O-MINIMAL PREPARATION THEOREMS L. VAN DEN DRIES AND P. SPEISSEGGER To Angus Macintyre, on his 60th birthday

1. Introduction Macintyre, Marker and Van den Dries [6] use model theory to show that globally subanalytic functions are piecewise given by terms in a certain language. Adding symbols for the exponential and logarithm function to this language, they also show that functions definable from exponentiation and globally subanalytic functions are piecewise given by terms. This piecewise definability by terms inspired Lion and Rolin [9] to find geometric proofs for what they call preparation theorems, which are sometimes more useful in applications. (A prepared function of several variables depends in a piecewise simple way on any chosen variable. An earlier preparation theorem for subanalytic functions is due to Parusinski [11]. Still earlier, Denef [2],[3] obtained results of this kind in the p-adic algebraic setting; see also Cluckers [1] for a recent p-adic analytic version and a nice application.) In Section 2 we prove a preparation theorem for functions belonging to a polynomially bounded o-minimal structure. In Section 3 we extend this to functions that are logarithmic-exponential over a polynomially bounded o-minimal structure. For a survey of o-minimality emphasizing geometry we refer to [5]. The rest of the introduction updates aspects of this survey and fixes notation and terminology. Recall from [5] that a structure on R is a family S = (Sn )n∈N such that for each n: (S1) Sn is a boolean algebra of subsets of Rn with Rn ∈ Sn , (S2) the graphs of addition and multiplication on R belong to S3 , (S3) A ∈ Sn implies A × R ∈ Sn+1 and R × A ∈ Sn+1 , (S4) A ∈ Sn+1 implies π(A) ∈ Sn , where π : Rn+1 → Rn is the projection map defined by π(x1 , . . . , xn , xn+1 ) = (x1 , . . . , xn ). Date: February 2003. 1991 Mathematics Subject Classification. Primary 03C64, 14P15. Key words and phrases. O-minimal structures, Valuation property, Preparation Theorems. Supported in part by NSF grants DMS-0100979 and DMS-9988453. 1

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In this definition we view R as an ordered field, not just as an ordered set. Such an S is said to be o-minimal if in addition: (O) the sets in S1 are exactly the subsets of R = R1 that have only finitely many connected components, that is, the finite unions of intervals and points. A key example of an o-minimal structure on R is the system of semialgebraic sets, another is the system of globally subanalytic sets (the subsets of the spaces Rn , n ∈ N, that are subanalytic in the ambient projective space Pn (R)). For other examples, summarized in an inclusion diagram, see [5], p. 146. Since then, Rolin, Speissegger and Wilkie [12] established a new source of o-minimal structures, namely those generated by quasianalytic Denjoy-Carleman classes. They solve several open problems listed on p.147 of [5], namely Problem 2 (first part), Problem 3, and Problem 5. Problem 6 of [5] asks whether Ecalle’s fonctions analysables [8] generate an o-minimal structure on R. This problem is still open, and a solution might throw light on the part of Hilbert’s 16th problem that deals with limit cycles. The theory of o-minimal structures aims at a general framework for tameness results in real analytic geometry. This includes uniform finiteness phenomena such as conjectured by Hilbert in his 16th problem on limit cycles. Indeed, Roussarie’s more precise conjectures [13] seem very much in the spirit of o-minimality. Some terminological and notational conventions: l, m, n, p and q will range over N = {0, 1, 2, 3, . . . }; given a structure S on R, a set S ⊆ Rm is said to belong to S (or to be in S) if S ∈ Sm , and a map F : S → Rn with S ⊆ Rm is said to belong to S (or to be in S) if its graph (a subset of Rm+n ) belongs to S. A covering of a set S is a collection C of subsets of S with union S. A structure S on R is said to be polynomially bounded if for each function f : R → R in S there exists an a > 0 and an n such that |f (x)| ≤ xn for all x > a. The system of semialgebraic sets, the system of globally subanalytic sets, the o-minimal structures generated by functions defined by series that are multisummable in the positive real direction (see [7]), and those generated by quasianalytic DenjoyCarleman classes (see [12]) are all polynomially bounded. All known non-polynomially bounded o-minimal structures on R are obtained by performing certain o-minimality preserving operations on polynomially bounded o-minimal structures; we refer to [5] for details.

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2. A preparation theorem for polynomially bounded o-minimal structures We fix a polynomially bounded o-minimal structure S on R. Let Λ be the set of all λ ∈ R such that the function x 7→ xλ : (0, ∞) → R is in S. Then Λ, a subfield of R, is called the field of exponents of S. Theorem 2.1. Let f1 , . . . , fl : Rn+1 → R be in S. Then there is a finite covering C of Rn+1 by sets in S, and for each set S ∈ C there are exponents λ1 , . . . , λl ∈ Λ and functions θ, a1 , . . . , al : Rn → R and u1 , . . . , ul : Rn+1 → R, all in S, such that graph θ is disjoint from S and for i = 1, . . . , l and all (x, y) = (x1 , . . . , xn , y) ∈ S we have fi (x, y) = |y − θ(x)|λi ai (x)ui (x, y),

1 |ui (x, y) − 1| < . 2

Remarks. (1) In the globally subanalytic case this holds with a more precise form for the units ui , see [9]. This extra information has nice consequences for integrals of globally subanalytic functions in [10], similar to the p-adic integrations in [1], [2]. (2) We can replace the inequality |ui (x, y) − 1| < 12 for any given  > 0 by |ui (x, y) − 1| < , with a covering depending on . This follows from the proof of the theorem. (3) The theorem can be seen as a geometric equivalent of valuationtheoretic facts established in [4] and [7], and the proof below consists of model-theoretic arguments deriving it from those facts. (Except for this proof we only use direct analytic and geometric arguments in the rest of the paper.) e = (R, S) We introduce the model-theoretic framework by setting R e and viewing R in the usual way as an L-structure where L is the firstorder language of ordered rings together with an n-ary function symbol for each function Rn → R in S; “definable” will mean “definable with parameters”. Thus by the properties (S1)–(S4) of S, the sets and maps e are exactly the sets and maps in S. definable in R The valuation theoretic facts alluded to are first transformed into a result on one-variable functions defined in arbitrary models R of the e Since R e has a unique elementary embedding into theory T := Th(R). e via this embedding with an elementary any such model R, we identify R submodel of R. The underlying set of R is denoted by R. Lemma 2.2. Let R |= T and suppose f1 , . . . , fl : R → R are definable in R. Then there is a finite covering C of R by definable subsets, and

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for each S ∈ C there is θ ∈ R \ S, and there are λ1 , . . . , λl ∈ Λ and a1 , ..., al ∈ R such that for i = 1, . . . , l and all y ∈ S we have fi (y) = |y − θ|λi ai u

1 for some u ∈ R with |u − 1| < . 2

Proof. Let R0 be an elementary extension of R with underlying set R0 , and let t ∈ R0 . The function R0 → R0 defined in R0 by the same LR formula that defines f in R is also denoted by f . By model-theoretic compactness it suffices to show: (∗) there exist λ1 , . . . , λl ∈ Λ, θ, a1 , . . . , al ∈ R and u1 , . . . , ul ∈ R0 (all depending on t) such that θ 6= t and for i = 1, . . . , l we have fi (t) = |t − θ|λi ai ui

and

1 |ui − 1| < . 2

Note that if i is such that fi (t) = 0, then the equality and inequality above hold with ai = 0 and ui = 1 (and any θ 6= t and λi ). Thus we may as well assume that fi (t) 6= 0 for all i. Let Rhti as in [4] denote the elementary submodel of R0 generated by t over R. We give Rhti its natural valuation v, with valuation ring {b ∈ Rhti : |b| ≤ r for some real r > 0}. We identify the residue field of Rhti in the usual way with the real field R. By the Valuation Property [7], and by [4], we have v(Rhti× ) = v(R× ) + Λv(t − θ) for some θ 6= t in R. With such a θ we have for i = 1, . . . , l that v(fi (t)) = λi v(t − θ) + v(ai ) with λi ∈ Λ and ai ∈ R, hence fi (t) = |t − θ|λi ai ui as required in (∗), except that instead of |ui − 1| < 1/2 we have v(ui ) = 0. But v(ui ) = 0 means that the residue class of ui is a non-zero real number. Multiplying ai by that real number and ui by its inverse, we achieve that v(ui − 1) > 0, in particular |ui − 1| < 1/2.  Since the lemma holds for all models of T , it implies a parametric version by a familiar model-theoretic compactness argument. With e this gives the preparation theorem 2.1. parameters in R We can improve on this description as follows. This improvement is not used in the remainder of the paper, and for simplicity we only state it for a single function. Theorem 2.3. Let f : Rn+1 → R be in S. Then there is a finite covering C of Rn+1 by sets in S, and for each set S ∈ C there are λ, µ ∈ Λ with µ > 0, and functions θ, a, b : Rn → R and u : Rn+1 → R,

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all in S, such that graph θ is disjoint from S and for all (x, y) ∈ S, f (x, y) = |y − θ(x)|λ a(x)u(x, y),

1 and 2 |u(x, y) − 1| ≤ |y − θ(x)|µ |b(x)|. |u(x, y) − 1|
0.) Our aim is to give an explicit description of the structure S(exp) on R that is generated by the exponential function over S, that is, the smallest structure on R that contains all sets in S and the exponential function. We achieve this aim in Theorem 3.2 and Corollary 3.3, following the treatment by Lion and Rolin [9] of the case S = {globally subanalytic sets}. For efficient formulations of these results we need some terminology. Consider a family A = (An )n∈N such that for each n: (i) An is a set of functions from Rn to R; (ii) whenever f, g ∈ An , then −f , f + g and f g belong to An ; (iii) the function on Rn taking the constant value 1 belongs to An ; (iv) for each f ∈ An the functions (x1 , . . . , xn , xn+1 ) 7→ f (x1 , . . . , xn ) and (x1 , . . . , xn , xn+1 ) 7→ f (x2 , . . . , xn+1 ) on Rn+1 are in An+1 ; (v) the functions (x, y) 7→ x and (x, y) 7→ y on R2 belong to A2 . Note that A0 is a subring of R where we identify each function in A0 with its unique real value. A function f : Rn −→ R is said to be an A-function if f belongs to An . A basic A-set in Rn is a set of the form {x ∈ Rn : f (x) = 0, g1 (x) > 0, . . . , gk (x) > 0} , where f, g1 , . . . , gk : Rn −→ R are A-functions. An A-set in Rn is a finite union of basic A-sets in Rn . Thus the A-sets in Rn form a boolean algebra of subsets of Rn . An A-cylinder in Rn+1 is a subset of Rn+1 of one of the following forms:  (f1 , f2 )S := (x, y) ∈ Rn+1 : x ∈ S, f1 (x) < y < f2 (x) ,  (−∞, f )S := (x, y) ∈ Rn+1 : x ∈ S, y < f (x) ,  (f, +∞)S := (x, y) ∈ Rn+1 : x ∈ S, f (x) < y ,  graph(f )S := (x, y) ∈ Rn+1 : x ∈ S, y = f (x) ,

where S is an A-set in Rn and f, f1 , f2 : Rn −→ R are A-functions. For cylinders of the first type we can assume that f1 < f2 on S, by shrinking S if necessary. Note that A-cylinders in Rn+1 are A-sets in Rn+1 of a

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special form. The intersection of any two A-cylinders in Rn+1 is a finite union of A-cylinders in Rn+1 , and the complement of any A-cylinder in Rn+1 is a finite union of A-cylinders in Rn+1 ; hence the finite unions of A-cylinders in Rn+1 form a boolean algebra of subsets of Rn+1 . In the expression “A-cylinder in Rn+1 ” we often omit the specification “in Rn+1 ” if the ambient space is clear from the context. We say that A admits cylindrical decomposition if for each n each A-set in Rn+1 is a finite union of A-cylinders. Cylindrical decomposition is a very strong property, as shown by the next lemma, which follows easily from the definitions and remarks above. Lemma 3.1. Suppose A admits cylindrical decomposition. Then the family ({A-sets in Rn })n∈N is a structure on R. If in addition A0 = R, then this structure is o-minimal. We are going to apply this to the family LE(S) = (LE(S)n )n∈N ; each LE(S)n is the ring of all functions on Rn obtained by superposition of functions in S with exp and log. Here exp is the usual exponential function on R, and log its inverse, extended to all of R by setting log x := 0 for x ≤ 0. A precise definition of LE(S)n is as follows. Let LLE be the language of the previous section augmented by the unary function symbols exp and log. Then each LLE -term t(v1 , . . . , vn ) defines a function (x1 , . . . , xn ) 7→ t(x1 , . . . , xn ) on Rn , and LE(S)n is the set of all functions defined by such terms. Note that the conditions (i)—(v) imposed on A are satisfied by LE(S). Since the functions (x, y) 7→ max(x, y) and (x, y) 7→ min(x, y) on R2 belong to LE(S)2 , the intersection of any two LE(S)-cylinders in Rn+1 is an LE(S)-cylinder (not just a finite union of such cylinders). Theorem 3.2. LE(S) admits cylindrical decomposition. Taking into account the previous lemma, this theorem yields the promised explicit description of S(exp) as well as o-minimality: Corollary 3.3. S(exp)n = {LE(S)-sets in Rn } for each n. Corollary 3.4. S(exp) is o-minimal. Theorem 3.2 follows easily from Theorem B in [7], obtained there by valuation theory and model-theory. Here we prefer to derive Theorem 3.2 from two more precise preparation theorems for LE(S)-functions, namely Theorems 4.11 and 5.4, which are the main results of this paper.

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4. Preparation for Logarithmic Functions In this section we prepare the LE(S)-functions in n + 1 variables that are purely logarithmic in the last variable, Theorem 4.11. It is worth noting that no induction on n will be involved, so n will be fixed throughout. For the rest of the paper we keep the conventions of the previous section, and let (x, y) = (x1 , . . . , xn , y) range over Rn+1 . Lemma 4.1. Let E ⊆ Rn+1 be an LE(S)-cylinder, and g ∈ LE(S)n+1 such that g has constant sign on E ∩ ({x} × R), for each x ∈ Rn . Then E is the disjoint union of LE(S)-cylinders E< , E= , E> on each of which g has constant sign. The characteristic function χS : Rn → R of any LE(S)-set S ⊆ Rn belongs to LE(S)n . Proof. Suppose E = (f1 , f2 )S with S an LE(S)-set and f1 , f2 ∈ LE(S)n . Then f : Rn → R defined by f (x) = 21 (f1 (x) + f2 (x)) belongs to LE(S)n . Hence S∗ := {x ∈ S : g(x, f (x)) ∗ 0} is an LE(S)-set, for each ∗ ∈ {}. Thus the lemma holds with E∗ = (f1 , f2 )S∗ , with ∗ ∈ {}. Other cases, such as E = (f, +∞)S , are handled similarly. For the assertion on characteristic functions we just consider the case that S = {x ∈ Rn : f (x) = 0 or g(x) > 0} with f, g ∈ LE(S)n . (The general case is handled similarly.) The function α : R2 −→ R given by α(s, t) = 1 if s = 0 or t > 0, and α(s, t) = 0 otherwise, belongs to S, hence χS = α(f, g) belongs to LE(S)n .  Recall that η : Rn+1 −→ R is given by η(x, y) = y. Definition 4.2. Let E ⊆ Rn+1 be an LE(S)-set. A logarithmic scale on E is a tuple (η0 , . . . , ηr ) of functions ηi : Rn+1 → R such that: (i) for each i, either ηi > 0 on E, or ηi < 0 on E; (ii) there is θ0 ∈ LE(S)n and 0 < 0 < 1 such that η0 = η − θ0 , and either 0 < |η0 | ≤ 0 |η| on E, or θ0 = 0; (iii) for i = 1, . . . , r there is θi ∈ LE(S)n and 0 < i < 1 such that ηi = log |ηi−1 | − θi , and either 0 < |ηi | ≤ i | log |ηi−1 || on E, or θi = 0. (Thus each ηi ∈ LE(S)n+1 .) Given such a logarithmic scale (η0 , . . . , ηr ), we shall throughout write yi instead of ηi (x, y), in analogy with η(x, y) = y. Until Theorem 4.11 below we fix an LE(S)-set E in Rn+1 and a logarithmic scale (η0 , . . . , ηr ) on E, with r ≥ 1. We also let θ0 , . . . , θr ∈ LE(S)n be as in the definition above. Here are some useful consequences of the definitions.

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Lemma 4.3. Let j ∈ {1, . . . , r} and define κj : Rn+1 −→ Rn+1 by κj (x, y) = (x, yj ). Then we have the following. (1) κj is injective on E. (2) κj (E) is an LE(S)-set; if E is an LE(S)-cylinder, so is κj (E). (3) If C is an LE(S)-cylinder contained in κj (E), then E ∩ κ−1 j (C) is an LE(S)-cylinder contained in E. (4) Define ζ0 , . . . , ζr−j : Rn+1 −→ R by ζ0 (x, z) := z and ζi (x, z) := log |ζi−1 (x, z)| − θi+j (x) for 1 ≤ i ≤ r − j. Then ηi = ζi−j ◦ κj for i = j, . . . , r, and (ζ0 , . . . , ζr−j ) is a logarithmic scale on κj (E). (5) If (ζ 0 , . . . , ζ p ) is a logarithmic scale on κj (E) and η j+i := ζ i ◦ κj for 0 ≤ i ≤ p, then (η0 , . . . , ηj−1 , η j , . . . , η j+p ) is a logarithmic scale on E. Proof. We leave most of this to the reader, and only sketch the case j = 1; the general case then follows by induction on j. In (2) and (3) we distinguish the cases η0 > 0 on E and η0 < 0 on E. If η0 > 0 on E, then for (x, z) ∈ Rn+1 we have (x, z) ∈ κ1 (E) ⇐⇒ (x, θ0 (x) + ez+θ1 (x) ) ∈ E, and the inverse of the bijection (x, y) 7→ (x, y1 ) : E → κ1 (E) is given by (x, z) 7→ (x, θ0 (x) + ez+θ1 (x) ). If η0 < 0 on E, then we get similar explicit expressions. Now (2) and (3) follow by considering each type of cylinder separately.  Lemma 4.4. Let 0 ≤ i ≤ r and θ ∈ LE(S)n , and assume that E is an LE(S)-cylinder. Then the sets {(x, y) ∈ E : yi < θ(x)},

{(x, y) ∈ E : yi = θ(x)},

{(x, y) ∈ E : yi > θ(x)} are LE(S)-cylinders. This follows easily by induction on i. Given M > 1, we put E>M = E>M (η0 , . . . , ηr ) := {(x, y) ∈ E : |yi | > M for i = 1, . . . , r} , Ei,M = Ei,M (η0 , . . . , ηr ) := {(x, y) ∈ E : |yi | ≤ M } ,

1 ≤ i ≤ r.

By Lemma 4.4, if E is an LE(S)-cylinder, then E>M is a LE(S)cylinder, and each Ei,M is a union of two LE(S)-cylinders. Let f and g be real valued functions and S a set contained in both their domains. We write f ∼S g, if there exists  with 0 <  < 1 such (x) that for all x ∈ S we have g(x) 6= 0 and  ≤ fg(x) ≤ 1/ (in particular, sign(f (x)) = sign(g(x)) for all x ∈ S).

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Lemma 4.5. Let 1 ≤ i ≤ r. (1) There is a constant α > 1 such that if (x, y) ∈ E, and |yi−1 | > 1, then |yi−1 | ≥ α|yi | . (2) Given any M > 1 there is a function θ ∈ LE(S)n such that |ηi−1 | ∼Ei,M θ. Proof. Take a constant a > 0 such that |ηi | ≤ a| log |ηi−1 || on E. Let (x, y) ∈ E and |yi−1 | > 1. Then |yi |/a ≤ log |yi−1 |, hence |yi−1 | ≥ α|yi | , where α := exp(1/a) > 1. For the second assertion, let (x, y) ∈ Ei,M , so −M ≤ log |yi−1 | − θi (x) ≤ M , hence exp(−M + θi (x)) ≤ |yi−1 | ≤ exp(M + θi (x)). Thus θ := exp(θi ) has the required property.



In the next two lemmas we fix exponents λ1 , . . . , λr ∈ Λ. Lemma 4.6. Let c ∈ R. Then there is M > 1 such that r |y | X 1 for all (x, y) ∈ E>M . λi log |yi | ≤ c + 2 i=1

This follows easily from the first part of the previous lemma.

Lemma 4.7. Let ψ ∈ LE(S)n and M > 1. Then (1) If 1 ≤ i ≤ r and |ηi−1 | ∼E>M ψ, then there is an N > 1 such that |ηi | ∼E>N | log ψ − θi |. (2) If |η1 | ∼E>M ψ, then there are N > 1 and ξ ∈ LE(S)n such that |η1 |λ1 · · · |ηr |λr ∼E>N ξ. Proof. Let 0 < a < b be such that a|yi−1 | ≤ ψ(x) ≤ b|yi−1 | for all (x, y) ∈ E>M . Taking logarithms and subtracting θi (x) gives log a + yi ≤ log ψ(x) − θi (x) ≤ log b + yi , for all (x, y) ∈ E>M ; this, together with Lemma 4.5 implies (1) by taking absolute values. Part (2) follows easily from (1).  Next we define the functions in LE(S)n+1 that are “purely logarithmic in the last variable”. Definition 4.8. The subring L(S)n+1,s of LE(S)n+1 is obtained by recursion on s ∈ N as follows. Let f : Rn+1 → R. Then f ∈ L(S)n+1,0 iff there are F : Rm+1 → R in S and f1 , . . . , fm ∈ LE(S)n such that f (x, y) = F (f1 (x), . . . , fm (x), y) for all (x, y); for s > 0 we declare that f ∈ L(S)n+1,s iff there are F : Rm+l −→ R in S and f1 , . . . , fm+l ∈ L(S)n+1,s−1 such that for all (x, y) we have f (x, y) = F (f1 (x, y), . . . , fm (x, y), log fm+1 (x, y), . . . , log fm+l (x, y)).

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Note that by this definition, L(S)n+1,p ⊆ L(S)n+1,q for p ≤ q , and that ηi ∈ L(S)n+1,i for i = 0, . . . , r. For the next two lemmas, let α : Rm+r+1 → R be in S and φ1 , . . . , φm in LE(S)n , and define g : Rn+1 −→ R by g(x, y) = α(φ1 (x), . . . , φm (x), y0 , y1 , . . . , yr ). (Thus g ∈ L(S)n+1,r .) There are ηij ∈ L(S)n+1,j−i for 0 ≤ i < j ≤ r such that yj = ηij (x, yi ) for all (x, y), and below we fix such ηij . Lemma 4.9. Suppose |η0 | ∼ ξ ∈ LE(S)n . Then there exists E ξ, where h ∈ L(S)n+1,r−1 such that g E = h E. Proof. On E we have

y1 = log |y0 | − θ1 (x) = log

|y0 | + log ξ(x) − θ1 (x). ξ(x)

|y0 | Let 0 <  < 1 be such that  ≤ ξ(x) ≤ 1/ on E, and define log∗ : R −→ R by log∗ (t) = log t if  ≤ t ≤ 1/, and log∗ t = 0 otherwise. Then log∗ belongs to S. We modify η1 to η1∗ : Rn+1 → R by

η1∗ (x, y) := log∗

|y0 | + log ξ(x) − θ1 (x). ξ(x)

Then η1 and η1∗ are equal on E, but η1∗ ∈ L(S)n+1,0 . For 1 < j ≤ r we define ηj∗ : Rn+1 → R by ηj∗ (x, y) := η1j (x, η1∗ (x, y)). Then the function h : Rn+1 → R defined by h(x, y) := α(φ1 (x), . . . , φm (x), η0 (x, y), η1∗ (x, y), . . . , ηr∗ (x, y)) has the desired properties.



Let η = (η 0 , . . . , η q ) be a second logarithmic scale on E such that η0 = η 0 and q ≥ 1. Let θ0 , . . . , θq be the associated functions. Put E >M := E>M (η 0 , . . . , η q ),

E i,M := Ei,M (η 0 , . . . , η q ).

Lemma 4.10. Suppose |η0 | ∼E |η 1 |µ1 · · · |η q |µq B, where µ1 , . . . , µq ∈ Λ and B ∈ LE(S)n . Then there are M > 1 and h ∈ L(S)n+1,r−1 such that g = h on E >M . Proof. Note that B > 0 on E. Let 0 < a < b be such that a|y 1 |µ1 · · · |y q |µq B(x) ≤ |y 0 | ≤ b|y 1 |µ1 · · · |y q |µq B(x) for all (x, y) ∈ E. Taking logarithms and subtracting θ1 gives log a + L + log B(x) − θ1 (x) ≤ y 1 ≤ log b + L + log B(x) − θ1 (x)

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Pq for (x, y) ∈ E, where L := i=1 µi log |y i |. By Lemma 4.6, we can choose M > 1 such that | log a + L| ≤ |y 1 |/2 and | log b + L| ≤ |y 1 |/2 for all (x, y) ∈ E >M . It follows that 1/2 ≤ | log B(x)−θ1 (x)|/|y 1 | ≤ 3/2 for (x, y) ∈ E >M , hence |η 1 | ∼E >M | log B − θ1 |. Increasing M if necessary, Lemma 4.7(2) and η0 = η 0 yield |η0 | ∼E >M ξ with ξ ∈ LE(S)n . Now Lemma 4.9 provides h ∈ L(S)n+1,r−1 such that g = h on E >M .  Call U : Rk → R a unit if |U − 1| < 1/2 on Rk (hence U > 0 on Rk ). Theorem 4.11. Let f1 , . . . , fl ∈ L(S)n+1,s . Then there is a finite covering C of Rn+1 η6=0 by LE(S)-cylinders, and for each C ∈ C there are r ≤ s, a logarithmic scale (η0 , . . . , ηr ) on C, exponents λi0 , . . . , λir in Λ, functions Ai ∈ LE(S)n and units Ui ∈ LE(S)n+1 (i = 1, . . . , l) such that for each such i fi = |η0 |λi0 · · · |ηr |λir · Ai · Ui

on C.

Corollary 4.12. Let f1 , . . . , fl ∈ L(S)n+1,s and σ1 , . . . , σl ∈ {−1, 0, 1}. Then the set {(x, y) : y 6= 0, sign f1 (x, y) = σ1 , . . . , sign fl (x, y) = σl } is a finite union of LE(S)-cylinders in Rn+1 . Proof. Take C and all associated data as in Theorem 4.11. Then for each C ∈ C, sign fi (x, y) = sign Ai (x) for i = 1, . . . , l and all (x, y) ∈ C, since the other factors on the right hand side in the display are positive. Hence we can always refine the covering C in the theorem to a finite covering of Rn+1 η6=0 by LE(S)-cylinders such that each fi has constant sign on each cylinder of this refinement.  Proof of Theorem 4.11. By induction on s; the case s = 0 follows from Theorem 2.4. So assume that s > 0 and that the theorem holds for lower values of s. We first derive three claims from this inductive assumption. In all claims we are given r ∈ {1, . . . , s}, a logarithmic scale (η0 , . . . , ηr ) on an LE(S)-cylinder C ⊆ Rn+1 η6=0 and functions φ1 , . . . , φm ∈ LE(S)n . Put n φ := (φ1 , . . . , φm ) : R → Rm . Claim 1. Let α1 , . . . , αl : Rm+r −→ R be in S. Then there is a finite covering P of C by LE(S)-cylinders, and for each P ∈ P there are q ∈ {1, . . . , r}, a logarithmic scale (η 0 , . . . , η q ) on P with η 0 = η0 , exponents µi1 , . . . , µiq in Λ, a function Bi ∈ LE(S)n and a unit Ui ∈ LE(S)n+1 for i = 1, . . . , l such that for each such i αi (φ, η1 , . . . , ηr ) = |η 1 |µi1 · · · |η q |µiq · Bi · Ui

on P.

Proof. With (ζ0 , . . . , ζr−1 ) as in Lemma 4.3, we note that κ1 (C) is an LE(S)-cylinder. The function αi (φ, ζ0 , . . . , ζr−1 ) belongs to L(S)n+1,r−1

13

for each i, so the inductive hypothesis applies. This gives a finite covering Q of κ1 (C) by LE(S)-cylinders and for each Q ∈ Q a logarithmic scale (ζ 0 , . . . , ζ q−1 ) with 1 ≤ q ≤ r, exponents µi1 , . . . , µiq ∈ Λ, a function Bi ∈ LE(S)n and a unit Ui ∈ LE(S)n+1 for i = 1, . . . , l, such that for each such i αi (φ, ζ0 , . . . , ζr−1 ) = |ζ 0 |µi1 · · · |ζ q−1 |µiq Bi Ui

on Q

Claim 1 now follows from part (5) of Lemma 4.3.



For G ⊆ Rm+r+1 and S ⊆ C, put SG := {(x, y) ∈ S : (φ(x), y1 , . . . , yr , y0 ) ∈ G} . Claim 2. Let G ⊆ Rm+r+1 and α : Rm+r+1 −→ R belong to S and put g := α(φ, η0 , . . . , ηr ). Moreover, let S ⊆ C be an LE(S)-cylinder, q ∈ {0, . . . , r}, (η 0 , . . . , η q ) a logarithmic scale on S with η 0 = η0 , β : Rk+q −→ R in S and ψ1 , . . . , ψk ∈ LE(S)n , and assume that |η0 | ∼SG β(ψ, η 1 , . . . , η q ), where ψ := (ψ1 , . . . , ψk ) : Rn −→ Rk . Then there is a finite covering T of S by LE(S)-cylinders such that for each T ∈ T we have g|TG = h|TG for some h ∈ L(S)n+1,r−1 . Proof. Note that g ∈ L(S)n+1,r and β(ψ, η 1 , . . . , η q ) ∈ L(S)n+1,q . We proceed by induction on q. If q = 0, Claim 2 follows from Lemma 4.9, so we assume that q > 0 and Claim 2 holds for lower values of q. By Claim 1 with β in place of α1 , . . . , αk , and ψ and η 0 , . . . , η q in place of φ and η0 , . . . , ηr we reduce to the situation that we have exponents µ1 , . . . , µq ∈ Λ and a function B ∈ LE(S)n such that (4.1)

|η0 | ∼SG |η 1 |µ1 · · · |η q |µq · B.

(This reduction involves replacing S by each cylinder of some finite covering of S by LE(S)-cylinders, and (η 0 , . . . , η q ) by a logarithmic scale on each of those cylinders, without increasing q.) Hence Lemma 4.10 yields M > 1 such that g = h on (S >M )G for some h ∈ L(S)n+1,r−1 . Since S = S >M ∪ S 1,M ∪ · · · ∪ S q,M , it remains to show that for each j ∈ {1, . . . , q} the claim holds with S j,M in place of S. So after renaming we have j ∈ {1, . . . , q} such that |η j | ≤ M on S. Then Lemma 4.5(2) gives ξ ∈ LE(S)n such that |η j−1 | ∼S ξ. If j = 1, Claim 2 follows from Lemma 4.9 (because η0 = η 0 ), so we assume below that j ≥ 2. Define κ : Rn+1 −→ Rn+1 by κ(x, y) := (x, y j−1 ) and ζ 0 , . . . , ζ q−j+1 : Rn+1 −→ R by ζ 0 (x, z) := z and ζ i (x, z) := log |ζ i−1 (x, z)| − θi−j+1 (x) for 1 ≤ i ≤ q − j + 1. So η i = ζ i−j+1 ◦ κ for i = j − 1, . . . , q and

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(ζ 0 , . . . , ζ q−j+1 ) is a logarithmic scale on κ(S) by part (4) of Lemma 4.3. Then |ζ 0 | ∼κ(S) ξ, so Lemma 4.9 gives h ∈ L(S)n+1,q−j such that (4.2)

|ζ 0 |µj−1 · · · |ζ q−j+1 |µq B = h

on κ(S).

The inductive hypothesis of the theorem applied to h yields a reduction to the case where we have a p ≤ q − j, a logarithmic scale (ζb0 , . . . , ζbp ) b ∈ LE(S)n and a unit on κ(S), exponents ν0 , . . . , νp ∈ Λ, a function B b ∈ LE(S)n+1 such that U bU b on κ(S). (4.3) h = |ζb0 |ν0 · · · |ζbp |νp B

(This reduction uses part(3) of Lemma 4.3 and involves replacing S by each cylinder of some finite covering of S by LE(S)-cylinders.) Put ηbi := ζbi−j+1 for j−1 ≤ i ≤ j−1+p. Then (η 0 , . . . , η j−2 , ηbj−1 . . . . , ηbj−1+p ) is a logarithmic scale on S by part (5) of Lemma 4.3. By (4.1), (4.2) and (4.3) we have b |η0 | ∼S |η 1 |µ1 · · · |η j−2 |µq |b ηj−1 |ν0 · · · |b ηj−1+p |νp B. G

The function on the right hand side has the form b B, b η , . . . , η , ηbj−1 , . . . , ηbj−1+p ) β( 1

j−2

with βb : R1+(j−1+p) −→ R in S and j − 1 + p < q. Now the inductive assumption on q yields the desired conclusion.  Claim 3. Let D ⊆ Rm+r+1 be a cell in S. Then CD is a finite union of LE(S)-cylinders. Proof. Below, u = (u1 , . . . , um ) ranges over Rm and v = (v1 , . . . , vr , v0 ) over Rr+1 ; we set v 0 = (v1 , . . . , vr ). Put D0 = π(D), where π : Rm+r+1 −→ Rm+r is given by π(u, v) = (u, v 0 ), and put CD0 = {(x, y) ∈ C : (φ(x), y1 , . . . , yr ) ∈ D0 } . Let κ := κ1 and (ζ0 , . . . , ζr−1 ) be as in Lemma 4.3, with C in place of E. Define a : Rn+1 → R by a = χD0 (φ, ζ0 , . . . , ζr−1 ), so a ∈ L(S)n+1,r−1 . By the inductive hypothesis and (the proof of) Corollary 4.12 the set S := {(x, z) : a(x, z) > 0} = {(x, z) : a(x, z) = 1} is a finite union of LE(S)-cylinders. Since CD0 = C ∩ κ−1 (S), it follows by Lemma 4.3 that CD0 is a finite union of LE(S)-cylinders. Replacing C by each of these cylinders we may, for the purpose of proving the claim, assume that CD0 = C. (Further reductions of this kind are made later in the proof.) We distinguish several cases. Case a: D = graph(α|D0 ) where α : Rm+r → R belongs to S and α|D0 is continuous. Put g = gα := η0 − α(φ, η1 , . . . , ηr ) : Rn+1 → R. We will

15

show that there is a finite covering Pα of C by LE(S)-cylinders such that g has constant sign on each P ∈ Pα . (Then the claim follows by noting that CD is the union of those P ∈ Pα on which g vanishes.) We can further reduce to the case that α has constant sign on D0 . If α vanishes identically on D0 , then g = η0 on C, so g has constant sign on C. Suppose α > 0 on D0 . (The case α < 0 on D0 is handled similarly.) Let G := ( 12 α|D0 , 2α|D0 ), so D ⊆ G, CD ⊆ CG , and 12 α < η0 < 2α on CG ; in particular |η0 | ∼CG α(φ, η1 , . . . , ηr ). By Claim 2 we can reduce to the situation where g|CG = h|CG for some h ∈ L(S)n+1,r−1 . Now the inductive hypothesis of the theorem and (the proof of) Corollary 4.12 produce a finite covering Q of C by LE(S)-cylinders such that h has constant sign on each Q ∈ Q. The continuity of α|D0 and CD0 = C yield that g|C ∩({x}×R) is continuous for each x. Since CD = {(x, y) ∈ C : g(x, y) = 0} = {(x, y) ∈ CG : h(x, y) = 0} , g has constant sign on Q ∩ ({x} × R) for each Q ∈ Q and each x. Lemma 4.1 applied to g and each Q ∈ Q thus produces a refinement P of Q such that g has constant sign on each P ∈ P. This finishes the proof of Claim 3 for Case a. Case b: D = (α1 |D0 , α2 |D0 ) where α1 , α2 : Rm+r −→ R belong to S, have continuous restrictions to D0 , and α1 < α2 on D0 . By Case a, there are finite coverings Pα1 and Pα2 of C by LE(S)-cylinders such that gα1 has constant sign on each P ∈ Pα1 and gα2 has constant sign on each P ∈ Pα2 . Let P be any finite covering of C by LE(S)-cylinders that is compatible with each element of Pα1 ∪ Pα2 ; then CD is a finite union of elements of P. The other cases are treated similarly to Case b. This finishes the proof of Claim 3.  We now return to the proof of Theorem 4.11. Applying the inductive hypothesis to the functions in L(S)n+1,s−1 involved in expressing the functions fi as elements of L(S)n+1,s , we obtain a finite covering C of Rn+1 η6=0 by LE(S)-cylinders, and for each cylinder C ∈ C a number r ∈ {1, . . . , s}, a logarithmic scale (η0 , . . . , ηr−1 ) on C, an m, functions Fi : Rm+r+1 −→ R in S for i = 1, . . . , l and a tuple φ = (φ1 , . . . , φm ) ∈ (LE(S)n )m such that for i = 1, . . . , l and (x, y) ∈ C, fi (x, y) = Fi (φ(x), y1 , . . . , yr−1 , yr , y0 ),

where yr = log |yr−1 |.

We now focus attention on one particular C ∈ C with corresponding η0 , . . . , ηr−1 , m and F1 , . . . , Fl , φ as above, and we write yr = log |yr−1 |

16

and ηr = log |ηr−1 |. If r < s, then the above identities imply fi |C = gi |C with g1 , . . . , gl ∈ L(S)n+1,s−1 , and the inductive assumption applies to g1 , . . . , gl . Thus from now on we may and shall assume that r = s. As before u = (u1 , . . . , um ) ranges over Rm , v = (v1 , . . . , vr , v0 ) over Rr+1 , and v 0 := (v1 , . . . , vr ). By Lemma 4.4 and the inductive hypothesis, we may also assume that either ηr > 0 on C, or ηr < 0 on C; in particular, (η0 , . . . , ηr ) is a logarithmic scale on C. Theorem 2.4 for F1 , . . . , Fl produces a finite partition D of the set {(u, v) ∈ Rm+r+1 : vi 6= 0, i = 0, . . . , r} by cells in S such that for each D ∈ D there is θ : Rm+r −→ R in S with either θ = 0, or 0 < |v0 − θ(u, v 0 )| ≤ |v0 |/2 for all (u, v) ∈ D, and there are exponents µ1 , . . . , µl ∈ Λ, functions α1 , . . . , αl : Rm+r −→ R and U1 , . . . , Ul : Rm+r+1 −→ R, all in S, such that 1 Fi (u, v) = |v0 − θ(u, v 0 )|µi αi (u, v 0 )Ui (u, v), |Ui (u, v) − 1| < 2 for i = 1, . . . , l and all (u, v) ∈ D. By Claim 3 we reduce to the case that C = CD , where D ∈ D. Thus it remains to examine the following two cases: Case 1: 0 < |v0 − θ(u, v 0 )| ≤ |v0 |/2 for all (u, v) ∈ D. Since C = CD , it follows that |η0 | ∼C θ(φ, η1 , . . . , ηr ). Applying Claim 2 to each fi in place of g and taking a common refine ment of the coverings obtained, we reduce to the case that fi C = hi C with hi ∈ L(S)n+1,r−1 , for i = 1, . . . , l. Applying the inductive hypothesis once more to h1 , . . . , hl yields the desired conclusion. Case 2: θ = 0; so for i = 1, . . . , l we have λi0 ∈ Λ, αi : Rm+r −→ R in S and a unit Ui : Rm+r+1 −→ R in S such that Fi (u, v) = |v0 |λi0 αi (u, v 0 )Ui (u, v) for all (u, v) ∈ D. Since C = CD , it follows that for each i and all (x, y) ∈ C, fi (x, y) = |y0 |λi0 αi (φ(x), y1 , . . . , yr )Ui (φ(x), y0 , . . . , yr ). The theorem now follows from Claim 1 applied to α1 , . . . , αl .



5. Preparation for Functions in LE(S) The notations and conventions of the previous section remain in force, and n is fixed throughout. We need to consider the level of nestedness of exp in functions in LE(S)n+1 with respect to the last variable. To define such a level (5.3 below), we fix distinct formal variables v1 , . . . , vn+1 .

17

In this section we heavily use the syntax of terms. Throughout, “term” means “LLE -term in which no other variables than v1 , . . . , vn+1 occur”, and t denotes a term. We assume familiarity with the notion (for terms s) of “occurrence of s in t”, with the fact that different occurrences of s in t do not overlap, and with the operation of substituting a term s0 for all occurrences of s in t; if s occurs in t we call s a subterm of t. Note that t defines a function (x, y) 7→ t(x, y) in LE(S)n+1 , which we also denote by t when the context demands that t be a function. Similarly, when dealing with a function f : Rp → R in S, the p-ary function symbol of L that corresponds to the function f is denoted by f as well. Definition 5.1. The exponential level of t is the number e(t) ∈ N defined inductively as follows: (i) if t is a variable vi or a constant symbol, then e(t) = 0; (ii) if t = g(t1 , . . . , tm ) where g is an m-ary function symbol of L, m ≥ 1, and t1 , . . . , tm are terms, then e(t) = max {e(t1 ), . . . , e(tm )} ; (iii) if t = log(s), where s is a term, then e(t) = e(s); (iv) if t = exp(s), where s is a term, then ( e(s) + 1 if vn+1 occurs in s, e(t) = e(s) otherwise. Remark. If e(t) = 0, then the function t belongs to the ring [ L(S)n+1,s . L(S)n+1 := s

Every function in L(S)n+1 arises in this way from some t with e(t) = 0. We call t exponential if t = exp(s) for some subterm s of t in which vn+1 occurs. It follows by induction on e(t) that t has an exponential subterm if and only if e(t) > 0, in which case t has an exponential subterm s with e(s) = e(t). Put E(t) := {s : s is an exponential subterm of t and e(s) = e(t)} . Thus |E(t)| counts, without multiplicity of occurrence, the number of exponential subterms of t of maximal exponential level. Lemma 5.2. Suppose e(t) > 0, s1 , . . . , sq are distinct subterms of t, q ≥ 1, r1 , . . . , rp are terms, C ⊆ Rn+1 is an LE(S)-cylinder, and θ : Rp+q−1 −→ R belongs to S, such that

18

(i) exp si ∈ E(t) for i = 1, . . . , q, (ii) e(rj ) < e(t) for j = 1, . . . , p, and (iii) exp sq ∼C θ(r1 , . . . , rp , exp s1 , . . . , exp sq−1 ). Then there is a term t0 such that t C = t0 C, and either e(t0 ) < e(t), or e(t0 ) = e(t) and E(t0 ) ⊆ E(t) \ {exp sq }. Proof. Let σ be the term θ(r1 , . . . , rp , exp s1 , . . . , exp sq−1 ), and let 0 < a < b be such that log a ≤ sq (x, y)−log σ(x, y) ≤ log b for all (x, y) ∈ C. Let exp∗ : R → R be defined by exp∗ (z) := exp(z) if log a 0≤ z ≤ ∗ 0 log b and exp (z) := 0 otherwise. Then t C = t C where t is the term obtained from occurrences of exp sq in t by
t by replacing all ∗ ∗ σ · exp sq − log σ . Clearly e(σ · exp (sq − log σ)) < e(exp sq ).  We define the logarithmic depth ldt (s) ∈ N of a term s in t inductively as follows (where t1 , ..., tm , t0 denote terms): (i) if t is a variable vi or a constant symbol, then ldt (s) = 0; (ii) if t = g(t1 , . . . , tm ) where g is an m-ary function symbol of L, m ≥ 1, then ldt (s) = max {ldt1 (s), . . . , ldtm (s)}; (iii) if t = exp(t0 ), then ldt (s) = ldt0 (s); (iv) if t = log(t0 ), then ldt (s) = ldt0 (s) + 1 if s occurs as a subterm in t0 , and ldt (s) = 0 otherwise. It follows easily that ldt (s) = 0 if s is not a proper subterm in t, in particular, ldt (t) = 0. We need to measure the logarithmic depth of the exponential subterms of t of maximal exponential level; so we put l(t) := max {ldt (s) : s ∈ E(t)} if e(t) > 0,

l(t) := 0 if e(t) = 0

and LE(t) := {s ∈ E(t) : ldt (s) = l(t)} . Definition 5.3. We define the logarithmic-exponential nestedness of t with respect to vn+1 to be the quadruple
n(t) := e(t), |E(t)|, l(t), |LE(t)| ∈ N4 . We can now prove the main result of this section. Theorem 5.4. Suppose e(t) > 0. Then there is a finite covering C of Rn+1 η6=0 by LE(S)-cylinders such that (∗)t,C for each C ∈ C there exist terms σ, τ and u such that e(σ) = 0, e(τ ) < e(t) and for all (x, y) ∈ C, t(x, y) = σ(x, y) · exp τ (x, y) · u(x, y),

1 |u(x, y) − 1| < . 2

19

Proof. Let n(t) = (e, m, l, q) ∈ N4 , where N4 is lexicographically ordered. We assume inductively that the proposition holds for values of n(t) lower than (e, m, l, q). Since e > 0 we have m, q > 0. Case 1: l = 0. Then m = q, and there is F : Rp+q −→ R in S and there are subterms r1 , . . . , rp and distinct subterms s1 , . . . , sq of t, all of exponential level strictly less than e, such that LE(t) = E(t) = {exp s1 , . . . , exp sq } and
t(x, y) = F r1 (x, y), . . . , rp (x, y), exp s1 (x, y), . . . , exp sq (x, y)

for all (x, y) ∈ Rn+1 . Since n(t) = n(F (r1 , . . . , rp , exp s1 , . . . , exp
sq )), we assume below that t is the term F r1 , . . . , rp , exp s1 , . . . , exp sq . We also write r = (r1 , . . . , rp ), and we let z = (z1 , . . . , zp ) range over Rp , w = (w1 , . . . , wq ) range over Rq and put w0 = (w1 , . . . , wq−1 ). Using Theorem 2.4 we take a finite covering of Rp+q by cells A in S each of which satisfies one of the following three conditions: (A1) wq = 0 for all (z, w) ∈ A; (A2) there is a function θ : Rp+q−1 −→ R in S such that 0 < |wq − θ(z, w0 )| ≤ |wq |/2 for all (z, w) ∈ A; (A3) wq 6= 0 for all (z, w) ∈ A, and there are λ ∈ Λ and functions a : Rp+q−1 −→ R and U : Rp+q −→ R, both in S, such that F (z, w) = |wq |λ a(z, w0 )U (z, w) and |U (z, w) − 1| < 21 for all (z, w) ∈ A. For each A in this finite covering we put 
B := (x, y) ∈ Rn+1 r(x, y), exp s1 (x, y), . . . , exp sq (x, y) ∈ A . η6=0 : It clearly suffices to show that for each A in this finite covering the corresponding B has a finite covering CB by LE(S)-cylinders such that (∗)t,CB holds. We now fix one such A and its corresponding B. Claim. B is a finite union of LE(S)-cylinders. Proof. Assume that A = graph(g A0 ), where A0 ⊆ Rp+q−1 is a cell in S and g : Rp+q−1 −→ R is a function in S such that g A0 is continuous; other possibilities for A are handled similarly. Let h : Rp+q−1 −→ R be the characteristic function of A0 , so h is in S and n(h(r, exp s1 , . . . , exp sq−1 )) < n(t). Using the equivalence (z, w0 ) ∈ A0 ⇐⇒ h(z, w0 ) > 0, the inductive hypothesis and Corollary 4.12 yield a finite covering P of Rn+1 η6=0 by LE(S)-cylinders such that
 B 0 := (x, y) ∈ Rn+1 r(x, y), exp s1 (x, y), . . . , exp sq−1 (x, y) ∈ A0 η6=0 : is a union of cylinders in P. Since B ⊆ B 0 , it is enough to show that for each P ∈ P, the set B ∩ P is a finite union of LE(S)-cylinders. We

20

therefore fix a P ∈ P with P ⊆ B 0 . Then B ∩ P = {(x, y) ∈ P : t1 (x, y) > 0, t2 (x, y) = 0} , where t1 := g(r, exp s1 , . . . , exp sq−1 ) and t2 := log g(r, exp s1 , . . . , exp sq−1 ) − sq . Clearly n(ti ) < n(t) for i = 1, 2. By the inductive hypothesis and after replacing P by each of the cylinders of a suitable finite covering of P by LE(S)-cylinders, there are terms σ1 and σ2 such that e(σ1 ) = e(σ2 ) = 0 and B ∩ P = {(x, y) ∈ P : σ1 (x, y) > 0, σ2 (x, y) = 0} . Since σ1 , σ2 ∈ L(S)n+1 , Corollary 4.12 implies that B ∩ P is a finite union of LE(S)-cylinders.  To finish the treatment of Case 1, we use the claim and take a finite covering of B by nonempty LE(S)-cylinders; it suffices to show that each cylinder C in this covering has a finite covering C by LE(S)cylinders such that (∗)t,C holds. Fix such a cylinder C. Since C is not empty, condition (A1) is not satisfied. Suppose (A2) holds. Then by Lemma 5.2 we have t C = t0 C for some term t0 (v) such that n(t0 ) < n(t). The inductive hypothesis yields a finite covering C 0 of Rn+1 η6=0 by LE(S)-cylinders such that (∗)t0 ,C 0 holds. Since C is itself an LE(S)-cylinder, C 0 induces a finite covering C of C by LE(S)-cylinders such that (∗) as required. t,C holds, 0 Suppose (A3) holds. Then t C = t C, where t0 := exp(λsq ) · a(r, exp s1 , . . . , exp sq−1 ) · U (r, exp s1 , . . . , exp sq ).

Since t00 := a(r, exp s1 , . . . , exp sq−1 ) satisfies n(t00 ) < n(t), the inductive hypothesis gives a finite covering C of C such that (∗)t00 ,C holds. It follows immediately that (∗)t,C holds as well. This concludes the treatment of Case 1. Case 2: l > 0. Let s1 , . . . , sq be the distinct subterms of t such that LE(t) = {exp s1 , . . . , exp sq }, and let o(t) be the number of occurrences of exp sq in t; we proceed by an auxiliary induction on o(t). There are a subterm s of t, a function F : Rp+q −→ R in S and terms r1 , . . . , rp such that (i) e(rj ) < e for j = 1, . . . , p, and ldt (s) = l, (ii) exp sq is a subterm of s and log s is a subterm of t, and (iii) s(x, y) = F (r, exp s1 , . . . , exp sq )(x, y) for all (x, y) ∈ Rn+1 . Replacing all occurrences of s in t by F (r, exp s1 , . . . , exp sq ) does not change n(t) or o(t), so we may and shall assume that s is the term F (r, exp s1 , . . . , exp sq ).

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Using Theorem 2.4 we take a finite covering of Rp+q by cells A in S each of which satisfies one of the conditions (A1), (A2), (A3) as in Case 1. To each such A we associate a set B ⊆ Rn+1 as in Case 1. It suffices to show that for each A in this finite covering the corresponding B has a finite covering CB by LE(S)-cylinders such that (∗)t,CB holds. We now fix one such A and its corresponding B. As in Case 1 we obtain a finite covering of B by nonempty LE(S)-cylinders; it suffices to show that each cylinder C in this covering has a finite covering C by LE(S)-cylinders such that (∗)t,C holds. Fix such a cylinder C. Since C is not empty, condition (A1) is not satisfied. Suppose (A2) holds. Then by Lemma 5.2 we have s C = s0 C where s0 is a term such that e(s0 ) ≤ e(s) and E(s0 ) ⊆ E(s) \ {exp sq }. Let t0 be 0 the term obtained by replacing every occurrence of0 s in t0 by s . Then 0 t C = t C, and by the above, either exp sq ∈ LE(t ), n(t ) ≤ n(t) and o(t0 ) < o(t), or exp sq ∈ / LE(t0 ) and n(t0 ) < n(t). In both cases, we are done by the inductive hypothesis. Assume (A3) holds. Then log s C = s0 C where s0 := λsq +log a(r, exp s1 , . . . , exp sq−1 )+log U (r, exp s1 , . . . , exp sq ).

We let t0 be the term obtained from t by replacing every occurrence of log s in t by s0 . Since log U is in S, we are in a similar situation as in the previous paragraph: either exp sq ∈ LE(t0 ), n(t0 ) ≤ n(t) and o(t0 ) < o(t), or exp sq ∈ / LE(t0 ) and n(t0 ) < n(t). Either way, we are done by the inductive hypothesis.  Combining Theorems 4.11 and 5.4 yields: Corollary 5.5. Let t1 , . . . , tl be terms. Then there is a finite covering C of Rn+1 η6=0 by LE(S)-cylinders, and for each C ∈ C there are a logarithmic scale (η0 , . . . , ηm ) on C, exponents λi0 , . . . , λim in Λ, a term si , a function Ai ∈ LE(S)n and a unit ui ∈ LE(S)n+1 for i = 1, . . . , l such that for each such i ti (x, y) = |y0 |λi0 · · · |ym |λim Ai (x) exp(si (x, y))ui (x, y) for all (x, y) ∈ C, and e(si ) < e(ti ) if e(ti ) > 0, and si = 0 if e(ti ) = 0. 6. Proof of Theorem 3.2, and concluding remarks Theorem 3.2 asserts that each LE(S)-set in Rn is a finite union of LE(S)-cylinders in Rn . To prove this assertion we proceed by induction on n. The case n = 0 is trivial. Assume the assertion holds for a certain n. To derive the assertion for n replaced by n + 1 we consider

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any functions f1 , . . . , fl ∈ LE(S)n+1 and signs σ1 , . . . , σl ∈ {−1, 0, 1}; it suffices to show that then the set {(x, y) : sign f1 (x, y) = σ1 , . . . , sign fl (x, y) = σl }, where (x, y) = (x1 , . . . , xn , y) ranges over Rn+1 , is a finite union of LE(S)-cylinders in Rn+1 . Representing f1 , . . . , fl by terms t1 , . . . , tl in LLE , Corollary 5.5 yields a finite covering C of Rn+1 η6=0 by LE(S)cylinders, such that for each C ∈ C we have A1 , . . . , Al ∈ LE(S)n with sign fi (x, y) = sign Ai (x) on C for i = 1, . . . , l. Thus by the inductive hypothesis, the set {(x, y) ∈ Rn+1 η6=0 : sign f1 (x, y) = σ1 , . . . , sign fl (x, y) = σl } is a finite union of LE(S)-cylinders in Rn+1 . The inductive hypothesis also yields that the remaining part {(x, 0) : x ∈ Rn , sign f1 (x, 0) = σ1 , . . . , sign fl (x, 0) = σl } is a finite union of LE(S)-cylinders in Rn+1 . This finishes the proof of Theorem 3.2. Comparison with the proof by Lion and Rolin. Our original intent for Section 4 was to simply “o-minimalize” the proof of Theorem II in [9], while removing some ambiguities. In the course of doing so we found a gap in [9]: in part a) of the proof of Dr , it is assumed without justification that one can reduce to working on cylinders. In discussing this issue with Lion, he sketched a way to remedy the situation. The extra arguments needed led to the three claims in the proof of Theorem 4.11 and to various adjustments in the lemmas preceding the theorem. This explains the length of our treatment compared to [9]. The Valuation Property. In one sense the proof of Theorem 3.2 is not so different from that of the closely related Theorem B in [7]: both depend on the Valuation Property in a crucial way, here via its consequence Theorem 2.4. This Valuation Property was conjectured for power bounded o-minimal theories in [4], proved for the polynomially bounded case in [7], and recently established in full (power bounded) generality by James Tyne [14]. References [1] R. Cluckers, Analytic p-adic cell decomposition and p-adic integrals. Preprint, 2002. [2] J. Denef, The rationality of the Poincar´e series associated to the p-adic points on a variety, Inv. Math., 77 (1984), pp. 1–23. [3] J. Denef, p-adic semi-algebraic sets and cell decomposition, J. Reine Angew. Math., 369 (1986), pp. 154–166.

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[4] L. van den Dries, T -convexity and tame extensions II, J. Symbolic Logic, 62 (1997), pp. 14–34. Correction, J. Symbolic Logic, 63 (1998), p. 1597. [5] , o-minimal structures and real analytic geometry, in Current Developments in Mathematics, 1998, International Press, 1999, pp. 105–152. [6] L. van den Dries, A. Macintyre and D. Marker, The Elementary Theory of Restricted Analytic Fields with Exponentiation, Ann. Math., 140 (1994), pp. 183–205. [7] L. van den Dries and P. Speissegger, The field of reals with multisummable series and the exponential function, Proc. London Math. Soc. (3), 81 (2000), pp. 513–565. [8] J. Ecalle, Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac, Hermann, Paris, 1992. [9] J.-M. Lion and J.-P. Rolin, Th´eor`eme de pr´eparation pour les fonctions logarithmico-exponentielles, Ann. Inst. Fourier, 47 (1997), pp. 859–884. [10] J.-M. Lion and J.-P. Rolin, Int´egration des fonctions sous-analytiques et volumes des sous-ensembles sous-analytiques, Ann. Inst. Fourier, 48 (1998), pp. 755–767. [11] A. Parusinski, Lipschitz stratification of subanalytic sets, Ann. Scient. Ecole Norm. Sup. 27 (1994), pp. 661–696. [12] J.-P. Rolin, P. Speissegger, and A. J. Wilkie, Quasianalytic DenjoyCarleman classes and o-minimality. Preprint, 2003; to appear in J. Amer. Math. Soc. [13] R. Roussarie, Bifurcation of planar vector fields and Hilbert’s sixteenth problem, Birkh¨ auser Verlag, Basel, Progress in Mathematics 164, (1998). [14] J. Tyne, T-levels and T-convexity, PhD thesis, University of Illlinois at Urbana-Champaign, January 2003. University of Illinois, Department of Mathematics, 1409 W. Green Street, Urbana, IL 61801 E-mail address: [email protected] University of Wisconsin, Department of Mathematics, 480 Lincoln Drive, Madison, WI 53706 E-mail address: [email protected]