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THE FIELD OF REALS WITH MULTISUMMABLE SERIES AND THE EXPONENTIAL FUNCTION LOU VAN DEN DRIES and PATRICK SPEISSEGGER [Received 17 March 1998; revised 1 December 1999]

Introduction We begin by rephrasing a few de®nitions concerning model completeness and o-minimality in a setting suitable for this paper. This allows us to state our ®rst main result, Theorem A, in a way that is at the same time precise and widely accessible. Let F be a collection of functions f : R m ! R for various m 2 N :ˆ f0; 1; 2; . . .g. A set A Í R n is said to be 0-de®nable in the structure RF :ˆ …R; 12 p. (Here the assumption f > 12 p is essential.) Example. The integral Z 1 0

eÿ t dt :ˆ lim T !‡1 1 ‡ zt

Z 0

T

eÿ t dt 1 ‡ zt

converges for every z 2 Cn…ÿ1; 0† and de®nes a function in G…R; f; 1† for any R > 0 and f 2 …0; p†, whose Taylor series at 0 is the divergent power series P 1 n n n ˆ 0 …ÿ1† n! Z . Borel summation of this divergent power series returns the value of the integral for Re…z† > 0. Over the last twenty years this quasianalyticity property (the injectivity of the Taylor series homomorphism above) has been extended by Ramis, Ecalle, and others in the context of the theory of multisummability; see [2, 16]. We only need a small part of this theory and restrict our attention to the functions relevant for our purpose, namely those that are real valued and de®ned on a segment ‰0; RŠ Í R. Thus, for R > 0 we let G…R† be the set of all functions f : ‰0; RŠ ! R for which e > R, an angle f 2 … 1 p; p†, numbers k 1 ; . . . ; k n 2 …0; 1Š and there exists an R 2 e f; k1 †; . . . ; fn 2 G… R; e f; k n † such that f1 ; . . . ; fn are real valued functions f1 2 G… R;

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e and on ‰0; R† f …x† ˆ f1 …x† ‡ . . . ‡ fn …x† for 0 < x < R: (The Taylor series at 0 of these functions are multisummable in the positive real direction; cf. Balser [2].) It follows easily from this de®nition that G…R† contains all R-valued constant functions on ‰0; RŠ, and that if f 2 G…R†, then f is C 1 on ‰0; RŠ, real analytic on …0; RŠ, and f 0 2 G…R†. Much less obvious is that G…R† is a ring (under pointwise addition and multiplication) and that the Taylor map f 7! b f :ˆ

1 X f …n† …0† n X : G…R† ÿ! R[X ] n! nˆ0

is injective; see [2, 16, 22] for various proofs of this injectivity. This quasianalyticity property is the main fact from multisummability theory that we use in this paper without giving the proof. We now de®ne the one-variable functions of our class G to be those functions f : R ! R that are identically 0 outside ‰0; 1Š and satisfy f j‰0; 1Š 2 G…1†. We actually need a rather subtle generalization of these rings G…R† to several variables, which is due to Tougeron [22]. In § 2 we develop a simple version of Tougeron's idea which suf®ces for our purpose. There we associate to every m-tuple R ˆ …R1 ; . . . ; Rm † of positive real numbers a set G…R† of functions f : ‰0; R1 Š ´ . . . ´ ‰0; Rm Š ! R. It is crucial for us that the family of all G…R† has certain `substitution' properties, which allow the use of `blowings-up'. We establish these properties in §§ 4 and 5. The m-variable functions of our class G are then de®ned to be the functions f : R m ! R that are identically 0 outside ‰0; 1Š m and satisfy f j‰0; 1Š m 2 G…1; . . . ; 1†. We can now state the ®rst main result, proved in §§ 6 and 7. Theorem A. The structure R G is model complete and o-minimal. If « > 0 and the function f : …0; «† ! R is de®nable in R G , then there are integers p and d with d > 0 and a function g 2 G…R† for some R > 0 such that f …x† ˆ x p = d g…x 1 = d † for all suf®ciently small x > 0. The second part of the theorem gives a kind of Puiseux expansion for onevariable functions de®nable in R G . In particular, it implies (by a change of variables) that the structure R G is polynomially bounded. (The structure RF is said to be polynomially bounded if for each function f : R ! R that is de®nable in RF there exists a d 2 N and an R > 0 such that j f …x†j < x d for all x > R. See [12] for numerous consequences of polynomial boundedness.) All subsets of R n that are de®nable in R an are also de®nable in R G . By Nielsen [18, § 79] the function f on …1; 1† given by log G…x† ˆ …x ÿ 12 † log x ÿ x ‡ 12 log…2p† ‡ f…x† is de®nable in R G . Theorem B below implies that …R G ; exp† is model-complete and o-minimal, thus providing a model-complete and o-minimal expansion of the real ®eld in which the Gamma function on …0; 1† is de®nable. Similarly, in [13] we constructed a polynomially bounded, model complete and o-minimal expansion R an  of the real ®eld in which the function z…ÿ log x† on …0; 1† is de®nable, where z denotes the Riemann zeta function. Thus the Riemann zeta function on …1; 1† is de®nable in …R an  ; exp†, and Theorem B below implies that this structure is model complete and o-minimal. (This answers a question left open in [13].)

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Up to this point we have de®ned from scratch all notions needed to state our results, but we now adopt the usual model-theoretic framework, which allows stronger and more precise formulations. In particular `de®nable' will mean e denote any `de®nable with parameters' unless otherwise indicated. Let R polynomially bounded o-minimal expansion of …R; 0 the number x r . We can now state the other main result of this paper, proved in § 9. e Then the structure Theorem B. Suppose expj‰0; 1Š is de®nable in R. e … R; exp; log† admits quanti®er elimination and its complete theory is axiomatized by (i) T; (ii) the Ressayre axioms for exp; and (iii) for each r 2 K , the axiom "x…exp…r x† ˆ fr …exp…x†††. Remarks. In this theorem and below we make the convention that log is everywhere de®ned by setting log…x† :ˆ 0 for x < 0. The `Ressayre axioms for exp' (cf. [9, 20]) are the universal closures of the following formulas: (1) exp…x ‡ y† ˆ exp…x† exp… y†; (2) x < y ! exp…x† < exp… y†; (3) …x > 0 ! exp…log x† ˆ x† ^ …x < 0 ! log x ˆ 0†; (4) x > n 2 ! exp…x† > x n , for each n 2 N; (5) 0 < x < 1 ! exp…x† ˆ E…x†, e is given by where E is the unary function symbol of L whose interpretation in R E…x† ˆ exp…x† if 0 < x < 1 and E…x† ˆ 0 otherwise. Note that by the assumption preceding the theorem there is indeed a function symbol E as in (5) above. Also, the axioms (iii) for r 2 Q can be omitted in Theorem B, since they are deducible from the axioms (i) and (ii). Among the consequences of Theorem B are (see § 9): e exp† is given piecewise (1) any function f : R n ! R that is 0-de®nable in … R; by terms in the language L…exp; log†; e exp† is o-minimal; and (2) … R; e exp† is exponentially bounded, that is, for each function f : R ! R that (3) … R; e exp† there exists an n such that j f …x†j < exp n …x† for all is de®nable in … R; suf®ciently large x. (Here exp n denotes the n-fold iterate of exp.)

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Theorem B is clearly a generalization of the main result in [9] on …R an ; exp†. The proof there also goes through, except for one dif®culty: the simple axiomatization of Th…R an † given in [9, § 3] enabled us to embed its models into power series models and then read off an otherwise non-trivial (but crucial) e , not valuation-theoretic property of its models. This route is not available for R even in the special case of R G . Therefore, we have to establish this valuationtheoretic property in a different way. After doing so we can turn things around and use this property to show that some of the generalized power series ®elds from [9, § 3] have natural expansions to models of Th…R G †. This is carried out in § 10, where we also indicate how the results from [10] on power series models of …R an ; exp† extend to …R G ; exp†. (The extension is not straightforward.) In this introduction we have focused on a careful description of the main results of this paper. As to the underlying ideas, those of Tougeron [22] were essential for Theorem A, but for various complicated reasons we had to reformulate and strengthen (in §§ 1±4) several technical lemmas from [22]. Section 5 on compositional inversion is needed later for the second part of Theorem A. Beginning with § 6 we have all ingredients ready so that we can almost literally copy the long inductive proof of the main theorem of [13], which is analogous to Theorem A. Rather than repeating that proof and increasing the length of the paper by a third, we have just mentioned the needed modi®cations in a few sentences. Thus familiarity with [13], which served as a warm-up for the present paper, is required for §§ 6 and 7. Theorem B is independent of Theorem A, and is obtained in § 9, using results from [8], [6] and [9] in an essential way. The results in § 10 on power series models depend in addition on [10] and [9]. General conventions. We let m and n range over N :ˆ f0; 1; 2; . . . g. If A is any set, we denote by jAj the cardinality of A. Given a subset A of a topological space S we let cl…A†, int…A†, bd…A† :ˆ cl…A†nint…A† and fr…A† :ˆ cl…A†nA denote the closure, interior, boundary and frontier of A in S respectively, if the ambient space S is clear from context. (Usually we have S ˆ R n for some n.) All rings are assumed to be commutative with 1 6ˆ 0. If A is a ring, we denote by A[Y ] the ring of power series in Y ˆ …Y1 ; . . . ; Yn † with coef®cients in A. Similarly, if A is a normed ring, then, for s 2 …0; 1†, AfY gs denotes the subring P P of A[Y ] consisting of all power series a b Y b such that ja b js b < 1. 1. Blow-up height For the inductive proof of [13, § 8] to go through in the present setting, we shall need a modi®ed notion of blow-up height for power series, because we are only dealing here with integer exponents as opposed to real exponents in [13]. Let A be a ring and let X ˆ …X1 ; . . . ; X m † be a tuple of distinct indeterminates with m > 2. Given distinct i; j 2 f1; . . . ; mg, we de®ne the corresponding singular blow-up substitution to be the (injective) A-algebra homomorphism si j : A[X ] ! A[X ] given by  Xk if 1 < k < m and k 6ˆ i; si j …X k † :ˆ X i X j if k ˆ i:

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1.1. De®nition. We assign to every pair of monomials X a and X b with a; b 2 N m a pair b X …X a ; X b † 2 N 2 , called the blow-up height of the pair …X a ; X b †, also denoted by b…X a ; X b † if X ˆ …X1 ; . . . ; X m † is clear from context, as follows. Special case: gcd…X a ; X b † ˆ 1. (Here `gcd' stands for `greatest common divisor'.) Subcase 1: X a ˆ 1 or X b ˆ 1. Then we put b…X a ; X b † :ˆ …0; 0†. Subcase 2: X a 6ˆ 1 and X b 6ˆ 1. Let M :ˆ maxfa i : 1 < i < mg;

d :ˆ jfi: a i ˆ M gj;

N :ˆ maxfb j : 1 < j < mg;

e :ˆ jf j: b j ˆ N gj;

and put b…X a ; X b † :ˆ …M ‡ N; d ‡ e†. General case. This is reduced to the special case by setting b…X a ; X b † :ˆ b…X a ÿ q ; X b ÿ q †; where X q ˆ gcd…X a ; X b †. 1.2. Lemma. (1) We have b…X a ; X b † ˆ …0; 0† if and only if X a j X b or X b j X a . (2) If b…X a ; X b † ˆ …0; 0† then b…s i j …X a †; si j …X b †† ˆ …0; 0†. (3) b…X a ; X b † ˆ b…X b ; X a †. (4) If b…X a ; X b † 6ˆ …0; 0†, then there are distinct i; j 2 f1; . . . ; mg such that b…si j …X a †; si j …X b †† < b…X a ; X b † and b…sj i …X a †; sj i …X b †† < b…X a ; X b †: Proof. Parts (1), (2) and (3) are easy, so we prove (4). Let b…X a ; X b † 6ˆ …0; 0†. Assume ®rst that we are in Subcase 2 of the previous de®nition, and use the same notation. Take i; j 2 f1; . . . ; mg with a i ˆ M and b j ˆ N; note that i 6ˆ j, because gcd…X a ; X b † ˆ 1. Assume that M < N (the case N < M is handled in the same way). Then si j …X a † ˆ X a X jM and si j …X b † ˆ X b . Dividing X a X jM and X b by their gcd X jM , we see that b…si j …X a †; si j …X b †† < b…X a ; X b †. Also, sj i …X a † ˆ X a and sj i …X b † ˆ X b X iN . Dividing X a and X b X iN by their gcd X iM , we see that b…sj i …X a †; sj i …X b †† < b…X a ; X b †. In general, write X a ˆ X q X a ÿ q and X b ˆ X q X b ÿ q with X q :ˆ gcd…X a ; X b †, and take distinct i; j 2 f1; . . . ; mg such that and

b…si j …X a ÿ q †; si j …X b ÿ q †† < b…X a ; X b † b…sj i …X a ÿ q †; sj i …X b ÿ q †† < b…X a ; X b †:

The identity si j …X a † ˆ si j …X q †si j …X a ÿ q † implies that b…si j …X a †; si j …X b †† ˆ b…si j …X a ÿ q †; si j …X b ÿ q ††; hence b…si j …X a †; si j …X b †† < b…X a ; X b †. Similarly b…sj i …X a †; sj i …X b †† < b…X a ; X b †. A

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Next we consider a ®nite collection M ˆ fX a…1† ; . . . ; X a…K † g of K distinct monomials in X, and de®ne si j …M† :ˆ fsi j …X a…1† †; . . . ; si j …X a…K † †g: We associate to M the triple b X …M† 2 N 3 de®ned as follows: if b X …X a k ; X a l † 6ˆ …0; 0† for some k; l 2 f1; . . . ; K g, let p 2 N be the number of pairs …k; l† such that 1 < k < l < K and bX …X a k ; X a l † 6ˆ …0; 0† (so p > 0), and let q 2 N 2 be the lexicographic minimum of Then put

fbX …X a k ; X a l †: 1 < k < l < K; bX …X a k ; X a l † 6ˆ …0; 0†g:

bX …M† :ˆ



…0; 0; 0†

if bX …X a k ; X a l † ˆ …0; 0† for all k; l 2 f1; . . . ; K g;

… p; q†

otherwise.

Again, if X ˆ …X1 ; . . . ; X m † is clear from the context, we just write b…M† for b X …M†. Order N 3 lexicographically in what follows. Note that b…M† ˆ …0; 0; 0† means that M is linearly ordered by divisibility. 1.3. Lemma. (1) If M 0 Í M then b…M 0 † < b…M†. (2) If b…M† 6ˆ …0; 0; 0†, then there are distinct i; j 2 f1; . . . ; mg such that b…si j …M†† < b…M† and

b…sj i …M†† < b…M†:

Proof. This is proved along the lines of [13, Lemma 4.10]. . . . . . . . . . . . A To avoid too many nested parentheses, we will write s i j f instead of s i j … f † below. Consider a ®nite collection F Í A[X ] of power series. For distinct i; j 2 f1; . . . ; mg we put s i j …F† :ˆ fs i j f : f 2 Fg and let b X …F† :ˆ b X …M†, where M is the (®nite) set of monomials that are minimal (with respect to the divisibility relation among monomials in X ) among the monomials occurring in the members of F. The elements of M are called the minimal monomials of F, and b X …F† is the blow-up height of F. (As before we write b…F† if X is clear from context.) P q Note that each f 2 F can be written as f ˆ X gq , where each gq 2 A[X ] m and the sum is over the ®nitely many q 2 N with X q 2 M. The following is proved along the lines of [13, Lemma 4.12]. 1.4. Proposition. (1) If b…F† 6ˆ …0; 0; 0†, then there are distinct i; j 2 f1; . . . ; mg such that b…si j …F†† < b…F† and b…s j i …F†† < b…F†. (2) If b…F† ˆ …0; 0; 0†, then each non-zero f 2 F is of the form f ˆ X q g for some q 2 N m and g 2 A[X ] with g…0† 6ˆ 0. . . . . . . . . . . . . . . . . . . . . . . . . A 2. Gevrey functions in several variables This section and the next two are based largely on parts of a paper by Tougeron [22], adding here and there extra precision and elaboration when needed for our purpose. It is perhaps surprising that the domains of the m-variable functions introduced below are in general not cartesian products of 1-variable domains. This, and other curious features, are forced on us because we need our family of function algebras to be closed under various operations and substitutions.

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By convention, let arg…z† 2 …ÿp; pŠ for z 2 C with arg…0† :ˆ 0, and put z k :ˆ jzj k e i k arg…z† for z 2 C and k > 0. Fix m > 0. Notation. For k ˆ …k 1 ; . . . ; k m † 2 ‰0; 1†m and z ˆ …z 1 ; . . . ; z m † 2 C m we put Sk :ˆ k 1 ‡ . . . ‡ k m ; k ´ jarg…z†j :ˆ k 1 jarg…z 1 †j ‡ . . . ‡ k m jarg…z m †j; z k :ˆ z 1k 1 . . . z mk m 2 C; jzj :ˆ supfjz i j: i ˆ 1; . . . ; mg: For a non-empty set S Í C m and z 2 C m , we let d…z; S† :ˆ inffjz ÿ uj: u 2 Sg: We also write z 0 :ˆ …z 1 ; . . . ; z m ÿ 1 † 2 C m ÿ 1 for any z 2 C m, if m > 0. If a 2 N m , we put a! :ˆ a 1 ! . . . a m !. e 2 …0; 1†m are A tuple R ˆ …R1 ; . . . ; R m † 2 …0; 1†m is called a polyradius. If R; R e if R i < R e i for each i (and similarly with ` 1, then S …R; f† ˆ S…R; f; 1=k† È f0g, where S…R; f; 1=k† is the open sector de®ned in the introduction. The reason for allowing k i ˆ 0 is that we need our class of generalized sectors to be closed under taking cartesian products with discs D…r† Í C, where r > 0. For example, if 0 m > 0 and k ˆ …k 0 ; 0† with k 0 2 K m ÿ 1 , then S k …R; f† ˆ S k …R 0 ; f† ´ D…R m †. Next, for p 2 N we put (see Figure 2.1)   Rk k k ; Dp …R† :ˆ z 2 D…R†: jz j < p‡1 and Spk …R; f† :ˆ S k …R; f† È Dpk …R†: For any non-empty ®nite set K Í K m we let (see Figure 2.2) \ k S K …R; f† :ˆ S …R; f†; and for p 2 N we let

k2K

Spk …R; f† :ˆ

\ k2K

Spk …R; f†:

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Figure 2.1 (with 12 p < f < p, m ˆ 1).

Figure 2.2 (with 12 p < f < p, m ˆ 1).

For later use we observe that int…S K …R; f†† ˆ

\

int…S k …R; f††

k2K

and S K …R; f† Í cl…int…S K …R; f†††: Moreover, if z 2 SpK …R; f† and t 2 ‰0; 1Š, then t z 2 SpK …R; f†; in particular, SpK …R; f† is connected. Note also that S K …R; f† Ç R m ˆ ‰0; R† whenever K Í ‰1; 1† m . On the other hand, S K …R; f† ˆ D…R† if K ˆ f0g. Given a set U and a function f : U ! C, we let k f kU :ˆ sup j f …z†j 2 ‰0; 1Š: z2U

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2.2. De®nition. Let t ˆ …K ; R; r; f†, where K Í K m is non-empty and ®nite, R 2 …0; 1†m , r 2 …1; 1† and f 2 …0; p†. To simplify notation we put S…t† :ˆ S K …R; f†

and Sp …t† :ˆ SpK …R; f† for p 2 N;

and if t is clear from the context we even write S and Sp for S…t† and Sp …t†. (Similarly, if a tuple et as above is given, we write Se and Sep for S…et † and Sp …et †.) Note that …Sp †p 2 N is a descending sequence of open neighbourhoods of S. If for have a P bounded holomorphic function fp : Sp ! C such that P each p we p fp clearly converges uniformly on S to a (bounded p k fp kSp ´ r < 1, then P and continuous) function f : S ! C; we indicate this state of affairs by f ˆt fp . Clearly, f is then holomorphic on int…S†; in particular, if K ˆ f0g, then f : D…R† ! C is holomorphic. P fp for some We let Gt be the set of all functions f : S ! C such that f ˆt sequence … fp †, and for f 2 Gt we put nX o k f kt :ˆ inf k fp kSp ´ r p ; P fp . where the in®mum is over all sequences … fp † such that f ˆt With this notation we have the following lemma. 2.3. Lemma. (1) If g: D…R† ! C is bounded and holomorphic, then gjS belongs to Gt ; in particular, for each polynomial P 2 C‰X 1 ; . . . ; X m Š the function z 7! P…z†: S ! C belongs to Gt . (2) We have k f kS < k f kt for all f 2 Gt , and kgjSkt ˆ kgjSkS for all bounded and holomorphic g: D…R† ! C. (3) The pair …Gt ; k ? kt † is a complex Banach algebra under pointwise addition and multiplication of functions. The proof is routine and left to the reader. Next, we establish some basic facts about derivatives of functions in Gt . 2.4. De®nition. Let U Í C m be such that U Í cl…int…U ††. We shall say that a continuous function f : U ! C is C 1 if f jint…U † is holomorphic and for each multi-index a 2 N m the partial derivative ¶ a … f jint U †=¶z a extends to a continuous function f …a† : U ! C. Below we use the notation f …a† in this sense; thus, if U is open, then f …a† ˆ ¶ a f = ¶z a . Let m > 1. Along with t ˆ …K ; R; r; f† we now consider a tuple e < f. We keep these e where R e < R, 1 < er < r and 0 < f e er ; f†, e 2 …0; 1† m , R e t ˆ …K ; R; e assumptions about t and e t until Lemma 2.14; in particular S ˆ S…e t † Í S ˆ S…t†. e 2.5. Lemma. There is a d 2 …0; 1† such that d…z; bd S† > djzj for all z 2 S. e it suf®ces to show the existence of such a constant Proof. Since Se Í cl…int S†, e Multiplying z 2 int… S† e by a suitable positive scalar, we d for points z 2 int… S†. e with jzj > minf 1 R e may further restrict to points z 2 int… S† 2 i : i ˆ 1; . . . ; mg. If no

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such constant d were to exist for those z, then d…z; bd S† could assume arbitrarily small positive values for such z, which is impossible. . . . . . . . . . . . . . . . . . A In general a function f 2 Gt is not C 1 on its entire domain S (at least, we have no reason to think so), but below we show that its restriction f j Se is C 1 . We keep the notation from above and also put  maxf1=…Sk†: k 2 K ; k 6ˆ 0g if K 6ˆ f0g; k :ˆ 1 if K ˆ f0g: Thus 0 < k < 1 and k ´ Sk > 1 for all k 2 K nf0g. 2.6. Lemma. Let f 2 Gt . Then f j Se is C 1 , and there are constants A; B > 0 e independent of f such that for all a 2 N m and z 2 S, j f …a† …z†j < AB S a …Sa†k ´ S a k f kt ; a! e …a† …z†. where we have written f …a† …z† instead of … f j S† Proof. We ®rst derive a distance estimate. Put l :ˆ minfR i : i ˆ 1; . . . ; mg. We claim that l ; for all p 2 N: …2:1† d…0; bd…Sp †† > … p ‡ 1†k To see this, let p 2 N, z 2 C m with jzj < l=… p ‡ 1†k , and k 2 K; we have to show that z 2 Spk :ˆ Spk …R; f†. We clearly have jz i j < l < R i for i ˆ 1; . . . ; m, so z 2 D…R†. If k ˆ 0, then D…R† ˆ Spk , and therefore z 2 Spk . Suppose that k 6ˆ 0. Then jzj < l=… p ‡ 1†k gives jz k j ˆ jz 1 j k 1 . . . jz m j k m < ˆ

R 1k 1 R kmm . . . … p ‡ 1†k k 1 … p ‡ 1†k k m Rk Rk ; < … p ‡ 1†k ´ S k p ‡ 1

so z 2 D pk …R† Í Spk , as required. Next, with d as in Lemma 2.5 we claim that d…z; bd…Sp †† >

dl 2… p ‡ 1†k

e for z 2 S:

…2:2†

To see this, suppose ®rst that z 2 Se with jzj < l=2… p ‡ 1†k . Then by (2.1) we obtain d…z; bd…Sp †† > l=2… p ‡ 1†k > dl=2… p ‡ 1†k , as desired. Suppose next that z 2 Se with jzj > l=2… p ‡ 1†k . Then by Lemma 2.5, d…z; bd…Sp †† > d…z; bd…S†† > djzj >

dl ; 2… p ‡ 1†k

which establishes (2.2). P P fp and k f kSp ´ r p < 2k f kt . Now choose a sequence … fp † such that f ˆt

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e Then we have, for a 2 N m and z 2 S, k fp kS p j fp…a† …z†j < (by the Cauchy estimates for derivatives) a! d…z; bd…Sp ††ÿ S a  S a 2 … p ‡ 1†k ´ S a k fp kS p (by (2.2)) < dl  S a 2 … p ‡ 1†k ´ S a r ÿ p k f kt : ÿk; we claim that there is a constant L > 0 such that … p ‡ 1†k ´ S a s p < …L ´ k ´ Sa†k ´ S a

for all p and all a.

To see this, note that any constant L > 0 works if a ˆ 0. For a 6ˆ 0, the function x 7! …x ‡ 1† ´ s x = k ´ S a for x > 0 attains its maximum at x ˆ ÿ…k ´ Sa= log s† ÿ 1; evaluating the function for this value of x and raising to the power k ´ Sa now yields the desired inequality with L :ˆ ÿ1= log s. Therefore, the right-hand side of (2.3) is bounded above by AB S a …Sa†k ´ S a k f kt for positive reals A and B which do not depend on f , a or z . . . . . . . . . . . . A Remark. Each point z 2 S belongs to some subsector Se as above, and clearly f …z† does not depend on the choice of such a subsector. This justi®es our e …a† …z†. We shall also write ¶f =¶z i instead of f …a† if notation of f …a† …z† for … f j S† a i ˆ 1 and a j ˆ 0 for j 6ˆ i. …a†

2.7. Lemma. There is a constant « > 0 such that « for all p 2 N and z 2 Sep : d…z; bd…Sp †† > … p ‡ 1†k Proof. We choose for each k 2 K nf0g a constant «k > 0 such that e and p 2 N. (The existence of e f† d…z; bd…Spk †† > «k =… p ‡ 1†k for all z 2 S k … R; such an «k follows from the estimate (2.2) for K ˆ fkg.) Next, we choose a constant C > 0 such that jz k ÿ w k j < C jz ÿ wj for all z; w 2 D…R† and k 2 K . e i : i ˆ 1; . . . ; mg, « < «k for all Now let « > 0 be such that « < minfR i ÿ R k k e for all k 2 K nf0g. We show that the lemma k 2 K nf0g, and C« < R ÿ … R† holds for this value of «. Let p 2 N, z 2 Sep and jz ÿ wj < «=… p ‡ 1†k ; it suf®ces to show that w 2 Sp . e or z 2 Dpk … R†. e then e f† e If z 2 S k … R; e f†, First, let k 2 K nf0g. Then either z 2 S k … R; k k k e e and we use « < «k to obtain w 2 Sp :ˆ Sp …R; f†. If z 2 Dp … R†, then z 2 D… R†

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jz ÿ wj < «, so w 2 D…R† and jz k ÿ w k j
0 independent of f such that kgke t < C k f kt . (2) For every r 2 ‰0; R m † the function g: S 0 ! C de®ned by g…z 0 † :ˆ f …z 0 ; r† belongs to Gt 0 . P fp . De®ne gp : Sep ! C for each p by Proof. (1) Assume that f ˆt Z 1 ¶fp 0 …z ; t z m † dt: gp …z† :ˆ 0 ¶z m

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Then each gp is holomorphic Sp < k¶ fp = ¶z m kSp . From the previous P and kgp ke e ¶ f ˆ =¶z , lemma we get ¶f = ¶z m t m and the function g: S ! C de®ned by e Pp P g with kgk < k¶f = ¶z k < Ck f kt for some g…z† :ˆ gp …z† satis®es g ˆ e t p t m e t e e C > 0 not depending on f . Moreover, for all z 2 S and each p we clearly have fp …z† ÿ fp …z 0 ; 0† ˆ z m ´ gp …z†; and taking the sum over all p on both sides gives the desired equality. (2) Let r 2 ‰0; R m †. Then for k 2 K and z 0 2 C m ÿ 1 we have k ´ jarg…z 0 ; r†j ˆ k 0 ´ jarg…z 0 †j, so that z 0 2 S 0 implies …z 0 ; r† 2 S. Moreover, if 0 0 j…z 0 †k j < …R 0 †k =… p ‡ 1† with p 2 N, then 0

…R 0 †k ´ r k m Rk < ; j…z ; r† j ˆ j…z † j jr j < p‡1 p‡1 0

k

0 k0

km

P fp gives which shows that z 0 2 Sp0 implies …z 0 ; r† 2 Sp . Therefore, f ˆt P gp , where gp : Sp0 ! C is given by gp …z 0 † :ˆ fp …z 0 ; r†. . . . . . . . . . . . . A g ˆt 0 2.10. Remark. The following property is the reason for allowing the parameters k to range over K m rather than just ‰1; 1†m . Let t ˆ …K; R; r; f† be e R; e r; f†, where K e :ˆ f…k; 0†: k 2 K g and as above and R m ‡ 1 > 0; put et :ˆ … K; e and similarly z 0 2 Sp for e :ˆ …R1 ; . . . ; R m ; R m ‡ 1 †. Note that z 0 2 S for all z 2 S, R all z 2 Sep . Thus, for each f 2 Gt the function g: Se ! C de®ned by g…z† :ˆ f …z 0 † belongs to Ge t and satis®es kgke t < k f kt . 2.11. De®nition. We let Tm be the collection of all tuples t ˆ …K; R; r; f† with K Í K m non-empty and ®nite, R 2 …0; 1†m , r > 1 and f 2 … 12 p; p†. (We restrict f to be greater than 12 p to obtain quasianalyticity; see Proposition 2.18.) If e belong to Tm , we put et < t if and only if e R; e er ; f† t ˆ …K; R; r; f† and et ˆ … K; e e R e < R, er < r and f < f. Clearly < de®nes a partial order on Tm such K Í K, that if t; et 2 Tm , then there exists a j 2 Tm such that j < t and j < et ; that is, Tm is a directed set. t < t. Then SeP Í S, and if f 2 Gt with 2.12. t 2 Tm with e P Remark. Let t; e fp for some sequence … fp †, then f j Se ˆ e … fp j Sep †; in particular f ˆt t e e f j S 2 Ge t and k f j Ske t < k f kt . 2.13. De®nition. Let t; et 2 Tm with et < t. By the previous remark, the restriction map f 7! f j Se maps Gt into Ge t , and it follows from the Identity Theorem for holomorphic functions that this map is injective. The family …Gt : t 2 Tm † equipped with the partial order < on Tm and the restriction maps is a directed system. Since the restriction maps are injective, there is no harm in t < t. Under these identifying each Gt with its image in Ge t whenever e identi®cations the direct limit of the system …Gt : t 2 Tm † is simply given by [ Gm :ˆ Gt : t 2 Tm

Then Gm is a C-algebra with derivations ¶= ¶z i (by Lemma 2.8), and we extend each norm k ? kt to all of Gm by setting k f kt :ˆ 1 if f 62 Gt .

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2.14. Lemma. Let f 2 Gm . (1) If f …0† ˆ 0, then lim t k f kt ˆ 0, where the limit is over the downwarddirected set Tm . (2) If f …0† 6ˆ 0, then f is a unit in Gm . (3) If f …0† ˆ 0, then there are fi 2 Gm for i ˆ 1; . . . ; m such that f ˆ z 1 ´ f1 ‡ . . . ‡ z m ´ fm , where z i 2 Gt Í Gm denotes the coordinate function …z 1 ; . . . ; z m † 7! z i : S ! C for t 2 Tm . P fp ; subProof. (1) Let t ˆ …K; R; r; f† 2 Tm with f 2 Gt , and let f ˆt the case that fp …0† ˆ 0 for tracting from each fp the constant fp …0†, we reduce toP p 1 all p. Let « > 0, and take an N 2 N such that p > N k fp kSp ´ r < 2 «. By decreasing each fp by the corresponding restriction, we get in P R and replacing p 1 addition p < N k fp kSp ´ r < 2 «. Then k f kt < «, which proves (1). (2) We may assume that f …0† ˆ 1, so that f ˆ 1 ÿ g with g 2 Gm and g…0† ˆ 0. By (1) there is a t 2 Tm such that f ; g 2 Gt with kgkt < 1. Since Gt is a Banach algebra, f has inverse 1 ‡ g ‡ g 2 ‡ g 3 ‡ . . . in Gt , and hence in Gm . (3) This follows by induction on m, using Lemma 2.9. . . . . . . . . . . . . . . A 2.15. De®nition. The Taylor series at 0 of a function f 2 Gt …t 2 Tm † is b f …X † :ˆ

X f …k† …0† k X 2 C[X ]: k! k2Nm

The map f 7! b f : Gt ! C[X ] is clearly a C-algebra homomorphism. These sot 2 Tm , and thus extend called Taylor maps satisfy fd jS ˆ b f for f 2 Ge t and t < e b for each m to a C-algebra homomorphism f 7! f : Gm ! C[X ], which respects the d ˆ ¶b standard derivations: ¶f=¶z f =¶X i for i ˆ 1; . . . ; m. i The Taylor map Gt ! C[X ] is injective for t 2 Tm (here the condition f > 12 p is essential). This result is important for us, and it is easily obtained from the special case m ˆ 1, as we shall see. The case m ˆ 1, however, requires an equivalent description of the functions in G1 , so that we can invoke the fundamental quasianalyticity theorem from multisummability theory. The next two lemmas provide this equivalence, which is also needed to relate the onevariable functions in G as de®ned in the Introduction to the functions in G1 of this section. e < R, 1 < er < r, 2.16. Lemma. Let K ˆ fk 1 ; . . . ; k n g Í ‰1; 1†, n > 0, 0 < R e < f < p, and put t :ˆ …K; R; r; f†, et :ˆ …K; R; e Then for any f 2 Gt e er ; f†. 0 1. Write f ˆt that each sp extends to a continuous function on cl…Sp †. Put K 0 :ˆ K nfk n g, 0 e e and S  :ˆ S k n … R; e e er ; f†, e f† e f†. S 0 :ˆ S K … R; For each p we t 0 :ˆ …K 0 ; R; parametrize bd…Sp † counterclockwise, and we consider gp :ˆ bd…Sp † Ç Dpk n …R† and dp :ˆ bd…Sp †ngp with the induced parametrization. (Note that the lengths of gp and d p are uniformly bounded for p 2 N.) For each p we de®ne Z sp …w† 1 g p …z† :ˆ dw for z 2 S p0 2pi d p w ÿ z and Z s p …w† 1 dw for z 2 S p : h p …z† :ˆ 2pi g p w ÿ z By Cauchy's theorem s p …z† ˆ g p …z† ‡ h p …z† for all z 2 Sep . Using Lemma 2.7, we get, by an argument similar to the proof of Lemma 2.8, X X kh p k S p ´ er p < 1: kg p kS p0 ´ er p < 1 and P g p …z† belongs to Gt 0 , and Therefore, the function g: S 0 ! C given byPg…z† :ˆ h p …z† belongs to Gtn . The lemma the function h: S  ! C given by h…z† :ˆ now follows easily from the inductive hypothesis. . . . . . . . . . . . . . . . . . . . A The proofs of the next two results involve references to the literature on multisummability (where notation may differ somewhat from ours). e < f < p, and put e < R, 1 < er < r and 1 p < f 2.17. Lemma. Let k > 1, 0 < R 2 e e j :ˆ …fkg; R; er; f†. Then for any f 2 G…R; f; 1=k† there exists a g 2 Gj such that e 1=k†. e f; f …z† ˆ g…z† for all z 2 S… R; Proof. See Tougeron [22, Lemma 2.8 and Proposition 2.9]. . . . . . . . . . . . A f : Gt ! C[X ] is injective. 2.18. Proposition. For t 2 Tm the Taylor map f 7! b Proof. In view of Lemma 2.16, the case m ˆ 1 follows from the fundamental quasianalyticity result of multisummability theory; see [2, 16, 22]. We now proceed by induction on m. Let m > 1 and assume the f ˆ 0. proposition holds for lower values of m. Let t 2 Tm and f 2 Gt with b We have to derive the fact that f ˆ 0. Let t 0 :ˆ …K 0 ; R 0 ; r; f† 2 Tm ÿ 1 with K 0 :ˆ f…k 1 ; . . . ; k m ÿ 2 ; k m ÿ 1 ‡ k m †: k 2 K g and R 0 :ˆ …R 1 ; . . . ; R m ÿ 1 †, and let e 2 T1 with K e :ˆ fk 1 ‡ . . . ‡ k m : k 2 K g, R e < R m , er 2 …1; r† e R; e er ; f† e t :ˆ … K; e < f. and f By Lemma 2.8 and (repeated application of) Lemma 2.9(2), for each a 2 N m ÿ 1 the function ga : Se ! C de®ned by ga …z m † :ˆ f …a; 0† …0; . . . ; 0; z m † belongs to Ge t. d …a; 0† b ˆ 0, so that gc Moreover, f ˆ 0 implies that f a ˆ 0. The inductive hypothesis therefore gives ga ˆ 0.

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e the function On the other hand, by Lemma 2.9(2), for each t 2 ‰0; R† h t : S…t 0 † ! C de®ned by h t …z 0 † :ˆ f …z 0 ; t † belongs to Gt 0 . The above now …a† implies that h t …0† ˆ ga …t † ˆ 0 for each a 2 N m ÿ 1 , so that hbt ˆ 0. Thus, by the e and inductive hypothesis, h t ˆ 0. Therefore f …z† ˆ 0 for each z 2 S…t 0 † ´ ‰0; R†, by analytic continuation f ˆ 0.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Here is an easy consequence. 2.19. Corollary. Let t 2 Tm and f 2 Gt . Then f …‰0; R†† Í R

if and only if

b f 2 R[X ]:

Proof. The forward direction is clear. For the converse, note that S ˆ S…t† is closed under the conjugation map z ˆ …z 1 ; . . . ; z m † 7! z :ˆ …z 1 ; . . . ; z m † from C m onto itself, that the function f  : S ! C de®ned by f  …z† :ˆ f …z† belongs to Gt , P P ca X a . Hence, if b f 2 R[X ], then b f ˆ fb , and that if b f ˆ ca X a , then fb ˆ  which by Proposition 2.18 implies f ˆ f , and thus f …‰0; R†† Í R. . . . . . . . . . A e > 1 p and that f 2 Gt takes Remark. Suppose that in Lemma 2.16 we have f 2 real values on ‰0; R†. Then the f j in Lemma 2.16 can also be taken to be real e This follows from Proposition 2.18 by replacing each f j by valued on ‰0; R†.  1 e … f ‡ f †, which is real valued on ‰0; R†. j j 2 2.20. De®nition. Given a polyradius R ˆ …R 1 ; . . . ; R m † we let G…R† be the set of all functions f : ‰0; RŠ ! R for which there exist a tuple e 2 Tm with polyradius R e ; R; e er ; f† e > R and a function ef 2 G t such that e t ˆ …K e e f …x† ˆ f …x† for all x 2 ‰0; RŠ. Thus, by the results above, G…R† is a ring (under pointwise addition and multiplication of functions) which contains all real constant functions on ‰0; RŠ. Also, if f 2 G…R†, then f is of class C 1 and its derivatives ¶ f = ¶x i belong to G…R†. These G…R† were mentioned in the introduction. We even gave there a precise de®nition of G…R† for m ˆ 1, which by Lemmas 2.16, 2.17 and the remark following Corollary 2.19 is equivalent to the de®nition given here for m ˆ 1. Having available the rings G…R†, we now de®ne the class G of real-valued functions on R m …m ˆ 0; 1; . . . † as in the introduction. This leads to the structure R G :ˆ …R; er and f > f. be such that R > R, g p and

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lou van den dries and patrick speissegger

that each g p extends to a continuous function harmless in applications, since we can always Fix a q 2 N. By Taylor's Theorem with § 5.4]), we have, for each p 2 N, every a 2 Sep g p …a ‡ z† ˆ g p …a† ‡ g p0 …a†z ‡ . . . ‡ where we de®ne for a; b 2 Sep : R q g p …a; b† :ˆ

g p…q ÿ 1† …a† q ÿ 1 z ‡ z q ´ R q g p …a; a ‡ z†; …q ÿ 1†!

Z

1 2pi

on cl…Sp †. (This last assumption is achieve it by decreasing R and f.) remainder (see for example, [23, and every z 2 C with a ‡ z 2 Sep ,

bd…S p †

…2:4†

g p …w† dw; …w ÿ a†q …w ÿ b†

with bd…S p † oriented counterclockwise. In the following lemma we show how to take the sum over all p on both sides of (2.4) above. e 2 T2 with Let q :ˆ …K q ; R q ; er; f† K q :ˆ f…k; 0†: k 2 K g È f…0; k†: k 2 K g e and we obtain the following. e R†. e Then S…q† ˆ Se ´ S, and R q :ˆ … R; 2.22. Lemma. There is a function R q g 2 Gq such that for all non-zero a 2 Se and all z 2 C with a ‡ z 2 Se nf0g we have g…a ‡ z† ˆ g…a† ‡ g 0 …a†z ‡ . . . ‡

1 g …q ÿ 1† …a†z q ÿ 1 ‡ z q ´ R q g…a; a ‡ z†; …q ÿ 1†!

t. and which satis®es kR q gkq < C ´ kgkt , where C > 0 depends only on q, t and e Proof. Let k be the quantity associated to K in the paragraph following Lemma 2.5. By Lemma 2.7 there is an « > 0 (depending only on t and et ) such that jw ÿ zj > «=… p ‡ 1†k for all z 2 Sep and all w 2 bd…S p †. Hence for all a; b 2 Sep we have Z g p …w† 1 dw jR q g p …a; b†j ˆ q 2p bd…S p † …w ÿ a† …w ÿ b†   … p ‡ 1†k q ‡ 1 ; < Akg p kS p « where A > 0 is a constant depending only on t and et. Now choose a B > 0 such that …… p ‡ 1†k =«†q ‡ 1 < B…r = er † p for all p. Then X X kR q g p kS p …q† ´ er p < AB kg p kS p ´ r p < 1: p

p

Thus the lemma follows with R q g: Se ´ Se ! C de®ned by R q g…a; b† :ˆ P 1 p ˆ 0 R q g p …a; b†. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A 3. Mixed series Let X 1 ; . . . ; X m and Y1 ; . . . ; Yn be distinct indeterminates, and put X :ˆ …X 1 ; . . . ; X m † and Y :ˆ …Y1 ; . . . ; Yn †. We also let X 0 ˆ …X 1 ; . . . ; X m ÿ 1 † if m > 1 and

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Y 0 ˆ …Y1 ; . . . ; Yn ÿ 1 † if n > 1. Similarly, t ˆ …K; R; r; f† will denote an element of Tm and r an element of …0; 1†n ; we put S :ˆ S…t† and D :ˆ D…r†. This notation will be in force throughout this paper, unless we explicitly indicate otherwise. We let CfX gt be the image of the Taylor map Gt ! C[X ]. We transfer the f kt :ˆ k f kt , and we norm on Gt to its isomorphic copy CfX gt by setting k b transfer evaluation to CfX gt by setting b f …x† :ˆ f …x† for x 2 S…t†. It is also convenient to de®ne CfX gG to be the image of the Taylor map Gm ! C[X ], so CfX gG is the union of its subrings CfX gt for t 2 Tm . The goal of this and the next sections is to show that after making certain substitutions for the indeterminate X mPof a series f …X † 2 CfX gG with m > 1, we obtain a `mixed series' g…X 0 ; Y1 † ˆ n 2 N gn …X 0 †Y1n that is `convergent in Y1 ' and whose coef®cients gn …X 0 † belong to CfX 0 gG (all expressions in quotes will be made precise below). We have to choose the right collection of such mixed series, so that (among other things) all the series obtained by making these substitutions are among our collection, and the collection `has Weierstrass Preparation' (see Proposition 3.7). 3.1. De®nition. Given f 2 CfX gt [Y ], we write f …X; Y † ˆ with all cb belonging to CfX gt , and we put X kc b kt r b 2 ‰0; 1Š: k f kt; r :ˆ

P

b2Nn

c b …X †Y b

Thus the set of f 2 CfX gt [Y ] with k f kt; r < 1 is just the subalgebra CfX gt fY gr of the C-algebra CfX gt [Y ] (see `conventions' in § 1). It is in fact a complex Banach algebra with respect to the norm k ? kt; r , and we shall denote it by CfX; Y gt; r . We will also refer to each X i as a Gevrey variable and to each Yj as a convergent variable, and we use ` ; ' to indicate the separation between the two types of variables in CfX; Y gt; r (whereas ` , ' merely indicates a separation between variables of the same type). e t 2 Tm It follows from Remark 2.12 that CfX; Y g e t;e r Í CfX; PY gt; r whenever with et > t and e r 2 …0; 1†n with e r > r. Each f …X; Y † ˆ c b …X †Y b 2 CfX; Y gt; r de®nes a continuous function ft; r : S ´ D ! C given by X X c b …x† y b ˆ lim c b …x† y b : …3:1† ft; r …x; y† :ˆ n!1

Sb R, e er; f† Assume now that m > 0 and n > 0, and let e t ˆ …K; R; n e > f, and e er > r and f r 2 …0; 1† with e r > r. We shall also write Se in place of e S…et † and D in place of D…e r †. 3.2. Lemma. Let f 2 CfX ; Y g e t;e r . Then (1) for each i 2 f1; . . . ; mg the partial derivative ¶f = ¶X i belongs to CfX; Y gt; r and satis®es …¶f = ¶X i †…x; y† ˆ ¶ft; r = ¶x i …x; y† for all …x; y† 2 int…S† ´ D; (2) for each j 2 f1; . . . ; ng the partial derivative ¶f = ¶Yj belongs to CfX; Y gt; r and satis®es …¶f = ¶Yj †…x; y† ˆ ¶ft; r =¶yj …x; y† for all …x; y† 2 int…S† ´ D.

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(Note that ¶ft; r = ¶x i …x; y† and ¶ft; r =¶y j …x; y† in (1) and (2) are de®ned since ft; r is holomorphic on int…S† ´ D.) P Proof. Let f …X; Y † ˆ b c b …X †Y b . By Lemma 2.8 we have ¶c b =¶X i 2 CfX gt for each b 2 N n , and therePis a constant C > 0 such that k¶c b = ¶X i kt < C kc b ke t for each b. Thus k¶c b =¶X i kt ´ r b < C k f ke The equation t; e r < 1. …¶f = ¶X i †…x; y† ˆ ¶ft; r =¶x i …x; y† for …x; y† 2 int…S† ´ D obviously holds if f has ®nite support (as a power series in Y with coef®cients in C[X ]). Therefore, (1) follows for general f from (3.1) and the Weierstrass Convergence Theorem. Part (2) follows similarly, using standard arguments for convergent power series in place of Lemma 2.8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A 3.3. Remark and De®nition. Let f 2 CfX; Y g e t; e r . Then it follows easily from Lemma 3.2 that f t; r is C 1 (in the sense of De®nition 2.4). We de®ne [ fCfX; Y g e t > t and e r > rg; CfX; Y gt ‡ ; r ‡ :ˆ t; e r: e then CfX; Y gt ‡ ; r ‡ is a C-subalgebra of CfX; Y gt; r , and CfX; Y gt ‡ ; r ‡ is closed under the operators ¶= ¶X i and ¶= ¶Yj , in contrast to CfX; Y gt; r . Next, we put CfX; Y gG :ˆ

[

fCfX; Y gt; r : t 2 Tm and r 2 …0; 1†n g;

and we extend each norm k ? kt; r to all of CfX; Y gG by putting k f kt; r :ˆ 1 whenever f 62 CfX; Y gt; r . This de®nition also makes sense when m ˆ 0 or n ˆ 0, and for n ˆ 0 it just gives CfX gG with the maps k ? kt as de®ned in the beginning of this section. So below we do allow again m and n to be 0, and we partially order the cartesian product Tm ´ …0; 1†n by the product ordering. Generalizing Lemma 2.14, we then obtain the following. 3.4. Lemma. Let f 2 CfX; Y gG . (1) If f …0; 0† ˆ 0, then lim …t; r† k f kt; r ˆ 0, where the limit is over the downward directed set Tm ´ …0; 1†n . (2) If f …0; 0† 6ˆ 0, then f is a unit in CfX; Y gG .. . . . . . . . . . . . . . . . . . . A The next lemma says in particular that no new series have been introduced by the mixed series, that is, CfX; Y gG Í CfX; Y gG . If m > 1 we let t 0 :ˆ …K 0 ; R 0 ; r; f† with K 0 :ˆ f…k 1 ; . . . ; k mÿ 1 †: k 2 K g and R 0 :ˆ …R 1 ; . . . ; R m ÿ 1 †, and we let r :ˆ …R m ; r†. 3.5. Lemma. Assume that m > 1. Then CfX 0 ; X m ; Y gt 0 ; r  Í CfX; Y gt; r ; and k f kt; r < k f kt 0 ; r  for f 2 CfX 0 ; X m ; Y gt 0 ; r  . P Proof. Let f …X; Y † ˆ c n; b …X 0 †X mn Y b 2 CfX 0 ; X m ; Y gt 0 ; r  . By Remark 2.10, each c n; b belongs to CfX gt and satis®es kc n; b kt < kc n; b kt 0 . By Lemma 2.3, each X mn belongs to CfX gt and satis®es kX mn kt < R mn . Hence, for each b 2 N n

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533

we have X n

kc n; b ´ X mn kt
0 and f is regular in Yn of order d. Then the series f factors uniquely as f ˆ gh, where g 2 CfX; Y gG is a unit and h 2 CfX; Y 0 gG ‰Yn Š is monic in Yn of degree d. . . . A 4. Substitutions In this section we show that the algebras CfX; Y gG are closed under certain substitutions needed in carrying out the arguments of § 6. Of particular importance is the fact that the `regular blow-up substitutions' and the `shift substitutions' as de®ned below change a Gevrey variable into a convergent variable. Throughout this section we let i range over f1; . . . ; mg and j over f1; . . . ; ng. 4.1. De®nition. A substitution j: …X; Y † ! C[Z ], where Z ˆ …Z 1 ; . . . ; Z p † is a tuple of distinct indeterminates, is by de®nition a map j: fX 1 ; . . . ; X m ; Y1 ; . . . ; Yn g ! C[Z ]: If j…X i †…0† ˆ 0 and j…Yj †…0† ˆ 0 for each i and j, then j extends to a unique C-algebra homomorphism C[X; Y ] ! C[Z ] which we also denote by j. We then usually write j f in place of j… f † for f 2 C[X; Y ]; in other words, if f ˆ f …X; Y † 2 C[X; Y ], then j f ˆ f …j…X 1 †; . . . ; j…X m †; j…Y1 †; . . . ; j…Yn ††. If all j…X i † and j…Yj † lie in a subring A of C[Z ], then we also refer to j as a substitution j: …X; Y † ! A.

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Let V ˆ …V1 ; . . . ; Vm †, W ˆ …W1 ; . . . ; Wn †, e t 2 Tm and e r 2 …0; 1†n , and write e :ˆ D…e Se :ˆ S…et † and D r †. If j: …X; Y † ! CfV; W g e t; e r is a substitution, then j m‡n e e : S ´ D ! C de®ned by gives rise to a map j e t; e r je t; e r …v; w† :ˆ …j…X 1 †…v; w†; . . . ; j…Yn †…v; w††: e e Once more we shall simply write j…v; w† in place of j e t; e r …v; w† for …v; w† 2 S ´ D.

4.2. Lemma. Let j: …X; Y † ! CfV ; W g e t ‡; e r ‡ be a substitution, and assume e Í S ´ D, j…X i †…0† ˆ 0 for each i and j…Yj †…0† ˆ 0 for each j. Then that j… e S ´ D† 1 e and the on e S ´ D, for every f 2 CfX; Y gt ‡ ; r ‡ the function ft; r ± j e t; e r is C is given by j f. Taylor series at 0 of ft; r ± j e t; e r 1 e is clear, so we prove the on e S´D Proof. The fact that ft; r ± j e t; e r is C second assertion. Let f 2 CfX; Y gt ‡ ; r ‡ . For each d 2 N write X X a Y b ´ h a; b ; f …X; Y † ˆ Pd …X; Y † ‡ Sa‡Sbˆd

where Pd …X; Y † 2 C‰X; Y Š is of degree at most d and each h a; b belongs to CfX; Y gt ‡ ; r ‡ . Since the lemma clearly holds with Pd in place of f , it follows that the Taylor series of ft; r ± j e t; e r equals X  j Pd ‡ j…X 1 †a 1 . . . j…Yn † b n ´ h a; b Sa‡Sbˆd

 h a; b

for certain 2 C[X; Y ]. By letting d tend to in®nity we obtain the desired result. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .A To illustrate the process of showing that our class of algebras is closed under a given substitution, we ®rst consider a fairly easy case. Assume that m > 1 and let the substitution j: …X; Y † ! C[X 0 ; Y ] be given by j…X i † :ˆ X i

for i ˆ 1; . . . ; m ÿ 1;

j…X m † :ˆ X m ÿ 1 ; j…Yj † :ˆ Yj

for each j:

Thus for f ˆ f …X; Y † 2 C[X; Y ] we have j f ˆ f …X 1 ; . . . ; X m ÿ 1 ; X m ÿ 1 ; Y †: e ; R; e r; f† 2 Tm ÿ 1 with We now let et :ˆ … K and

e :ˆ f…k 1 ; . . . ; k m ÿ 2 ; k m ÿ 1 ‡ k m †: k 2 K g K e :ˆ …R 1 ; . . . ; R m ÿ 2 ; minfR m ÿ 1 ; R m g†: R

Clearly j: …X; Y † ! CfX 0 ; Y g e t ‡; r‡ . 4.3. Lemma. We have j… e S ´ D† Í S ´ D, and if f 2 CfX; Y gt ‡ ; r ‡ then the series j f belongs to CfX 0 ; Y g e t ; r and satis®es …j f †e t ; r ˆ ft; r ± j e t ; r and < k f k . kj f ke t; r t; r

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535

Proof. We distinguish two cases. Case 1: n ˆ 0. We write e k :ˆ …k 1 ; . . . ; k m ÿ 2 ; k m ÿ 1 ‡ k m † for k 2 K . Note that for each k 2 K and all x 0 ˆ …x 1 ; . . . ; x m ÿ 1 † 2 C m ÿ 1 we have k j j…x 0 ; x m ÿ 1 † k j ˆ j…x 0 †e

Se k ˆ Sk; and

k ´ jarg x 0 j: k ´ jarg…x 0 ; x m ÿ 1 †j ˆ e 0 0 e S implies …x 0 ; x m ÿ 1 † 2 S, and Therefore, x 0 2 e P P x 2 S p implies …x ; x m ÿ 1 † 2 S p for each p. Thus, ft ˆt f p gives ft ± j e … fp ± j e t ˆe t t †. By Lemma 4.2 the Taylor series at 0 of ft ± j e t is given by j f, so Case 1 is proved. P Case 2: n > 0. We let f ˆ c b Y b with each c b 2 CfX gt ‡ . By Case 1 each 0 0 j c b belongs to CfX g e t with kj c b ke t < kc b kt . Hence j f belongs to CfX ; Y g e t; r and satis®es kj f ke t ; r < k f kt; r , and it remains to show that …j f †e t ; r ˆ ft; r ± j e t ; r. For each d 2 N we put X f d …X; Y † :ˆ c b …X †Y b 2 CfX; Y gt ‡ ; r ‡ : Sbd

Sb>d

S ´ D. Hence so that lim d ! 1 j f d …x ; y† ˆ j f …x ; y† for all …x ; y† 2 e 0

0

0

j f …x 0 ; y† ˆ lim j fd …x 0 ; y† d!1

0 0 ˆ lim …… fd †t; r ± j e t ; r †…x ; y† ˆ … ft; r ± j e t ; r †…x ; y† d!1

S ´ D, which completes the proof. . . . . . . . . . . . . . . . . . . . A for all …x ; y† 2 e 0

e Y e ] satisfy one of the 4.4. Lemma. Let the substitution j: …X; Y † ! C[ X; three conditions below: e ˆ X, Y e ˆ Y, and a permutation p of f1; . . . ; mg is given such that (1) X j…X i † ˆ X p…i † for each i and j…Yj † ˆ Yj for each j (note that for f ˆ f …X; Y † 2 C[X; Y ] we have j f ˆ f …X p…1† ; . . . ; X p…m† ; Y †); in this case we put e t :ˆ …p…K †; p…R†; r; f† with p…K † :ˆ f…k p…1† ; . . . ; k p…m† †: k 2 K g, r :ˆ r; p…R† :ˆ …R p…1† ; . . . ; R p…m† † and e e e (2) X ˆ X, Y ˆ Y, j…X i † ˆ a i X i with a i 2 …0; 1† for each i, and j…Yj † ˆ Yj for each j (note that for f ˆ f …X; Y † 2 C[X; Y ] we have j f ˆ f …a 1 X 1 ; . . . ; a m X m ; Y †); in this case we put et :ˆ …K ; R=a; r; f† with r :ˆ r; R=a :ˆ …R 1 =a 1 ; . . . ; R m =a m †, and e e e (3) X ˆ X, Y ˆ Z ˆ …Z 1 ; . . . ; Z p †, j…X i † ˆ X i for each i, and a polyradius s 2 …0; 1† p is given such that j…Yj † 2 CfZ gs with j…Yj †…0† ˆ 0 and kj…Yj †ks < r j , for each j (note that for f ˆ f …X; Y † 2 C[X; Y ] we have e :ˆ X, Y e :ˆ Z , j f ˆ f …X 1 ; . . . ; X m ; j…Y1 †; . . . ; j…Yn ††); in this case we put X e t :ˆ t and e r :ˆ s.

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e Í S ´ D, and if f 2 CfX; Y gt ‡ ; r ‡ then the series j f belongs to Then j… e S ´ D† e Y e g t ; r and satis®es …j f † t ; r ˆ ft; r ± j t ; r and kj f k t ; r < k f kt; r . Cf X; ee ee ee ee Proof. We obtain (1) and (2) as in the previous proof, while (3) is standard; we leave the details to the reader. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A The following application P of bLemma 4.4 is of interest in connection with 6ˆ 0, n > 0. Assume that Proposition 3.7. Let f ˆ c b Y 2 CfX; Y gG with fP there is an a 2 N m such that f ˆ X a ´ g with g ˆ g b Y b 2 CfX; Y gG and n g b …0† 6ˆ 0 for at least one b 2 N . (Note that this always holds if m ˆ 1.) Take a minimal d 2 N such that there is a b 2 N n with d ˆ Sb and g b …0† 6ˆ 0. Consider a linear substitution v: …X; Y † ! C[X; Y ] of the form v…X i † :ˆ X i

for each i,

v…Y j † :ˆ Yj ‡ c j Yn

with c j 2 C for j ˆ 1; . . . ; n ÿ 1,

v…Yn † :ˆ Yn : Then …vg†…0; 0; Yn † ˆ g…0; c 1 Yn ; . . . ; c n ÿ 1 Yn ; Yn † ˆ P…c 1 ; . . . ; c n ÿ 1 †Ynd ‡ terms of higher degree in Yn ; where P is a non-zero polynomial in c 1 ; . . . ; c n ÿ 1 depending only on f (but not on c 1 ; . . . ; c n ÿ 1 ). By Lemma 4.4(iii) we have vg 2 CfX; Y gG . We therefore get the following. 4.5. Lemma. Let f 1 ; . . . ; f l be non-zero series belonging to CfX; Y gG , and assume that f k ˆ X a k ´ g k for some suitable a k 2 N m and g k 2 CfX; Y gG satisfying g k …0; Y † 6ˆ 0, for k ˆ 1; . . . ; l. Then there is a linear substitution v as above such that vf k ˆ X a k ´ h k with each h k 2 CfX; Y gG regular in Yn . . . . . . A We now return to dealing with substitutions. Assume that m > 1 and consider the singular blow-up substitution s: …X; Y † ! C[X; Y ] given by s…X i † :ˆ X i

for i ˆ 1; . . . ; m ÿ 1;

s…X m † :ˆ X m ÿ 1 X m ; s…Yj † :ˆ Yj

for each j.

Thus for f ˆ f …X; Y † 2 C[X; Y ] we have sf ˆ f …X 1 ; . . . ; X m ÿ 1 ; X m ÿ 1 X m ; Y †: p e i :ˆ minfR i ; R i g for each i and e R; e r; f† 2 Tm with R Let et :ˆ … K; e :ˆ f e K k: k 2 K g

with e k :ˆ …k 1 ; . . . ; k m ÿ 2 ; k m ÿ 1 ‡ k m ; k m †:

4.6. Lemma. We have s… e S ´ D† Í S ´ D, and if f 2 CfX; Y gt ‡ ; r ‡ then the series and satis®es …sf † e sf belongs to CfX; Y g e t ;r t ; r ˆ ft; r ± s e t ; r and ksf k e t ; r 0. Given f 2 CfX; Y gG and 0 < d 2 N, the series f …X 0 ; X md ; Y † also belongs to CfX; Y gG : by Remark 2.10 we have f 2 CfX; X m ‡ 1 ; Y gG , so by repeated application of the above lemma f …X 0 ; X md ÿ 1 X m ‡ 1 ; Y † 2 CfX; X m ‡ 1 ; Y gG , and then f …X 0 ; X md ; Y † 2 CfX; Y gG by Lemma 4.3. Next, assume that m > 1, let l > 0 and consider the regular blow-up substitution r l : …X; Y † ! C[X; Y ] given by r l …X i † :ˆ X i

for i ˆ 1; . . . ; m ÿ 1;

r l …X m † :ˆ X m ÿ 1 …l ‡ X m †; r l …Yj † :ˆ Yj

for each j:

Thus for f ˆ f …X; Y † 2 C[X; Y ] we have j f ˆ f …X 1 ; . . . ; X m ÿ 1 ; X m ÿ 1 …l ‡ X m †; Y †: 1 e for all v 2 D…2r 0 † e Let f 2 … 2 p; f† and r0 > 0 be such that k m jarg…l ‡ v†j < f ÿ f e and r 0 exist.) Let l :ˆ maxfk m : k 2 K g and and all k 2 K. (Note that such f e e e e t :ˆ … K; R; r; f† 2 Tm ÿ 1 , where e :ˆ f e K k: k 2 K g with e k :ˆ …k 1 ; . . . ; k m ÿ 2 ; k m ÿ 1 ‡ k m †;  e e e R :ˆ …R 1 ; . . . ; R m ÿ 2 ; R m ÿ 1 † with R m ÿ 1 :ˆ min R m ÿ 1 ; R m ; S Í Cmÿ1 and let e r :ˆ …r 0 ; r†. Thus e each k 2 K.

 Rm ; …l ‡ 2r 0 †l e Í C 1 ‡ n . Note that Sk ˆ S e and D k for

e Í S ´ D, and if f 2 CfX; Y gt ‡ ; r ‡ then the 4.7. Lemma. We have r l … e S ´ D† 0 series r l f belongs to CfX ; X m ; Y g e t; e r and satis®es …r l f † e t; e r ˆ ft; r ± …r l † e t; e r and kr l f k e t; e r < 2k f kt; r . Proof. We ®rst consider the special case n ˆ 0. In this case we have the following. Claim 1.

S ´ D…2r 0 †† Í S. r l… e

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Proof. It is enough to consider the case K ˆ fkg. Let u 2 e S and v 2 D…2r 0 †. e so with k 0 :ˆ …k 1 ; . . . ; k m ÿ 1 † we get By de®nition e k ´ jarg…u†j < f, k ´ jarg…r l …u; v††j ˆ k 0 ´ jarg…u†j ‡ k m jarg…u m ÿ 1 …l ‡ v††j < k 0 ´ jarg…u†j ‡ k m jarg…u m ÿ 1 †j ‡ k m jarg…l ‡ v†j e < f; 0, d 2 N, d > 0 and l > 0, and consider the substitution j: …X; Y † ! C[X; T; Y ] given by j…X i † :ˆ X i

for i ˆ 1; . . . ; m ÿ 1;

j…X m † :ˆ X md …l ‡ T †; j…Yj † :ˆ Yj

for each j:

(Note that we have j f ˆ f …X 1 ; . . . ; X m ÿ 1 ; X md …l ‡ T †; Y † for f ˆ f …X; Y † 2 C[X; Y ].)

the reals with multisummable series

541

e 2 … 1 p; f† and r 0 > 0 be such that 1 p < f e < f and k m jarg…a ‡ v†j < f ÿ f e Let f 2 2 e e e e for all v 2 D…2r 0 †. (Note that such f and r 0 exist.) Let et :ˆ … K; R; r; f† 2 Tm e :ˆ f…k 1 ; . . . ; k m ÿ 1 ; dk m †: k 2 K g, R e :ˆ …R 1 ; . . . ; R m ÿ 1 ; R e m † with with K  1=d  e m :ˆ min R m ; R m1 = d ; R m ; R m ; R l ‡ 2r 0 l ‡ 2r 0 and let e r :ˆ …r 0 ; r†. e Í S ´ D, and if f 2 CfX; Y gt ‡ ; r ‡ then the 5.2. Lemma. We have j… e S ´ D† series j f belongs to CfX; T; Y g e t; e r and satis®es …j f † e t; e r ˆ ft; r ± j e t; e r and kj f k e t; e r < 2k f kt; r . Proof. We proceed along the lines of the proof of Lemma 4.7, giving only an outline here. As in the proof of Lemma 4.3 we ®rst consider the case n ˆ 0. In this case we de®ne h: e S ´ D…2r 0 † ! C by h…x; v† :ˆ f …x 0 ; x md …l ‡ v††. Then for every ®xed x 2 e S the function h x : D…2r 0 † ! C de®ned by h x …v† :ˆ h…x; v† is holomorphic, and its Taylor series at the origin is given by X h…n† x …0† n T : hcx …T † ˆ n! n2N Each function x 7! h n …x† :ˆ …1=n!†h…n† x …0† with n 2 N belongs to G e t and satis®es n kh n k e t < k f kt =…2r 0 † , that is, 1 X n kh n k e t ´ r 0 < 2k f kt : nˆ0 P Moreover j f ˆ hbn T n , so the lemma is proved for n ˆ 0. The case n > 0 is now an easy consequence.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A We next consider a g 2 CfT gG such that g…T † ˆ T d …l ‡ h…T †† for some d 2 N, l > 0 and h 2 CfT gG with h…0† ˆ 0, and we let j: …X; Y † ! C[X; Y ] be given by j…X i † :ˆ X i for i ˆ 1; . . . ; m ÿ 1; j…X m † :ˆ g…X m †; j…Yj † :ˆ Yj

for each j:

If d ˆ 0, we assume in addition that l < R m . r 2 …0; 1†n such that 5.3. Proposition. There are a e t 2 Tm and a e e Í S ´ D; and j… e S ´ D† j: …X; Y † ! CfX; Y g e t ‡; e r‡ and such that if f 2 CfX; Y gt ‡ ; r ‡ then there is a series j f 2 CfX; Y g e t; e r satisfying …j f † e t; e r ˆ f t; r ± j e t; e r. Proof. If g…0† ˆ 0, then we apply Lemmas 5.2 and 5.1. If g…0† 6ˆ 0, then by Lemma 4.8 there is a series t l f 2 CfX 0 ; X m ; Y gG such that t l f is the Taylor series at 0 of f t; r ± t l . Now apply Lemma 5.1 with h in place of g to t l f . . . A Finally, we prove that a compositional inverse exists in CfT gG for certain elements of CfT gG . First, let the substitution j: …T; Y † ! C[T; Y2 ; . . . ; Yn ] be

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given by j…T † :ˆ T; j…Y1 † :ˆ g…T †; j…Yj † :ˆ Yj

for j ˆ 2; . . . ; n;

where g 2 CfT gt is such that g…0† ˆ 0 and kgkt < r1 . (Note that for e :ˆ …Y2 ; . . . ; Yn † f ˆ f …X; Y † 2 C[X; Y ] we have j f ˆ f …T; g…T †; Y2 ; . . . ; Yn †.) Let Y and put e r :ˆ …r 2 ; . . . ; r n †. e Í S ´ D, and e gt; r and j… e S ´ D† 5.4. Lemma. We have j: …T; Y † ! CfT ; Y e e and satis®es whenever f 2 CfT ; Y gt; r , then j f belongs to CfT ; Y gt; e r …j f †t; e r ˆ ft; r ± jt; e r and kj f kt; e r < k f kt; r . Proof. For simplicity we the case n ˆ 1 (so r 2 R and Y is a single P just consider n c Y 2 CfT ; Y gP indeterminate). Let f ˆ 1 nˆ0 n t; r . Since CfT gt is a Banach c n …T †g…T †n belongs to CfT gt algebra and kgkt < r, we ®nd that j f …T † ˆ with kj f kt < k f kt; r . Similarly to Case 2 in the proof of Lemma 4.3 we get …j f †t ˆ ft; r ± jt . (Again, we do not use Lemma 4.2 here.) . . . . . . . . . . . . . . . . . . . A 5.5. De®nition. Let g 2 C[T ] be such that g…0† ˆ 0, and let j: …T † ! C[T ] be the substitution de®ned by j…T † :ˆ g…T †. Given f 2 C[T ], we shall simply write f ± g instead of j f , and we call f ± g the composition of f with g. These compositions behave as expected; for example, if f 2 C[T ] with f …0† 6ˆ 0, then 1=… f ± g† ˆ …1=f † ± g. Below we shall freely use facts of this nature. 5.6. Proposition. Let f …T † ˆ T d …l ‡ h…T †† 2 CfT gG be such that d 2 N is non-zero, l > 0 and h 2 CfT gG with h…0† ˆ 0. Then there is a g 2 CfT gG such that g…0† ˆ 0 and f ± g ˆ T d . Proof. If … f = l† ± g 1 ˆ T d , where g 1 2 CfT gG with g 1 …0† ˆ 0, then f ± g 1 ˆ lT d , and hence f ± g ˆ T d , where g :ˆ g 1 …lÿ1 = d T † belongs to CfT gG by Lemma 4.4(2). Thus upon replacing f by f = l, we may as well assume that l ˆ 1. Choose i 2 T1 such that f ; h 2 CfT gi and kT ki < 1. There are t < i, s 2 …0; 1† and « n ; dn 2 CfT gt for each n, such that P …S†n «n …0† ˆ 0, k« n kt < s n and kdn kt < s n ‡ 1 , and with g n :ˆ niˆ 0 T d ÿ 1 « i we have f ± T…1 ‡ g n † ˆ T d …1 ‡ T d d n †:

Claim.

Assume for the that the claim holds. By the completeness of CfT gt P moment dÿ1 the series g :ˆ 1 « n belongs to CfT gt , so that f ± T…1 ‡ g† 2 CfT gG nˆ0 T by Proposition 5.3. By the claim, we have f ± T…1 ‡ g n † 2 CfT gt for each n, and hence, in CfT gt , we have lim … f ± T…1 ‡ g n †† ˆ lim …T d …1 ‡ T d d n †† ˆ T d :

n!1

n!1

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543

On the other hand, we get from the claim the fact that lim n ! 1 f …t…1 ‡ g n …t ††† ˆ f …t…1 ‡ g…t ††† for all suf®ciently small t > 0, which together with the above and Proposition 2.18 implies that f ± T…1 ‡ g† ˆ T d . Proof of the Claim. Note that f 0 ˆ T d ÿ 1 ef with ef 2 CfT gG and ef …0† ˆ d > 1, so 1= ef belongs to CfT gG as well. Decreasing i if necessary, we may assume that f , f 0 and 1= ef belong to CfT gi and satisfy k f ki < 1

and k1= ef ki < 14 :

…5:1†

Next, by Lemma 2.22, there are an q 2 T2 and a series R 2 f 2 CfX 1 ; X 2 gq such that f …X 1 ‡ X 2 † ˆ f …X 1 † ‡ f 0 …X 1 †X 2 ‡ X 22 ´ …R 2 f †…X 1 ; X 1 ‡ X 2 †: Applying Lemma 5.2 to f , 1= ef and R 2 f in place of f , we obtain a t < i and a r 2 …0; 1† such that f ± T…1 ‡ Y1 † 2 CfT ; Y1 gt; r 1

ef ± T …1 ‡ Y1 †

2 CfT ; Y1 gt; r

with k f ± T …1 ‡ Y1 †kt; r < 2k f ki ;



1

< 2k1= ef ki

with ef ± T …1 ‡ Y1 † t; r

…5:2†

and …R 2 f †…T …1 ‡ Y1 †; T …1 ‡ Y2 †† 2 CfT ; Y1 ; Y2 gt; … r; r† with k…R 2 f †…T …1 ‡ Y1 †; T …1 ‡ Y2 ††kt; … r; r† < 4kR 2 f kq : Next, we decrease r if necessary and choose s 2 …0; r† such that d ÿ 1  1 s 1 < r and s < < 2; : 1ÿr 1ÿs 4kR 2 f kq

…5:3†

…5:4†

Decreasing t further if necessary, we may also assume that khkt < s 2 < s. We now proceed by induction on n. Initial Step. We put « 0 :ˆ 0 and d 0 :ˆ h; then k« 0 kt ˆ 0 and kd 0 kt < s 2 . This proves …S†0 . Inductive Step. Let n > 0 and assume that we are given d i ; « i 2 CfT gt for i ˆ 0; . . . ; n ÿ 1 such that …S†i holds for each i < n. Put « n :ˆ

ÿT ´ d n ÿ 1 1 ´ : ef ± T …1 ‡ g n ÿ 1 † …1 ‡ g n ÿ 1 †d ÿ 1

By the inductive hypothesis, (5.1) and (5.4), we have kg n ÿ 1 kt < s ‡ s 2 ‡ . . . ‡ s n ÿ 1 < r; so by (5.2) and Lemma 5.4 the series 1=… ef ± T …1 ‡ g n ÿ 1 †† belongs to CfT gt and satis®es k1=… ef ± T …1 ‡ g n ÿ 1 ††kt < 2k1= ef ki . Moreover, by (5.4),

dÿ1

1

< …1 ‡ kg n ÿ 1 kt ‡ kg n ÿ 1 k t2 ‡ . . .†d ÿ 1

…1 ‡ g † t nÿ1  d ÿ 1 1 < < 2; 1ÿr

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which shows that « n 2 CfT gt . Now we see from the inductive hypothesis (respectively, from the Initial Step if n ˆ 1) that ( s 2 if n ˆ 1; …5:5† k« n kt < kd n ÿ 1 kt ´ 12 ´ 2 ˆ kd n ÿ 1 kt < s n if n > 1: In particular, kg n kt < r. Note that T d =f 0 ˆ T = ef , so that «n ˆ

ÿT d d n ÿ 1 : f 0 ± T …1 ‡ g n ÿ 1 †

Hence, by Lemma 5.4 and the inductive hypothesis, we have f ± T …1 ‡ g n † ˆ f ± T …1 ‡ g n ÿ 1 ‡ T d ÿ 1 « n † ˆ f ± T …1 ‡ g n ÿ 1 † ‡ … f 0 ± T …1 ‡ g n ÿ 1 †† ´ T d « n ‡ T 2 d « n2 ´ …R 2 f †…T …1 ‡ g n ÿ 1 †; T …1 ‡ g n †† ˆ T d …1 ‡ T d d n ÿ 1 ‡ … f 0 ± T …1 ‡ g n ÿ 1 †† ´ « n ‡ T d « n2 ´ …R 2 f †…T …1 ‡ g n ÿ 1 †; T …1 ‡ g n ††† ˆ T d …1 ‡ T d « n2 ´ …R 2 f †…T …1 ‡ g n ÿ 1 †; T …1 ‡ g n †††: Putting d n :ˆ « n2 ´ …R 2 f †…T …1 ‡ g n ÿ 1 †; T …1 ‡ g n ††, we therefore get f ± T …1 ‡ g n † ˆ T d …1 ‡ T d d n † with d n 2 CfT gt . Moreover, by (5.3) and (5.5), we have kd n kt < k« n k 2t ´ k…R 2 f †…T …1 ‡ g n ÿ 1 †; T …1 ‡ g n ††k t < s n ‡ 1 ; which completes the proof of the claim.. . . . . . . . . . . . . . . . . . . . . . . . . . A 6. Gevrey semianalytic sets and model completeness In this section we prove model completeness and o-minimality of R G . We also e of show that R G admits analytic cell decomposition. (An o-minimal expansion R the ordered ®eld of real numbers is said to admit analytic cell decomposition if e there is a decomposition of R m into for any A1 ; . . . ; A k Í R m de®nable in R, e analytic cells de®nable in R and compatible with each A i .) We ®x m and n, and let r 2 …0; 1†m ‡ n be a polyradius. We put Im; n; r :ˆ ‰0; r1 Š ´ . . . ´ ‰0; r m Š ´ ‰ÿr m ‡ 1 ; r m ‡ 1 Š ´ . . . ´ ‰ÿr m ‡ n ; r m ‡ n Š Í R m ‡ n ; and we write Im; n; « instead of Im; n; …«;...; «† for positive real «, and Im; n; 1 :ˆ ‰0; 1†m ´ R n : 6.1. De®nition. We let RfX; Y gG; r :ˆ

[ t; j

CfX; Y gt; j Ç R[X; Y ];

where the union is taken over all t ˆ …K ; R; r; f† 2 Tm with R i > r i for each i ˆ 1; . . . ; m and all j 2 …0; 1†n with j j > r m ‡ j for each j ˆ 1; . . . ; n. It follows

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from § 3 that RfX; Y gG; r is a subalgebra of the R-algebra R[X; Y ] closed under the operators ¶= ¶X i and ¶= ¶Yj . Correspondingly, we let Gm; n; r be the set of all functions …x; y† 7! f …x; y†: Im; n; r ! R with f 2 RfX; Y gG; r . Note that Gm; n; r contains all R-valued constant functions on Im; n; r and is a ring of R-valued functions on Im; n; r under pointwise addition and multiplication of functions. (In contrast, Gt; r consists of C-valued functions on a subset of C m ‡ n .) If « > 0, we write RfX; Y gG; « and Gm; n; « instead of RfX; Y gG; …«;...;«† and Gm; n; …«;...;«† . Next, we put [ RfX; Y gG; r : RfX; Y gG :ˆ r 2 …0; 1†m ‡ n

1

Note that each f 2 Gm; n; r is C on Im; n; r and real analytic on int…Im; n; r †, and that its partials ¶f = ¶x i and ¶f = ¶y j also belong to Gm; n; r . The properties in §§ 3, 4 and 5 of the algebras CfX; Y gt ‡ ; r ‡ are easily seen to imply corresponding properties for the algebras RfX; Y gG; r and Gm; n; r . For n ˆ 0 we just write RfX gG; r instead of RfX; Y gG; r . Note that Gm; 0; r ˆ G…r†, where the latter is de®ned as in De®nition 2.20. 6.2. De®nition. A set A Í Im; n; r is called a basic Gm; n; r -set if there are f ; g 1 ; . . . ; g k 2 Gm; n; r such that A ˆ fz 2 I m; n; r : f …z† ˆ 0; g 1 …z† > 0; . . . ; g k …z† > 0g: A Gm; n; r -set is a ®nite union of basic Gm; n; r -sets. Note that the Gm; n; r -sets form a boolean algebra of subsets of Im; n; r . Given a point a ˆ …a 1 ; . . . ; a m ‡ n † 2 R m ‡ n and a choice of signs j 2 fÿ1; 1gm , we let h a; j : R m ‡ n ! R m ‡ n be the bijection given by h a; j …z† :ˆ …a 1 ‡ j1 z 1 ; . . . ; a m ‡ jm z m ; a m ‡ 1 ‡ z m ‡ 1 ; . . . ; a m ‡ n ‡ z m ‡ n †: Note that the maps h a; j (with a 2 R m ‡ n and j 2 fÿ1; 1gm ) form a group of permutations of R m ‡ n . 6.3. De®nition. If a 2 R m ‡ n , then a set X Í R m ‡ n is Gm; n -semianalytic at a if there is an « > 0 such that for each j 2 fÿ1; 1gm there is a Gm; n ; « -set A j Í Im; n; « with X Ç h a; j …Im; n; « † ˆ h a; j …A j †: A set X Í R m ‡ n is Gm; n -semianalytic if it is Gm; n -semianalytic at every point a 2 R m ‡ n . For convenience, if X Í R m is Gm; 0 -semianalytic, we also simply say that X is Gm -semianalytic. 6.4. Remark. (1) If X; Y Í R m ‡ n are Gm; n -semianalytic at a, then so are X È Y, X Ç Y and X nY. (2) Let X Í R m ‡ n be Gm; n -semianalytic, a 2 R m ‡ n and j 2 fÿ1; 1gm . Then the set h a; j …X † is Gm; n -semianalytic. Moreover, for each l 2 …0; 1†m ‡ n the set E l …X † is Gm; n -semianalytic, where E l : R m ‡ n ! R m ‡ n is given by E l …z† ˆ …l 1 z 1 ; . . . ; l m ‡ n z m ‡ n † (here we use Lemma 4.4(2)).

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(3) If X Í R n is semianalytic, then X is G0; n -semianalytic. Below we write 0 for the point …0; . . . ; 0† 2 R m ‡ n . The following lemma is now proved just as in [13, § 7] with obvious changes: `R... -set' is replaced by `G... -set', RfX  ; Y g by RfX; Y gG and the algebras Rm; n;... by Gm; n;.... Also, the results from §§ 4, 5 and 6 there need to be replaced by the corresponding results of §§ 2, 3 and 4 here. 6.5. Lemma. (1) Let A Í R m ‡ n be Gm; n -semianalytic at 0 and let j be a permutation of f1; . . . ; mg. Then j…A† :ˆ f…x j…1† ; . . . ; x j…m† ; x m ‡ 1 ; . . . ; x m ‡ n †: x 2 Ag is Gm; n -semianalytic at 0. (2) If n > 1, then each Gm; n -semianalytic subset of R m ‡ n is Gm ‡ 1; n ÿ 1 semianalytic. (3) Every Gm; n; r -set A Í Im; n; r is Gm; n -semianalytic. . . . . . . . . . . . . . . . . A Note that Remark 6.4(3) and Lemma 6.5(2) imply in particular that every semianalytic subset of R m ‡ n is Gm; n -semianalytic. We now consider the system L ˆ …L p †p 2 N , where each L p is the collection of subsets of I p with I :ˆ ‰ÿ1; 1Š de®ned by L p :ˆ fA Í I p : A is Gp -semianalyticg: Note that if A Í I p is Gm; n -semianalytic with m ‡ n ˆ p, then A is also Gp semianalytic by Lemma 6.5(2), so A 2 L p . We refer the reader to [13] for the notion of L-set and the L-Gabrielov Property. The system L is easily seen to satisfy Axioms (I)±(III) listed in [13, § 2]; the next proposition establishes Axiom (IV). 6.6. Proposition. Every L-set has the L-Gabrielov Property. Proof. The proof proceeds almost literally as in [13, § 8], with the obvious changes indicated earlier, as well as the following. Replace RfX  ; Y g . . . there by RfX; Y gG ; . . . here. The quadruples b… f † 2 N 4 there become quintuples b… f † 2 N 5 here, de®ned from the blow-up height of § 1 here rather than § 4 g g there. The g > 0 in the notation s m; m ÿ 1 and r l there (for singular and regular g g blow-up substitutions) is simply equal to 1 here, and s m; m ÿ 1 ; r l there correspond to s, r l here as de®ned in § 4. The facts of the present section here are used in place of the corresponding facts from § 7 there. It remains to note that Remark 8.11 there becomes slightly simpler here: while the generalized power series rings are only closed under the differential operators X i ´ …¶= ¶X i †, the rings we are dealing with here are actually closed under ¶= ¶X i . A Recall that by the remarks after De®nition 6.1, R G ˆ …R; m and a bounded Gn -semianalytic set B Í R n such that A ˆ P m …B†, where P m : R n ! R m is the projection on the ®rst m coordinates. More precisely, there are an n > m and bounded Gn -semianalytic sets B 1 ; . . . ; B k Í R n such that A ˆ P m …B 1 † È . . . È P m …B k † and each B i is a connected analytic (embedded) submanifold of R n for which P m jB i is an immersion into R m . . . . . . . . . . . . A We use this proposition to show that R G admits analytic cell decomposition. Since Proposition 6.8 holds for R a n  , Corollary 6.10 below also holds with R a n  in place of R G . 6.9. Lemma. Let A Í R m be de®nable in R G . Then A is a ®nite union of analytic manifolds that are de®nable in R G . Proof. This is by induction on d :ˆ dim…A†; the case d ˆ 0 is trivial, so we assume that d > 0 and the lemma holds for lower values of d. Using an analytic, semialgebraic diffeomorphism mapping R m onto …ÿ1; 1†m , we may assume that A is bounded. Thus by the previous proposition, we may assume that A ˆ P m …B†, where B Í R n with n > m is a Gn -semianalytic set and an analytic submanifold of R n for which P m jB is an immersion. Using cell decomposition, we now partition B into cells B 1 ; . . . ; B k de®nable in R G . By the inductive hypothesis, it suf®ces to show that if i 2 f1; . . . ; kg is such that dim…P m B i † ˆ d, then P m …B i † is an analytic manifold. Fix such an i; since P m jB is an immersion, we have dim…B† ˆ d, so that dim…B i † ˆ dim…P m B i †. Since B i is a cell, P m jB i is a homeomorphism onto P m …B i †. Also, B i must be open in B, and hence is itself an analytic manifold. Thus the image P m …B i † of the analytic immersion P m jB i is an analytic manifold. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A 6.10. Corollary. The structure R G admits analytic cell decomposition. Proof. We show by induction on m that if A is a ®nite collection of subsets of R m de®nable in R G , then there is a decomposition of R m into analytic cells de®nable in R G that is compatible with each member of A. The cases m ˆ 0; 1 are trivial, so we assume that m > 1. Let f : A ! R be a C 1 function de®nable in R G with A Í R m ÿ 1 a C 1 cell. It suf®ces to partition A into analytic cells A 1 ; . . . ; A K de®nable in R G , such that f jA j is analytic for each j. This follows easily from the inductive hypothesis by applying Lemma 6.9 to G… f †. . . . . . . A

548

lou van den dries and patrick speissegger 7. Characterizing de®nable 1-variable functions

In this section we prove the second part of Theorem A, which characterizes de®nable 1-variable functions. We proceed along the lines of [13, § 9]. Below we let d and e range nf0g, and we let R……t Q †† be the ®eld of all formal P over N g with each a g 2 R and such that fg 2 Q: a g 6ˆ 0g is power series g2Q ag t well ordered. 7.1. De®nition. For a single indeterminate T we let and we put

RfT 1 = d gG :ˆ f f …T 1 = d †: f 2 RfT gG g Í R……t Q ††; P…RfT gG † :ˆ

[ d >1

RfT 1 = d gG :

Note that P…RfT gG † is a valuation ring whose maximal ideal consists of those f in P…RfT gG † satisfying f …0† ˆ 0. We shall consider the fraction ®eld of P…RfT gG † as a sub®eld of R……t Q ††. Every g 6ˆ 0 in the fraction ®eld of P…RfT gG † is of the form g ˆ T r h with r 2 Q and h 2 P…RfT gG † satisfying h…0† 6ˆ 0, and we make this fraction ®eld into an ordered ®eld by putting g > 0 if h…0† > 0. 7.2. Lemma. The local ring P…RfT gG † is henselian, that is, given any f …T; W † ˆ W n ‡ a 1 …T †W n ÿ 1 ‡ . . . ‡ a n …T † 2 P…RfT gG †‰W Š with f …0; 0† ˆ 0 and …¶f = ¶W †…0; 0† 6ˆ 0, there is an a 2 P…RfT gG † such that a…0† ˆ 0 and f …T; a…T †† ˆ 0. Proof. First we observe that RfT gG is henselian. Let f …T; W † 2 RfT gG ‰W Š be as in the statement of the lemma. Considering f …T; W † as an element of RfT ; W gG , this means that f is regular in W of order 1. Hence, by the Weierstrass Preparation Theorem 3.7 we have f …T; W † ˆ u…T; W †…W ÿ a…T †† for some unit u 2 RfT ; W gG and some a 2 RfT gG , so f …T; a…T †† ˆ 0. Next, each ring RfT 1 = d gG for non-zero d 2 N is isomorphic to RfT gG , and hence is henselian. Therefore P…RfT gG † is henselian. . . . . . . . . . . . . . . . . . A 7.3. Corollary. The fraction ®eld of P…RfT gG † is real closed. Proof. This follows from the previous lemma and the remarks preceding it, using [19]. 7.4. Lemma. Let r 2 …0; 1†m ‡ n , s 2 …0; 1†, f 2 RfX; Y gG; r and g 1 ; . . . ; g m ‡ n 2 RfT gG; s be such that g i > 0 for each i ˆ 1; . . . ; m and jg j …t †j < r j for all t 2 ‰0; sŠ and each j ˆ 1; . . . ; m ‡ n. Then there is an h 2 RfT gG such that, for every suf®ciently small t > 0, h…t † ˆ f …g1 …t †; . . . ; g m ‡ n …t ††. Proof. For j ˆ 1; . . . ; n let « j :ˆ ÿ1 if g m ‡ j …0† < 0 and « j :ˆ 1 otherwise. Then the series f …X; « 1 Y1 ; . . . ; « n Yn † belongs to RfX; Y gG; r as well, so replacing each g m ‡ j by « j ´ g m ‡ j we may assume that g m ‡ j …0† > 0 for each j. Now use Lemmas 5.1, 5.3 and 4.4(1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A

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Given g 2 P…RfT gG † we choose d; r > 0 and e g 2 RfT gG; r such that 1=d 1=d †, and we put g…x† :ˆ e g …x † for 0 < x < r d . (This is unambiguous gˆe g …T in the sense that if we make another such choice, then the same value of g…x† results for those x where g…x† is de®ned for both choices.) 7.5. Proposition (Curve Selection). Let B Í R n be de®nable in R G , and let 0 2 bd…B†. Then there exists a g ˆ …g 1 ; . . . ; g n † 2 P…RfT gG †n such that g…0† ˆ 0 and g…t † 2 B for all suf®ciently small t > 0. Proof. The proof goes like the proof of [13, 9.6], with the obvious changes indicated in the proof of Proposition 6.6, and using Corollary 7.3 and Lemma 7.4 (together with the previous remark) here in place of 9.2 and 9.4 there. . . . . . . A We can now deduce the second part of Theorem A. 7.6. Theorem. Let « > 0 and let f : …0; «† ! R be de®nable in R G . Then there exist a g 2 P…RfT gG † and an r 2 Q such that f …t † ˆ t r g…t† for all suf®ciently small t > 0. Proof. Assume ®rst that lim t ! 0 f …t † ˆ 0. Then …0; 0† 2 fr…G… f ††, so by Proposition 7.5 there are g 1 ; g 2 2 P…RfT gG † such that …g 1 …t †; g 2 …t †† 2 G… f † for g1; e g 2 2 RfT gG all small enough t > 0 and g 1 …0† ˆ g 2 …0† ˆ 0. Take e > 0 and e such that g 1 ˆ e g 1 …T 1 = e † and g 2 ˆ e g 2 …T 1 = e †. By Lemma 5.6 there are h 2 RfT gG and d > 0 such that h > 0, h…0† ˆ 0 and e g 1 ± h ˆ T d . Then it is clear that the desired result holds for g :ˆ …e g 2 ± h†…T 1 = d †. If lim t ! 0 f …t † ˆ c < 1, then the theorem follows easily from the case above by considering f ÿ c in place of f . If lim t ! 0 j f …t †j ˆ 1, then the theorem follows similarly from the ®rst case by considering 1=f in place of f . . . . . . . . . . . . A 7.7. Corollary. The expansion R G of the real ®eld is polynomially bounded. A 8. Examples of de®nable and non-de®nable functions We indicate here two functions (already mentioned brie¯y in the introduction) that are de®nable in R G , and we also give a large class of functions that are not de®nable in R G . 8.1. Example.

From Binet's second formula (see for instance [23]), we have

log G…z† ˆ …z ÿ 12 † log z ÿ z ‡ 12 log…2p† ‡ f…z† for z 2 C n…ÿ1; 0Š; and the function f has an asymptotic expansion at ‡1 that is referred to in the literature as Stirling's series. The estimates on the remainder term of Stirling's series found in [18, § 79] show that the function w…z† :ˆ f…1=z† belongs in fact to G…R; f; 1† for every R > 0 and f 2 …0; p†. Also, since G is meromorphic on C, the restriction Gj…0; 1† is de®nable in R a n . Corollary. The function x 7! log G…x† ÿ …x ÿ 12 † log x: …1; 1† ! R is de®nable in R G . Therefore, Gj…0; 1† is de®nable in …R G ; exp†. . . . . . . . . . A

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8.2. Example.

Consider the function f : ‰0; 1† ! R given by Z 1 eÿt dt: f …x† :ˆ 1 ‡ xt 0 (In [10] it is shown that f is not de®nable in …R a n ; exp†.) As mentioned in the function f introduction, f j‰0; 1Š belongs to G…1† and thus is de®nable in P R G . The has as its Taylor series at 0 the divergent power series …ÿ1† n n!X n and is actually the Borel sum on ‰0; 1† of this power series. (For more on this, see Hardy [15].) The function f is not polynomially bounded at ‡1, and thus f itself is not de®nable in R G . But as noted in [10], its restriction f j‰1; 1† is de®nable in …R a n ; exp†. Combining these facts, we obtain the following. Corollary. The function f is de®nable in …R G ; exp†. . . . . . . . . . . . . . . A For the remainder of this section we assume familiarity with [2], and we freely use its de®nitions and notation. Following a common abuse of notation (as in [2]), we shall write z ˆ r e i f for the point on the Riemann surface C log of the logarithm that has modulus r > 0 and argument f 2 R; thus r e i…2p ‡ f† is a ck and Lk the formal and analytic, different point on C log . We denote by L b k and Bk denote respectively, Laplace transforms of index k > 0, and similarly B the formal and analytic Borel transforms of index k > 0. (Such an analytic transform is de®ned on a subset of C log .) 8.3. Example. Let T be a single indeterminate. By Proposition 2.17 the algebra CfT gG consists exactly of those power series in C[T ] that are ksummable in the positive real direction for some ®nite sequence k ˆ …k 1 ; . . . ; k n † in …0; 1Š, as de®ned in [2]. Conversely, every power series f …T † 2 C[T ] that is ksummable in the positive real direction for some arbitrary ®nite sequence k in …0; 1† is of the form f …T † ˆ g…T 1 = d † for some g 2 CfT gG and some non-zero d 2 N. (Such a series f is also referred to simply as multisummable in the positive real direction.) On the other hand, Proposition 2.17 and [2] imply that for every f 2 CfT gt with t ˆ …K; R; r; f† 2 T1 the Borel sum fB of f , as de®ned in [2], contains in its domain some interval …0; «† with 0 < « < R and agrees there with ft (where we identify a positive real number s with the point se i ´ 0 on C log ). We therefore get another corollary. Corollary. Let f 2 R[T ] be multisummable in the positive real direction. Then there is an « > 0 such that fB is de®ned on the interval …0; «† and fB j…0; «† is de®nable in R G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Finally, using multisummability theory and Theorem 7.6, we show that no P divergent power series a n X n with all a n > 0 can be the Taylor series at 0 of a de®nable C 1 function f : ‰0; «† ! R. If h 2 C[T ] is a convergent power series (that is, converges in a neighbourhood of 0 2 C) with radius of convergence a > 0, we denote here (as in [2]) by Sh: D…a† ! C the analytic function de®ned by h on the disk D…a† Í C. P 8.4. Lemma. Let f ˆ a n T n be such that a n > 0 for all n. Assume that f has in®nite radius of convergence and that A; B; P k > 0 are such that f …x† < A exp…Bx k † ck f :ˆ G…1 ‡ n=k†a n T n is convergent and for all x > 0. Then the power series L ck f †…x† ˆ Lk …Sf †…x† for all suf®ciently small x > 0. satis®es S…L

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Proof. The entire function Sf lifts (when restricted to C nf0g) to a holomorphic function on C log , and we denote this lifting by Sf as well. The growth condition on f implies that Sf is of exponential size at most k on any sector S…0; a† Í C log with a > 0. Hence by Theorem 1 of [2, Chapter 2] with k 1 ˆ ‡1 and f in place of b f there, for every b > 0 there exists an r ˆ r…b† > 0 ck f on the sector S :ˆ S…0; b; r† Í C log . such that Lk …Sf † >k L Now choose a b > 2p ‡ p=k and let r :ˆ r…b†. For z ˆ re i f 2 S 0 :ˆ S…p; b ÿ p; r† Í C log we put z 0 :ˆ re i…f ÿ 2p† 2 C log , and we de®ne an analytic function g: S 0 ! C by g…z† :ˆ …Lk …Sf ††…z† ÿ …Lk …Sf ††…z 0 †: Then g >k 0, and thus by Theorem 2 of [2, Chapter 2] with g in place of f , Bk g >1 0, that is, Bk g ˆ 0. Since b ÿ p > p=k, we conclude from Theorem 3 of [2, Chapter 2] with g in place of f that g ˆ …Lk ± Bk †g ˆ Lk 0 ˆ 0. Therefore, the function h: D…r†n…ÿr; 0Š ! C de®ned by h…z† :ˆ …Lk …Sf ††…z† extends to an analytic function e h: D…r† ! C (here D…r† is the disk in C de®ned in § 2). Since ck f , this shows that the latter is convergent and the Taylor series of e h is equal to L ck f †…x† ˆ Lk …Sf †…x† for all suf®ciently small x > 0. . . . . . . . . A satis®es S… L P 8.5. Lemma. If f ˆ a n T n is divergent and satis®es a n > 0 for all n, then f 62 RfT gG . Proof. Assume for a contradiction that f belongs to CfT gG . Take q minimal such that f 2 CfT gt for some t ˆ …K; R; r; f† 2 T1 with K ˆ ft 1 ; . . . ; t q g and t 1 < . . . < t q . (We use t i here in place of the usual k i in order to make the notation below agree with [2].) Note that t q > 0, since f is divergent. Furthermore, if t 1 ˆ 0, then by the claim in Lemma 2.16 we have f ˆ f 1 ‡ f 2 with f 1 2 CfT gt 0 , t 0 :ˆ …K 0 ; R; r; f†, K 0 :ˆ ft 2 ; . . . ; t q g, and f 2 absolutely convergent on D…R†. By Lemma 2.3, f 2 belongs to CfT gt 0 as well, so f ˆ f1 ‡ f 2 2 CfT gt 0 . The minimality of q therefore implies that t 1 > 0. Write k i :ˆ 1=t i for i ˆ 1; . . . ; q, and put k :ˆ …k 1 ; . . . ; k q †. By Proposition 2.17 and the remarks in Example 8.3, the series f is k-summable in the positive real direction (corresponding to the multidirection d ˆ …d 1 ; . . . ; d q † ˆ …0; . . . ; 0† in [2]). We de®ne k ˆ …k 1 ; . . . ; k q † by k 1 :ˆ k 1 and 1=k i :ˆ 1=k i ÿ 1=k i ÿ 1 for i ˆ 2; . . . ; q, and we let bk ± . . . ± B bk ± B bk † f : gbq :ˆ …B q

2

1

By Theorem 1 of [2, Chapter 7] with f in place of b f (notice the misprint there: b b b b bk b g q :ˆ Bk q ± . . . ± B f '), the series b g q is ` gbq :ˆ Bk 1 ± . . . ± Bk q f ' should be `b 1 g q † extends to a real analytic function on convergent and the function g q :ˆ S…b …0; 1† of exponential size at most k q . ck b ck b g is convergent and satis®es S… L g †…x†ˆLk q g q …x† We claim that the series L q q q q for all suf®ciently small x > 0. In fact, by the hypothesis on f and the de®nition of P b n T n with b n > 0 for all n. Thus, if a > 0 is the formal Borel transform, b gq ˆ the radius of convergence of b g q , then lim x ! a ÿ g q …x† ˆ ‡1. Since g q extends to a real analytic function on …0; 1†, we get a ˆ ‡1. But g q is of exponential size at g q in place of f ) now implies the claim. most k q , so the previous lemma (with b b b b ck b g ˆ … B On the other hand, L k q ÿ 1 ± . . . ± B k 2 ± Bk 1 † f , so we get from the q q

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claim and Theorem 1 of [2, Chapter 7] that f is …k 1 ; . . . ; k q ÿ 1 †-summable in the positive real direction. Hence by Proposition 2.17, f belongs to CfT g e t for some e e e e e t ˆ … K; R; er ; f† with K ˆ ft 1 ; . . . ; t q ÿ 1 g, which again contradicts the minimality of q. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A f : ‰0; «† ! R, with « > 0, that is 8.6. Corollary. There is no C 1 function P a n T n at 0 is divergent with all a n > 0. de®nable in R G and whose Taylor series for a contradiction that there is such an f . Then PProof.n Assume a n T ˆ g…T 1 = d † for some g 2 RfT gG and some non-zero d 2 N. But then P gˆ a n T dn , which contradicts Lemma 8.5. . . . . . . . . . . . . . . . . . . . . . . A P P n!T n is not the Taylor series at Thus in contrast to …ÿ1† n n!T n , the series 1 0 of any C function ‰0; «† ! R, with « > 0, that is de®nable in R G . 9. Expanding by the exponential function e exp†. The only obstacle The goal of this section is to prove Theorem B on … R; in just copying the proof for …R a n ; exp† in [9] is that we ®rst need to know a e For R e ˆ Ran purely valuation-theoretic property of elementary extensions of R. this valuation-theoretic fact was established in [9, § 3] by showing that its elementary extensions could be `nicely' embedded into certain power series e and even for R G , this method is unavailable. Below in models. For general R, Proposition 9.2 we obtain this valuation-theoretic property in another way. e is a polynomially bounded From now on in this section we assume that R o-minimal expansion of the real ®eld. (The assumption that exp j‰0; 1Š is de®nable e is only needed later.) Let R be a model of T :ˆ Th… R† e and V a convex in R subring of R. We let V ˆ V =m be the (ordered) residue ®eld, and v: R ! G È f1g the induced valuation with valuation ring V. In the next lemma we also assume that V 6ˆ R. Then the structure …R; V † is a model of the theory Tconvex in the notation of [6, 8]; also, given a function f : V ! R that is de®nable in …R; V †, there exists by [6, Proposition 4.2] an element g 2 G È f1g such that v… f …x†† ˆ g for all suf®ciently large x 2 V, that is, for all x 2 V greater than some element a 2 V (which may of course depend on f ). Denote this (unique) element g by v… f †. It turns out that we need something better. 9.1. Lemma. Let f : V ! R be de®nable in …R; V †. Then the subset Gf :ˆ fv… f ÿ c†: c 2 Rg of G È f1g has a largest element. Proof. If f is ultimately constant on V, we have 1 2 Gf , so we may assume f is not ultimately constant on V. By [6, Corollary 2.8], and replacing f by ÿf if necessary, we may assume that f …x† > 0 for all suf®ciently large x 2 V. Similarly, replacing f by 1=f if necessary, we may also assume that f is ultimately strictly increasing on V. Take p 2 V such that f is positive and strictly increasing on Vp :ˆ fx 2 V: x > pg. Suppose (for a contradiction) that Gf has no largest element. Consider the map x 7! v… f ÿ f …x††: Vp ! Gf :

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This map is clearly increasing, and we claim that it has co®nal image in Gf . Otherwise there would be an element c 2 R with v… f ÿ c† > v… f ÿ f …x†† for all x 2 Vp . Then take a 2 Vp such that v… f … y† ÿ c† ˆ v… f ÿ c† for all y > a; in particular v… f …a† ÿ c† ˆ v… f ÿ c†. Hence v… f … y† ÿ f …a†† > v… f ÿ c† for all y > a, so v… f ÿ f …a†† > v… f ÿ c† > v… f ÿ f …a††, a contradiction. Since the above map is increasing and has co®nal image, it follows by the `Fact' in the proof of [6, Proposition 2.10] that cofinality…V † ˆ cofinality…Vp † ˆ cofinality…Gf †: Because T is polynomially bounded, the results from [6] imply that there is an elementary extension of …R; V † that has the same value group G, but whose residue ®eld is any given elementary extension of V (viewed as model of T ). By passing to such an elementary extension we do not change Gf and can arrange that the residue ®eld has co®nality greater than cofinality…Gf †, which contradicts the equality above. (A similar argument is carried out in greater detail in the proof of Proposition 4.2 of [6].) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A e Recall that K denotes the ®eld of exponents of R. Let now R a Rhai be models of T, and let V and W be convex subrings of R and Rhai respectively, with W Ç R ˆ V. (We allow V ˆ R, W ˆ Rhai.) We denote the valuations on R and Rhai with valuation rings V and W by v and w, and their value groups by Gv and Gw . With the usual identi®cations we have Gv Í Gw (as ordered K-linear spaces), w extends v and V Í W (as ordered ®elds), where V and W are the residue ®elds of V and W. 9.2. Proposition (Valuation Property). r 2 R such that w…a ÿ r† 62 Gv .

Suppose Gv 6ˆ Gw . Then there exists an

In other words, the value group extension is witnessed by an element of the form a ÿ r with r 2 R. Proof. We ®rst note that V ˆ W by [6, Corollary 5.6], and hence that W is the convex hull of V in Rhai. Case 1 (easy case): V ˆ R. Then Gv ˆ f0g. If w…a† 6ˆ 0, then we can take r ˆ 0. If w…a† ˆ 0, then a :ˆ a ‡ m…W † 2 W ˆ V, so there exists an r 2 V with a ˆ r, so w…a ÿ r† > 0, and we have ®nished. Case 2 (main case): V 6ˆ R and there exists a b 2 Rhai with V < b < jRnV j and w…b† 62 Gv . Then we take a function f : R ! R that is de®nable in R such that f …b† ˆ a. Let us denote the restriction of f to V also by f. In the notation of the previous lemma there is then an element s 2 R for which v… f ÿ s† is maximal. After replacing f by f ÿ s and a by a ÿ s we reduce to the following situation: v… f ÿ c† < v… f † < 1 for all c 2 R:

…9:1†

Taking c 2 R with v…c† ˆ v… f † and replacing f by f =c and a by a=c, we may assume in addition that v… f † ˆ 0: …9:2† We now take elements p 2 V and q 2 R with q > V such that f is continuous,

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monotone and of ®xed sign on the interval … p; q†: positive if a ˆ f …b† > 0, negative if a ˆ f …b† < 0. Take an elementary submodel R 0 of R such that V ˆ R 0 ‡ m…V †. By increasing p, we may assume that p 2 R 0 , p > 0. De®ne the function f : … p; ‡1†R 0 ! R 0 by f …x† ˆ st R 0 … f …x††. Since v… f † ˆ 0, the function f only takes non-zero values, after increasing p if necessary. Because R 0 is tame in R, the function f is de®nable in R 0 by [6, Corollary 1.5]. Taking p large enough we may further assume that f is continuous, monotone, and that for all x 2 V with x > p we have …9:3† f …x† ÿ f …st R 0 …x†† 2 m…V †: By polynomial boundedness there exist a l 2 K and a non-zero c 2 R 0 such that f …x† ˆ cx l ‡ o…x l † as x ! ‡1 in R 0 , where o…x l †=x l ! 0 as x ! ‡1 in R 0 . Subcase (i): l 6ˆ 0. From j f …x† ÿ cx l j < 12 jcx l j for large x > 0 in R 0 and (9.3) it follows that j f …x† ÿ cx l j < 12 jcx l j for large x > 0 in V, say for all x > p in V, after increasing p if necessary. By `overspill' (using the fact that f is de®nable in R and that V has no supremum in R), this last inequality continues to hold `beyond' V, that is, after decreasing our element q > V suitably we have j f …x† ÿ cx l j < 12 cx l for all x 2 R with p < x < q. Since R a Rhai, this remains true for x ˆ b, and thus w…a† ˆ w… f …b†† ˆ w…cb l † ˆ lw…b† ‡ v…c† 62 Gv , and we have ®nished. Subcase (ii): l ˆ 0. Then f …x† ! c as x ! ‡1 in R 0 . If f …x† ˆ c for all suf®ciently large x 2 R 0 , then by (9.3) we have v… f ÿ c† > 0, which contradicts (9.1) and (9.2) above. Hence f …x† 6ˆ c for all suf®ciently large x 2 R 0 . But then there are some m < 0 in K and a non-zero d 2 R 0 such that f …x† ˆ c ‡ d x m ‡ o…x m † as x ! ‡1 in R 0 . Then we are back in Subcase (i) with f ÿ c, a ÿ c, m instead of f , a, l, and the result from that subcase gives w…a ÿ c† 62 Gv . This completes the proof in Case 2. In the remainder of the proof we assume V 6ˆ R and take b 2 Rhai with w…b† 62 Gv . Case 3. The set D v :ˆ fg 2 Gv : jgj < jw…b†jg is a subgroup of Gv . Note that D v is then even a convex K-linear subspace of Gv . Also, D w :ˆ fg 2 Gw : jgj < jw…b†jg is a convex K-linear subspace of Gw , and D w Ç Gv ˆ D v . Put Gv D :ˆ Gv =D v and Gw D :ˆ Gw =D w . Then we have Gv D Í Gw D as ordered K-linear spaces after the obvious identi®cation. The composition v R ´ ÿ! Gv ÿ! Gv D is a valuation on R (a `coarsening' of v), which we shall denote by v D . Similarly w gives rise to the valuation w D : Rhai´ ÿ! Gw D , and clearly w D extends v D , the valuation rings VD and WD of v D and w D are convex subrings of R and Rhai with VD ˆ WD Ç R. Replacing b by 1=b if necessary, we may assume that w…b† < 0. Since g < wD …b† < 0 for all negative g 2 GD , we are back in Case 1 or Case 2 (with VD , WD instead of V, W ). Hence there exists an r 2 R such that w D …a ÿ r† 62 Gv D , and thus w…a ÿ r† 62 Gv .

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Next we make three observations. Observation 1. Let …Rhai; W; R; V † denote the model Rhai of T with predicates for the subrings W, R and V. Consider any elementary extension …  …Rhai†;  W;  R;  V † of …Rhai; W; R; V †. We denote the valuation and value group of the valued ®eld …  R;  V † by  v and G v , and we denote the valuation and value group of its valued ®eld extension …  Rhai;  W Ç  Rhai† by  w and G w . Then G v Í G w and  w extends  v. From Rhai ˆ Rhbi and w…b† 62 Gv we obtain  Rhai ˆ  Rhbi and  w…b† 62 G v ; in particular, G v 6ˆ G w . Now note that if there exists r 2  R such that  w…a ÿ r† 62 G v , then there exists r 2 R such that w…a ÿ r† 62 Gv . Observation 2. Let …R 0 ; V 0 † be an elementary substructure of …R; V †, and let f : R ! R be de®nable in …R; V † with parameters from R 0 such that f …b† ˆ a. Suppose r 2 R 0 is such that w…a ÿ r† 62 v……R 0 †´ †. Then we claim that w…a ÿ r† 62 Gv . The assumption and [6, Lemma 5.4] imply that w…R 0 hai´ † ˆ v……R 0 †´ †  K ´ w…a ÿ r†; thus, the claim follows from R 0 hai ˆ R 0 hbi and w…b† 62 Gv . Observation 3. Suppose Gv ˆ G1  G2  . . .  Gn (internal direct sum of Klinear subspaces of Gv ), where each Gj > R (isomorphism of ordered K-linear spaces) and g1 < g2 < . . . < gn whenever 0 < g j 2 Gj for each j. Then w…a ÿ r† 62 Gv for some r 2 R. To see this, we let j ˆ j…b† 2 f0; . . . ; ng be maximal such that G1  . . .  Gj < jw…b†j (here G1  . . .  Gj ˆ f0g if j ˆ 0), and we proceed by induction on j (simultaneously for all b 2 Rhai with w…b† 62 Gv ). Note that if j ˆ n, or if j < n and jw…b†j < g for all positive g 2 Gj ‡ 1 , then we are in Case 3, with D v ˆ G1  . . .  Gj . We therefore assume that j < n and g < jw…b†j < d for some positive g; d 2 Gj ‡ 1 . In this case, Gj ‡ 1 > R gives a c 2 R such that v…c† 2 Gj ‡ 1 and jw…b=c†j ˆ jw…b† ÿ w…c†j < g for all positive g 2 Gj ‡ 1 . If j ˆ 0, or if j > 0 and Gj < jw…b=c†j, then we are in Case 3 with b=c in place of b. Otherwise we must have j…b=c† < j, so the inductive hypothesis applies. These three observations will now be used to ®nish the proof. By the ®rst observation, we can assume that the ordered K-linear space Gv is À0 -saturated. We also ®x a function f : R ! R de®nable in …R; V † such that f …b† ˆ a. We let …R 0 ; V 0 † be an elementary substructure of …R; V † such that f is de®nable with parameters from R 0 , and R 0 is a ®nitely generated model of T. Then v……R 0 †´ † ˆ G1  G2  . . .  Gn (internal direct sum of K-linear subspaces of Gv ), where each Gj is an archimedean ordered K-linear space and g1 < g2 < . . . < gn whenever 0 < gj 2 Gj for each j. By adjunction of suitable elements from Gv to the value group v……R 0 †´ † (see [6, Corollary 5.6]), we enlarge …R 0 ; V 0 † in such a way that v……R 0 †´ † still has the structure just indicated, but satis®es in addition Gj > R for each j (isomorphism of K-linear spaces). Now Observation 3 applied to …R 0 ; V 0 † in place of …R; V † implies that w…a ÿ r† 62 v……R 0 †´ † for some r 2 R 0 , and then Observation 2 implies that w…a ÿ r† 62 Gv . Remark. The Valuation Property was stated as an open problem in [6, 7.4] in the general setting of power bounded o-minimal theories. The proof here does not generalize to that setting. We also refer the reader to [6] for information about previously known special cases of the Valuation Property and some of its interesting consequences (not needed here); for instance, it is shown there that the Valuation Property fails for exponential o-minimal theories. We note here that preprints of this paper contained an incorrect `proof ' of Proposition 9.2. We thank James Tyne for pointing out the error.

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Proof of Theorem B. We can now almost copy the proof in [9, § 4] to e exp† from the Introduction. The only changes are the establish Theorem B on … R; following. The role of Ta n is taken over by T, and the value groups that occur should now be considered as ordered K-linear spaces instead of ordered Q-linear spaces. The Valuation Property is needed to make Lemma 4.2 there go through here. To make Lemmas 4.3 and 4.4 there go through here, we also use the fact that if …R; V † and …Rhai; W † are as at the beginning of this section, and b 2 Rhai is any element with w…b† 62 Gv , then Gw ˆ Gv  Kw…b†; this is established in [6, § 5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A The ®rst consequence of Theorem B that we mentioned in the Introduction, e exp† are piecewise given by namely that functions that are 0-de®nable in … R; terms in L…exp; log†, now follows exactly as in [9, Corollaries 4.7 and 2.15]. (The same holds for de®nable functions and terms in that language with real parameters.) From this fact we obtain the following. e admits analytic cell decomposition, so does … R; e exp†. 9.3. Corollary. If R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .A e exp†, follows The second consequence of Theorem B, the o-minimality of … R; as in [9, § 5]. The third consequence, the exponential boundedness, requires more work and will be done below. From Theorems A and B and the corollary in Example 8.1, we get immediately the following result. De®ne G> 0 : R ! R by G> 0 …x† :ˆ G…x† if x > 0 and G> 0 …x† :ˆ 0 otherwise. 9.4. Corollary. The expansion …R G ; exp† of the ®eld of real numbers is model complete and o-minimal and admits analytic cell decomposition. In particular, the expansion …R; 0 † of the ordered real ®eld is o-minimal. . . . . A In a similar way, Theorem B together with some remarks made in the Introduction of [13] imply a corresponding result for the Riemann zeta function z . De®ne z > 1 : R ! R by z > 1 …x† :ˆ z…x† if x > 1 and z…x† :ˆ 0 otherwise. 9.5. Corollary. The expansion …R a n  ; exp† of the ®eld of real numbers is model complete and o-minimal and admits analytic cell decomposition. In particular, the expansion …R; 1 † of the ordered real ®eld is o-minimal. . A For the rest of this section we assume that the hypotheses of Theorem B hold, e in particular that exp j‰0; 1Š is de®nable in R. e exp† 9.6. Proposition. For each function f : R ! R that is de®nable in … R; there exists an n such that j f …x†j < exp n …x† for all suf®ciently large x. Proof. Let H be the Hardy ®eld of germs at ‡1 of functions f : R ! R that e and let H  be the Hardy ®eld of germs at ‡1 of functions are de®nable in R, e exp†. Let LE…H † be the smallest sub®eld of H  that f : R ! R de®nable in … R; contains H and contains exp… f † whenever it contains f , and log… f † whenever it

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contains f and f > 0. Then LE…H † is a Hardy ®eld and is differentially algebraic over H , and the value group of H is archimedean (in fact, isomorphic to K, the ®eld of exponents). Hence by [21] there exists for each g 2 LE…H † an n such that jgj < exp n …x†. (Here x denotes the germ at ‡1 of the identity function on R.) Therefore it suf®ces to show that LE…H † is co®nal in H  . By Consequence (1) of Theorem B the ®eld H  consists exactly of the germs at ‡1 of the functions R ! R de®ned by some one-variable L…exp; log†-term with parameters from R. e Also, by [9, § 5] the Hardy ®eld H  is an R-®eld, and as such is an elementary e extension of R. Hence the desired result follows by an elementary chain argument from the two claims below. Claim 1. Let F be a Hardy ®eld, H Í F Í H  , and suppose that for all f 2 F there is a g 2 LE…H † such that j f j < jgj. Let a 2 F, a > 0. Then for all f 2 F…exp …a†† and for all f 2 F…log…a†† there exists a g 2 LE…H † such that j f j < jgj. Proof. Replacing F by its real closure in H  if necessary, we may assume that F is real closed. Clearly for all f 2 F‰exp…a†Š and for all f 2 F‰log…a†Š there exists a g 2 LE…H † with j f j < jgj. For f 2 H  , write f , 0 if j f j < jgj for all g 2 LE…H †nf0g. It is now enough to show that for no f 2 F‰exp…a†Šnf0g we have f , 0, and similarly for no f 2 F‰log…a†Šnf0g. Suppose on the contrary that there exists an f 2 F ‰exp…a†Š nf0g with f , 0. It follows that exp…a† 62 F. Take a polynomial p…T † 2 F ‰T Šnf0g of minimal degree d such that p…exp…a†† , 0. We now apply the well-known `Hardy trick': p…0† 6ˆ 0 since p…T † is of minimal degree, so after dividing by p…0† we may assume that p…T † ˆ 1 ‡ c 1 T ‡ . . . ‡ c d T d

with c 1 ; . . . ; c d 2 F:

Taking the derivative of p…exp…a†† , 0 gives …c 10 ‡ c 1 a 0 † exp…a† ‡ . . . ‡ …c d0 ‡ dc d a 0 †…exp…a††d , 0: Dividing by exp…a† gives q…exp…a†† , 0, where q…T † ˆ …c 10 ‡ c 1 a 0 † ‡ . . . ‡ …c d0 ‡ dc d a 0 †T d ÿ 1 2 F ‰T Š: But q…T † is of degree d ÿ 1 in T : c d0 ‡ dc d a 0 6ˆ 0, because otherwise c d0 =c d ˆ …log jc d j†0 ˆ …ÿda†, so that ÿda ˆ log jc d j ‡ some real constant, and hence exp…ÿda† ˆ …some positive real constant† ´ jc d j; and thus exp…a† 2 F, a contradiction. This contradicts the minimality property of d. For log…a† we proceed in the same way, except that, instead of the constant term, we make the leading coef®cient equal to 1 before taking the derivative. This ®nishes the proof of Claim 1. Claim 2.

Let R a Rhbi o T. Then the ®eld R…b† is co®nal in Rhbi.

Proof. This follows easily from the Valuation Property 9.2, but here is a more elementary argument. Let V be the convex hull of R…b† in Rhbi. Then V is a convex valuation ring of Rhbi. The canonical map x 7! x: V ! V sends R…b† isomorphically onto the sub®eld R…b† of V. Note that V is naturally a model of

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T with R a V, and R…b† Í Rhbi Í V. The elements b and b realize cuts in R and R that correspond under the isomorphism R ! R. Hence this isomorphism extends to an isomorphism Rhbi ! Rhbi (of models of T ) that sends b to b. Since R…b† is co®nal in V, so is R…b† in V, and hence in Rhbi. Thus R…b† is co®nal in Rhbi. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A 10. Power series models There is a natural way to expand the power series ®eld R……t G †† to an L G structure, for any divisible ordered abelian group G. We indicate in this section how to do this. We would like to show that these expansions are elementary extensions of R G , as was done with R an instead of R G (and L an instead of L G ) in [9, § 3]. However, the argument in [9] breaks down for R G , because we lack a simple axiomatization of its elementary theory. Thus we have only been able to obtain the desired result when G is subject to certain restrictions (being archimedean, for instance). We proceed rather indirectly, by ®rst establishing some facts about arbitrary polynomially bounded o-minimal expansions of the real ®eld. These are then used in studying the natural L G -expansions of the power series ®elds R……t G ††. At the end of this section we show that our partial results on power series models are nevertheless suf®cient to extend some theorems from [10] about …R an ; exp† to …R G ; exp†. We keep the notation and conventions of the previous section. In particular R a Rhai o T, and we are given convex subrings V and W of R and Rhai with W Ç R ˆ V. Here is an analogue for residue ®elds of the Valuation Property: residue ®eld extension is witnessed by an af®ne transform of the generator a . 10.1. Proposition (Residue Property). Suppose the residue ®eld V is isomorphic to the ®eld of real numbers, and let V 6ˆ W. Then a has an af®ne transform a 0 ˆ ra ‡ s with r; s 2 R, r 6ˆ 0, such that a 0 2 W and a 0 62 V. Proof. Note that Gv ˆ Gw by [6, § 5]. Suppose ®rst that we are in the trivial case that V ˆ R. Then Gv ˆ Gw ˆ f0g, so W ˆ Rhai and a 0 ˆ a has itself the desired property. So from now on we may assume that V 6ˆ R. Since V is isomorphic to the real ®eld and V 6ˆ W, there exists a b 2 W with b > V, so V < b < jR nV j. In other words, we are now in a similar situation as in Case 2 of the proof of the Valuation Property. We make the same reductions as in that proof, also introducing f , p, q, f , l, and c. We also make the same case distinctions. Subcase (i): l 6ˆ 0. The same argument as in the proof of the Valuation Property gives j f …b† ÿ cb l j < 12 jcb l j. If l > 0, then this implies f …b† ˆ a 2 W and jaj > V, so a 0 ˆ a has the desired property. If l < 0, then we also obtain a 2 W and 0 < jaj < r 0 for all positive r 0 2 R 0 , and again a 0 ˆ a has the desired property. Subcase (ii): l ˆ 0. We reduce this to Subcase (i) as in the proof of the Valuation Property, after appealing to formula (9.3) to ®rst exclude the possibility that f …x† ˆ c for all suf®ciently large x 2 R 0 . . . . . . . . . . . . . . . . . . . . . . . A e 4 R with 10.2. Lemma. Let G be an ordered K-linear space and let R G R Í R……t ††, where R denotes the underlying ordered ®eld of R. Then the ordered ®eld R……t G †† can be expanded to a model of T in such a way that R 4 R……t G ††.

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Proof. We may assume that R 6ˆ R……t G ††. Claim. There exists a model S of T such that S is a proper elementary extension of R and R Í S Í R……t G ††, where S denotes the underlying ordered ®eld of S. Once this claim is established, an application of Zorn's Lemma gives the required result. Let v: R……t G ††´ ! G be the usual valuation. To establish our claim we distinguish two cases. Case 1: v…R ´ † 6ˆ G. Then we take some element g 2 G nv…R ´ † and consider an elementary extension Rhai of R such that a and t g realize the same cut in the ordered ®eld R. We now have an isomorphism i: R…a† ! R…t g † of ordered ®elds such that i is the identity on R and i…a† ˆ t g . The idea is now to extend i to a valued ®eld embedding Rhai ! R……t G †† for a suitable valuation on Rhai, and let S be the `image' of Rhai under this embedding. Let V be the (convex) valuation ring of vjR ´ . We ®rst extend vjR ´ to a valuation va : R…a†´ ! v…R ´ †  Zg Í G by setting va …a† :ˆ g. Then i is also an isomorphism from the valued ®eld …R…a†; va † onto the valued ®eld …R…t g †; vjR…t g †´ †. In particular, the valuation ring Va of va is a convex subring of R…a†. Let W be the convex hull of Va in the underlying ordered ®eld Rhai of Rhai, and let w be a valuation on this underlying ®eld that extends va and has valuation ring W. So W is a convex subring of Rhai and W Ç R ˆ V. Since w…a† ˆ g 62 v…R ´ †, we have w…Rhai´ † ˆ v…R ´ †  K g (internal direct sum) and W ˆ V > R by [6, § 5]. Hence we have an embedding j: w…Rhai´ † ! G of ordered K-linear spaces such that j is the identity on the common subspace v…R ´ † and j…g† ˆ g. The universal property of the valued ®eld R……t G †† implies that i extends to a valued ®eld embedding h: …Rhai; w† ! …R……t G ††; v†. We now expand the ordered sub®eld h…Rhai† of R……t G †† to a model S of T in such a way that h is an isomorphism from Rhai onto S. Then S has the property of our claim. Case 2: v…R ´ † ˆ G. Then we take any element b 2 R……t G ††nR and take an elementary extension Rhai of R such that a and b realize the same cut in the ordered ®eld R. As in Case 1 we extend vjR ´ to a valuation va on R…a† such that we have an ordered and valued ®eld isomorphism i: …R…a†; va † ! …R…b†; vjR…b†´ † with ijR ˆ id R and i…a† ˆ b. Since …R……t G ††; v† is an immediate extension of …R; vjR ´ †, so is …R…b†; vjR…b†´ †, and hence …R…a†; va † is an immediate extension of …R; vjR ´ †. As in Case 1 we extend va further to a valuation w on the underlying ordered ®eld Rhai of Rhai whose valuation ring W is the convex hull of Va in Rhai. The previous remark, the Valuation Property and the Residue Property imply that …Rhai; w† is an immediate extension of …R; vjR ´ †. Hence i can be extended to a valued ®eld embedding h: …Rhai; w† ! …R……t G ††; v†. Just as in Case 1 this produces an S as required. . . . . . . . . . . . . . . . . . . . . . . . . A Note the following special case of this lemma. 10.3. Corollary. If G is an ordered vector space over K, ®eld R……t G †† can be expanded to a model of T that contains submodel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

then the ordered e as elementary R . . . . . . . . . .A

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We now turn our attention to the expansion R G . Below we let G denote a divisible ordered abelian group, or equivalently, an ordered vector space over Q. We are going to expand the power series ®eld K :ˆ R……t G †† to a structure for the language L G . Let m be the set of in®nitesimals of K. For an ordinary formal power series F…Z † 2 R[Z ], where Z ˆ …Z 1 ; . . . ; Z N †, and for « ˆ …« 1 ; . . . ; « N † 2 m N , we de®ne the element F…«† 2 K as usual. Consider a power series F…X † 2 RfX gG; 1 where X ˆ …X 1 ; . . . ; X m †. As in [9, § 2] we take F as an m-ary function symbol of the language L G . We expand the ordered ®eld K to an L G -structure KG ˆ R……t G ††G by interpreting this function e K m ! K. symbol F as the following function F: m Let x ˆ …x 1 ; . . . ; x m † 2 K . If 0 < x i < 1 for all i, we take the unique point a ˆ …a 1 ; . . . ; a m † 2 R m such that x ˆ a ‡ « for some in®nitesimal « 2 m m ; then 0 < a i < 1 for all i, so that we have by Corollary 4.9 a well-de®ned power series Fa …X † :ˆ

X a2Nm

1 ¶ S aF …a†X a 2 RfX gG ; a! ¶X a

e e :ˆ 0. and we put F…x† :ˆ Fa …«†. If x i < 0 or x i > 1 for some i, we put F…x† G Note that the L G -structure R……t ††G contains R G as a substructure. 10.4. Proposition. Suppose G is archimedean. Then R G 4 R……t G ††G . Proof. By the previous lemma we already know that K can be expanded to an L G -structure K ] such that R G 4 K ] . We claim that then K ] ˆ KG . Let F…X † 2 RfX gG; 1 , considered as an m-ary function symbol of L G , interpreted as e in KG . We only have to show that the function F ] in K ] and as the function F e as functions on K m . then F ] ˆ F e for « 2 m m with each « i > 0. (The general Let us just verify that F ] …«† ˆ F…«† case is quite similar but notationally awkward to spell out.) For each n 2 N we can write X F…X † ˆ Pn …X † ‡ X a Ha …X †; Saˆn‡1

where Pn …X † is a real polynomial in X of degree at most n, and Ha …X † 2 RfX gG for a 2 N m with Sa ˆ n ‡ 1. Take g > 0 in G such that g < v…« i † for all e ˆ Pn …«† ‡ bn , where v…a n † > ng i ˆ 1; . . . ; m. Then F ] …«† ˆ Pn …«† ‡ a n and F…«† and v…bn † > ng. Hence the sequence …Pn …«††n converges in the valuation topology e e Therefore F ] …«† ˆ F…«†. . . . . . . . . . . . . . . . . .A both to F ] …«† and to F…«†. A similar argument will give the following more general result. 10.5. Proposition. Suppose D is a convex Q-linear subspace of G and R G 4 R……t D ††G , and G=D is archimedean. Then R G 4 R……t G ††G . Proof. Put k :ˆ R……t D †† and K ˆ R……t G ††. Since R G 4 k G by assumption, we know from Lemma 10.2 that K can be expanded to an L G -structure K ] such that k G 4 K ] . We claim that then K ] ˆ KG . As in the last proof we consider a power series F…X † 2 RfX gG; 1 and view it as an m-ary function symbol interpreted as the e in KG . Let a 2 K m with 0 < a i < 1 for function F ] in K ] and as the function F

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e i ˆ 1; . . . ; m. We have to show that then F ] …a† ˆ F…a†. To simplify notation we assume that each a i is in®nitesimal. (The argument in this case works essentially also in general, after an initial reduction.) Write a i ˆ b i ‡ « i with b i 2 k and « i a kin®nitesimal, and put b ˆ …b 1 ; . . . ; b m † and « ˆ …« 1 ; . . . ; « m †. Note that 0 < b i < 1, but it can happen that « i < 0 if b i > 0. Take some g 2 G such that g > D and v…« i † > g for all i. Let U ˆ …U1 ; . . . ; Um † be a tuple of m distinct indeterminates (not among the X i ). For each n we have a formal power series identity in R[X; U ]: X 1 Fa …X †U a ‡ terms of degree greater than n in U; F…X ‡ U † ˆ a! Sa ng. Since G=D is archimedean, the sequence …ng†n 2 N is co®nal in G. e in the valuation topology of K. Hence the sequence …Pn …b; «††n converges to F…a† ] f ˆ F…X † For F …a† we argue as follows: the (unique) function f 2 Gm; 0; 1 with b 1 m …a† m d …a† for a 2 N satisfy f ˆ Fa …X † 2 RfX gG; 1. is C on I , and its partials f Hence by `Taylor expansion with remainder' there is for each n 2 N a positive real constant C n such that jF…x ‡ u† ÿ Pn …x; u†j < C n ´ juj n ‡ 1 for all x 2 ‰0; 12 Š m Ì R m and all u 2 R m with ju i j < x i for each i ˆ 1; . . . ; m. Since R G 4 K ] , this implies that F ] …a† ˆ F ] …b ‡ «† ˆ Pn …b; «† ‡ d n , where d n 2 K with v…d n † > ng. Hence the sequence …Pn …b; «††n converges also to F ] …a† in the e valuation topology of K. Thus F ] …a† ˆ F…a†. . . . . . . . . . . . . . . . . . . . . . .A Repeated use of the last proposition gives the following result. 10.6. Corollary. If dim Q …G† < 1, then R G 4 R……t G ††G . . . . . . . . . . . . . A For arbitrary G we de®ne R……t G ††f d to be the set of elements of R……t G †† whose support is contained in some ®nite-dimensional Q-linear subspace of G. (The subscript `fd' stands here for `®nite-dimensional'.) Note that R……t G ††f d is a sub®eld of R……t G ††; in fact, it is the underlying ®eld of an LG -substructure of R……t G ††G which we shall denote by R……t G ††f d; G . Then the corollary above is easily seen to imply the following. 10.7. Corollary. R G 4 R……t G ††f d; G . . . . . . . . . . . . . . . . . . . . . . . . . . A It would be nice to know if R……t G ††G is always an elementary extension of R G . We know of one more result in this direction, which is as follows. 10.8. Proposition. Let b be an ordinal, b 6ˆ 0, and …Ga †a < b a family of Qlinear subspaces of G such that R G 4 R……t Ga ††G for all a < b and GaS1 is a convex subspace of Ga 2 whenever a 1 < a 2 < b. Assume also that G ˆ a Ga . Then R G 4 R……t G ††G . S Proof. Note that R G 4 a R……t Ga ††G . By Lemma 10.2 we know that then K ˆ R……t G †† can be expanded to an L G -structure K ] that elementarily extends

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R……t Ga ††G . It suf®ces to show that then K ] ˆ KG . This follows S along the lines of the proofs of the two propositions above, using the fact that a R……t Ga †† is dense in K in the valuation topology. . . . . . . . . . . . . . . . . . . . . . . . . . . A a

We now apply these results to the ®eld R……t ††LE of logarithmic-exponential series. The construction of this ordered exponential ®eld (see [10, § 2] for details) consists of two parts, which we now brie¯y review. In the ®rst part one obtains the sub®eld R……t ††E of R……t ††LE consisting of the S Gn exponential series as a union n K n , where K n ˆ R……t †† and …Gn †n 2 N is a certain increasing sequence of divisible ordered abelian groups. Each Gn is actually also an additive subgroup of the ordered ®eld K n ÿ 1 (with the induced ordering) and contains Gn ÿ 1 as a convex subgroup. For n ˆ 0 we make this true by setting K ÿ 1 :ˆ R (the ®eld of real numbers), G0 :ˆ R (the ordered additive group of real numbers), and Gÿ 1 ˆ 0. Assuming inductively that Gn is an additive subgroup of K n ÿ 1 and Gn ÿ 1 is a convex subgroup of Gn , we obtain Gn ‡ 1 as follows: put Jn ˆ fz 2 K n : supp…z† < Gn ÿ 1 g, an additive subgroup of K n , and put Gn ‡ 1 :ˆ Jn  Gn , ordered as an additive subgroup of K n . (This makes sense, since Gn Í K n ÿ 1 Í K n with Jn Ç Gn ˆ 0. It is also easy to see that Gn is convex in Gn ‡ 1 .) We refer to [10, § 2] for the de®nition S of the ! K , which give rise to the exponential map E ˆ maps E n : K n n ‡ 1 n E n on S R……t ††E ˆ n K n . In the second part of the construction of R……t ††LE we `close under logarithms': this consists of building an increasing chain L 0 Ì L 1 Ì L 2 Ì . . . of ordered exponential ®elds, where each L i comes equipped with a speci®c isomorphism h i : L i ! R……t ††E of ordered exponential ®elds, and where h 0 is the identity on L 0 ˆ R……t ††E . The details of how this is done (see [10, § 2]) guarantee that the positive elements of L i have a logarithm in L i ‡ 1 . It turns out that then the ordered S exponential ®eld R……t ††LE ˆ i 2 N L i is an elementary extension of the ordered exponential ®eld of real numbers. We now adapt this to our situation. It is not enough to restrict to the exponential series of `®nite-dimensional support', that is, to replace R……t Gn †† in the construction above by R……t Gn ††f d. It will be necessary to impose the stronger restriction of `hereditarily ®nite-dimensional support'. (This does not affect any potential applications we can think of.) The sub®eld R……t ††E; f of R……t ††E consisting of the exponential S series of hereditarily ®nite-dimensional support is again obtained as a union n K nf where f K nf :ˆ R……t Gn ††f d , and …Gnf †n 2 N is a certain increasing sequence of divisible ordered abelian groups. Each Gnf is actually a subgroup of Gn (with the induced ordering) as well as an additive subgroup of the ordered ®eld K nf ÿ 1 (also with the induced ordering) and contains Gnf ÿ 1 as a convex subgroup. f ˆ 0. For n ˆ 0 we make this true by setting K ÿf 1 :ˆ R, G0f :ˆ R, and Gÿ1 Assume inductively that (1) Gnf is an additive subgroup of K nf ÿ 1 , (2) Gnf Í Gn as ordered groups, and Gnf ÿ 1 ˆ Gn ÿ 1 Ç Gnf . (In particular, Gnf ÿ 1 is a convex subgroup of Gnf .) Then we de®ne Gnf ‡ 1 as follows: put Jnf ˆ fz 2 K nf : supp z < Gnf ÿ 1 g, an additive subgroup of K nf , and put Gnf ‡ 1 :ˆ Jnf  Gnf , ordered as an additive subgroup of K nf .

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(This makes sense, since Gnf Í K nf ÿ 1 Í K nf and Jnf Ç Gnf ˆ 0.) It is easy to see that (1) and (2) above remain true when n is replaced by n ‡ 1. This ®nishes the inductive construction of the sequences …Gnf † and …K nf †. It is also clear from this construction that E…K nf † Ì K nf ‡ 1 for all n, and we may therefore consider R……t ††E; f as an ordered exponential sub®eld of R……t ††E . Moreover, [ f f …K n †G R……t ††E; G :ˆ n

is clearly an elementary extension of R G , by Corollary 10.7. It is easy to check that F…R……t ††E; f † Í R……t ††E; f , where F: R……t ††E ! R……t ††E is the map de®ned in [10, 2.2]. Moreover, the proof of [10, Lemma 2.6] shows that each positive element of F…R……t ††E; f ) has its logarithm in R……t ††E; f . Because of these two facts we can now close under logarithms as before. Put E; f : L fi :ˆ hÿ1 i R……t †† is an ordered exponential sub®eld of L i , and each positive element of Then L fi has its logarithm in L fi ‡ 1 . Next we expand the ordered ®eld L fi to the L G onto the L G structure …L fi †G so that h i maps this expansion S isomorphically f LE; f f . Finally, let R……t †† :ˆ …L † , an ordered exponential structure R……t ††E; i G G G sub®eld of R……t ††LE and at the same time an L G -structure. Then we have the following analogue of the corresponding result [10, Corollary 2.8] for R an .

L fi

f 10.9. Proposition. The …L G ; exp†-structure R……t ††LE; G Th…R G ; exp†, hence an elementary extension of …R G ; exp†.

is

a

model

of

Proof. It is routine to check that R……t ††LE; f as an ordered exponential ®eld satis®es the Ressayre axioms as given in the Introduction. The Proposition is then an immediate consequence of Theorem B and earlier results in this section.. . . A This result provides an explicit example of a proper elementary extension of …R G ; exp†. The Hardy ®eld H…R G ; exp† of the germs at ‡1 of the functions f : R ! R de®nable in …R G ; exp† is also naturally a proper elementary extension of …R G ; exp†; in fact, it is generated as such over the standard model …R G ; exp† by the germ of the identity function; see [9, § 5]. Hence there is a unique f of …L G ; exp†-structures that is elementary embedding H…R G ; exp† ! R……t ††LE; G the identity on R and sends the germ of the identity function to x :ˆ 1=t . One can view this embedding as providing a logarithmic-exponential series expansion at ‡1 for the one-variable functions de®nable in …R G ; exp†. Sections 3 and 5 (up to Corollary 5.5) of [10] now go through with the following changes. Replace R……t ††E by R……t ††E; f and R……t ††LE by R……t ††LE; f (and make all S other corresponding changes as indicated above). Put FG :ˆ m 2 N RfX 1 ; . . . ; X m gG , and let HG; exp be the smallest FG -closed sub®eld of R……t ††LE; f containing P R…t † and closed under log P and exp. Finally, in 5.1±5.5 of [10], replace ` a n X n converges near 0' by ` a n X n belongs to RfX gG '. (Note that Lemma 5.1 there goes through by Corollary 7.4 here.) Combining this with Lemma 8.5, we get the following. 10.10. Corollary. Suppose f : …0; «† ! R isP de®nable in …R G ; exp† and has P an asymptotic expansion an T n P at 0. Then a n T n belongs to RfT gG . In n a n T is convergent. . . . . . . . . . . . . . A particular, if a n > 0 for all n, then

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lou van den dries and patrick speissegger P Thus the series n!T n is not the Taylor series at 0 of any C 1 function …0; «† ! R, with « > 0, that is de®nable in …R G ; exp†. Finally, note that every element y 2 R……t ††E; f , when considered as an element S of R……t G †† with G :ˆ n Gnf , has the property that supp… y† is contained in a ®nite-dimensional Q-linear subspace of G. As in Corollary 5.14 of [10], we therefore obtain the following corollary. 10.11. Corollary. The restriction of the Riemann zeta function to …1; ‡1† is not de®nable in …R G ; exp†. On the other hand, this function is de®nable in …R an  ; exp†. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Concluding remarks 1. It is well known that R an , the ®eld of real numbers with restricted analytic functions, admits elimination of quanti®ers in its natural language augmented by a symbol for the function x 7! x ÿ 1 (with 0ÿ1 :ˆ 0 by convention); see [4]. We do not know if this is the case for R G . We suspect it is not, even though R G is in many respects quite similar to R an . 2. At about the same time (summer 1996) as we proved Theorem B, Wilkie [24] established a completely different method for obtaining the o-minimality of certain expansions of the real ®eld. His result (together with the fact that R G is o-minimal and generated by total C 1 -primitives) implies for example that the expansion of R G by (suitably de®ned) Pfaf®an functions is o-minimal. This gives in particular the o-minimality of …R G ; exp†, which is also among our conclusions. Nevertheless, the method of [24] does not (to our knowledge) provide the other results concerning …R G ; exp† in our paper: the quanti®er elimination and axiomatization relative to R G and its various consequences such as exponential boundedness. 3. Theorem A implies that any function f : R ! R de®nable in R G is analytic on R nF for some ®nite set F Í R. We do not know whether this conclusion can be strengthened as follows: given a C 1 function f : R ! R de®nable in R G , is f necessarily analytic? An af®rmative answer to this question would imply the following extension to several variables: if a C 1 function f : U ! R, with U Í R n open, is de®nable in R G , then f is analytic. (Proceed by induction on n, using part (3) of the theorem in [1].) References 1. S. S. Abhyankar and T. T. Moh, `A reduction theorem for divergent power series', J. Reine Angew. Math. 241 (1970) 27±33. 2. W. Balser, From divergent power series to analytic functions, Lecture Notes in Mathematics 1582 (Springer, New York, 1994). 3. C. Chang and H. Keisler, Model theory (North-Holland, Amsterdam, 1973). 4. J. Denef and L. van den Dries, `P-adic and real subanalytic sets', Ann. of Math. 128 (1988) 79±138. 5. L. van den Dries, `o-minimal structures', Logic: from foundations to applications (ed. W. Hodges et al., Oxford University Press, 1996). 6. L. van den Dries, `T-convexity and tame extensions II', J. Symbolic Logic 62 (1997) 14±34. 7. L. van den Dries, Tame topology and o-minimal structures, London Mathematical Society Lecture Note Series 248 (Cambridge University Press, 1998).

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8. L. van den Dries and A. Lewenberg, `T-convexity and tame extensions', J. Symbolic Logic 60 (1995) 74±102. 9. L. van den Dries, A. Macintyre and D. Marker, `The elementary theory of restricted analytic ®elds with exponentiation', Ann. of Math. 140 (1994) 183±205. 10. L. van den Dries, A. Macintyre and D. Marker, `Logarithmic-exponential power series', J. London Math. Soc. (2) 56 (1997) 417±434. 11. L. van den Dries and C. Miller, `On the real exponential ®eld with restricted analytic functions', Israel J. Math. 85 (1994) 19±56. 12. L. van den Dries and C. Miller, `Geometric categories and o-minimal structures', Duke Math. J. 84 (1996) 497±540. 13. L. van den Dries and P. Speissegger, `The real ®eld with convergent generalized power series is model complete and o-minimal', Trans. Amer. Math. Soc. 350 (1998) 4377±4421. 14. A. Gabrielov, `Projections of semianalytic sets', Funct. Anal. Appl. 2 (1968) 282±291. 15. G. H. Hardy, Divergent series (Clarendon Press, Oxford, 1963). 16. J. Martinet and J.-P. Ramis, `Elementary acceleration and multisummability I', Ann. Inst. Henri Poincare 54 (1991) 331±401. 17. F. Nevanlinna, `Zur Theorie der asymptotischen Potenzreihen', Ann. Acad. Sci. Fenn. Math. XII (1919) 1±81. 18. N. Nielsen, Die Gammafunktion (Chelsea, New York, 1965). 19. A. Prestel, Lectures on formally real ®elds, Lecture Notes in Mathematics 1093 (Springer, Berlin, 1984). 20. J.-P. Ressayre, `Integer parts of real closed exponential ®elds', Arithmetic proof theory, and computational complexity (ed. P. Clote and J. KrajõÂcÏek), Oxford Logic Guides 23 (Oxford University Press, New York, 1993) 278±288. 21. M. Rosenlight, `The rank of a Hardy ®eld', Trans. Amer. Math. Soc. 280 (1983) 659±671. 22. J.-C. Tougeron, `Sur les ensembles semi-analytiques avec conditions Gevrey au bord', Ann. Sci. EÂcole Norm. Supl. (4) 27 (1994) 173±208. 23. E. Whittaker and G. Watson, A course of modern analysis (Cambridge University Press, 1927). 24. A. J. Wilkie, `A theorem of the complement and some new o-minimal structures', Selecta Math. (N.S.) 5 (1999) 397±421. 25. A. J. Wilkie, `Model completeness results for expansions of the ordered ®eld of real numbers by restricted pfaf®an functions and the exponential function', J. Amer. Math. Soc. 9 (1996) 1051±1094.

Lou van den Dries Department of Mathematics University of Illinois 1409 West Green Street Urbana IL 61801 USA [email protected]

Patrick Speissegger Department of Mathematics University of Wisconsin 480 Lincoln Drive Madison WI 53706 USA [email protected]