( K ) \ K such that F ∩ B ( x ) = { b } . The set { x ∈ T ( K ) \ K : ( ∃ γ ∈ Y )( x ∈ A γ and x < x ) } is therefore definable and linearly ordered. Letting Y (cid:48) = { γ ∈ Y : ( ∃ x ∈ A γ )( x < x ) } and x γ be the unique element in A γ such that x < x γ , the set X (cid:48) = (cid:91) γ ∈ Y (cid:48) { γ } × B ( x γ )is a dnf of balls. By Proposition 2.1.5, X (cid:48) has non-empty intersection. But the only elementin the intersection must be b , so b ∈ K . But then b belongs to the intersection of X , since theintersection is a closed set. Case 2:
No point b ∈ F is isolated. Let us show this case does not occur. Note that thecardinality of F is at least the cofinality of Γ K . Let µ be a variable of value group sort and S µ (Γ K ) denote the set of all types in the variable µ over Γ K . Note that since Γ K is stablyembedded (Theorem 1.0.1), the restriction map σ : S µ ( K ∪ Γ K ) → S µ (Γ K ) is a bijection. Let S ∞ (Γ K ) be the subset of S µ (Γ K ) consisting of all completions of the partial type at infinityover Γ K (i.e. the partial type containing the formulas { µ > γ : γ ∈ Γ K } ). An element p ( µ ) ∈ S ∞ (Γ K ) is determined by the congruences µ ≡ k (mod n ) it contains, where k, n arepositive integers. This yields that the cardinality of S ∞ (Γ K ) is 2 ℵ . For each b ∈ F , let p b ( µ )be an element of S µ ( K ∪ Γ K ) containing the set of formulas { ( ∃ x ∈ A µ )( x > y ) : y ∈ Br( b ) } ∪ { µ > γ : γ ∈ Γ K } . Let q b ∈ S ∞ (Γ K ) be the image of p b under σ . By possibly working in a large elementaryextension, we may suppose that | Γ K | is regular and strictly bigger than 2 ℵ . We obtain acontradiction by showing that | S ∞ (Γ K ) | ≥ | Γ K | > ℵ . Assume there is an increasing chain( F i ) i< | Γ K | of subsets of F such that(1) | F i | < | Γ K | for each i < | Γ K | ;(2) if b, b (cid:48) ∈ F i are different, then q b (cid:54) = q b (cid:48) . EFINABLE COMPLETENESS OF P -MINIMAL FIELDS AND APPLICATIONS 13 Setting F (cid:48) := (cid:83) i< | Γ K | F i , we have that | F (cid:48) | (cid:62) | Γ K | and q b (cid:54) = q b (cid:48) for any two elements in F (cid:48) , which shows the above bound. It remains to build the chain. Fix some element b ∈ F and set F = { b } . Suppose F j has been defined for all j < i . If i is a limit ordinal, weset F i = (cid:83) j γ . Since x ∈ X γ , there is γ (cid:48) ≥ γ such that x ∈ cl ( f − ( γ (cid:48) ))which contradicts that f ( x ) = γ . (cid:3) The following theorem corresponds to [9, Theorem 4.1].
Theorem 2.1.7.
Assume that ( K, L ) is P -minimal and satisfies the extreme value property.Then every definable set X ⊆ Γ dK × K is L ring -definable, for every d ≥ . (cid:3) Theorems (C) and (D) are direct corollaries of Theorem (B) and Theorems 2.1.7 and 1.1.7,the latter two due to Darni`ere and Halupczok in [9].We finish with a short question. In view of the clustered cell decomposition theorem forgeneral P -minimal fields, can one provide an analogue of cell preparation for general P -minimal fields? Acknowledgements:
P. Cubides Kovacsics was partially supported by the ERC projectTOSSIBERG (Grant Agreement 637027) and individual research grant
Archimedische undnicht-archimedische Stratifizierungen h¨oherer Ordnung , funded by the DFG. F. Delon waspartially supported by the Idex Universit´e de Paris.
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E-mail address : [email protected] Franc¸oise Delon, Universit´e de Paris and Sorbonne Universit´e, CNRS, Institut de Math´ematiquesde Jussieu-Paris Rive Gauche, F-75006 Paris, France.
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