Model Theory and Exponentiation David Marker

odel theory is a branch of math- interested in showing that there is an algorithm ematical logic in which one studies for deciding first-order sentences about the real mathematical structures by con- numbers and hence showing the decidability of sidering the first-order sentences elementary Euclidean geometry. true of those structures and the Tarski’s decidability of the real numbers setsM definable in those structures by first-order stands in stark contrast to Gödel’s undecid- formulas. ability of the natural numbers. The great free- The fields of real and complex numbers have dom we have to solve polynomial equations in long served as motivating examples for model the real numbers insures that the real solution theorists. Many model theoretic concepts arose sets of systems of polynomial equations are by abstracting classical algebraic phenomena to topologically well behaved. In the natural num- a more general setting (see, for example, [29]). bers such solution sets can behave very badly. In the past five years significant progress has This wild behavior allows considerable latitude been made in the other direction. Model theoretic for coding complicated combinatorial phenom- methods have been used to develop new in- ena, while, as we will see below, in the reals we sights into real analytic geometry. In particular, cannot even define the integers. this has led to a generalization of semialgebraic The natural language for studying the real and subanalytic geometry to a setting in which numbers is the language of ordered rings one studies global exponentiation on the reals. = +, , ,<,0,1 . In this language the basic - L { − · } L Tarski’s Theorem formulas are of the form p(v1,...,vn)= q(v ,...,v ) and p(v ,...,v )set of all -formulas is obtained by taking the were announced by Tarski in [33], but publica- L basic -formulas and closing under the Boolean tion was interrupted by the war, and the proofs L connectives , , and quantifiers and . For did not appear until [34]. Tarski was primarily ∧ ∨ ¬ ∃ ∀ example, (1) yx=y2 y David Marker is professor of mathematics at the Uni- ∃ − versity of Illinois at Chicago. His e-mail address is (2) x y (x = y2 x + y2 =0) [email protected]. ∀ ∃ ∨ This paper is based on an invited lecture presented in (3) w (x w 2 + xy =0 w<3y 4) October 1994 at the AMS Central regional meeting in ∃ − ∧ − Stillwater, Oklahoma. are legitimate -formulas. We say that an oc- L Partially supported by NSF grants DMS-9306159 and currence of a variable in a formula is free if it is INT-9224546, and an AMS Centennial Fellowship. not inside the scope of a quantifier; otherwise,

JULY 1996 NOTICES OF THE AMS 753 we say it is bound. For example, in (1) x is a free Step 1: (Elimination of Quantifiers) If variable and y is bound, while in (2) both x and φ(v1,...,vn) is an -formula, then there is an L y are bound. If a formula has free variables, we equivalent quantifier-free -formula L often denote the formula in a way that explic- (v1, . . . , vn). (Here by equivalent we mean that itly shows which variables are free. For example, φ holds of a = (a1, . . . , an) if and only if holds n we might denote (3) by φ(x, y). We think of a for- of a for all a R .) Moreover, there is an algo- ∈ mula φ(x1, . . . , xn) as expressing a relation that rithm that transforms φ to . holds among the variables x1, . . . , xn. For exam- Note that quantifier-free formulas are just fi- ple, nite Boolean combinations of p(x1, . . . , xn) = q(x , . . . , x ) and p(x ,...,x ) > q(x ,...,x ) (4) w xw 2 + yw + z = 0 1 n 1 n 1 n ∃ where p, q Z[X1, . . . , Xn] for some n N. ∈ ∈ expresses the relationship “x, y, z are coeffi- We are familiar with some natural examples cients of a solvable quadratic polynomial”, while of quantifier elimination. For example, (4) is equivalent to the quantifier-free formula 2 2 (5) x y ux + vxy + zy 0 2 ∀ ∀ ≥ y 4xz 0. − ≥ expresses that u, v, z are the coefficients of a Also, while the formula in free variables positive semidefinite binary quadratic form. vi,j : 1 i, j n that expresses that the ma- A formula is a sentence if it has no free vari- { ≤ ≤ } trix (vi,j ) has an inverse involves existential ables. If is an ordered ring, then any -sen- A L quantifiers, it is equivalent to the quantifier- tence φ will either be true or false for . For A free formula that asserts that the determinant example, (2) is true for the field of real numbers of (vi,j ) is nonzero. R but false for the field of rational numbers Q. Tarski’s original proof relied on techniques Formulas that are not sentences may hold for from elimination theory and Sturm’s algorithm. some assignments of the free variables but not Other proofs are based on model-theoretic ideas for others. For example, in Q (4) holds for (1,5,6) (see, for example, [24]) or cylindric decomposi- but fails for (1,0,-2). Of course, (4) holds for (1,0,- tion (see, for example, [2]). 2) in R. The first-order theory of R, T h(R), is the The second step is just an easy observation. set of all -sentences that are true about R. L Many important properties of R can be expressed Step 2: Quantifier-free sentences are finite Boolean as -sentences. For example, (2) expresses that combinations of ones of the form m = n and L for any x either x or x is a square. For each n m > n where m and n are integers. These can − be decided easily. the sentence Thus, to decide if a sentence φ is true in R, a, b w0, . . . , wn [(a < b ∀ ∀ ∧ n n we first transform it into a quantifier-free sen- i i tence and then test to see if the quantifier-free ( wia )( wib ) < 0) → sentence is true. iX=0 iX=0 n While Tarski’s result proves the decidability i x(a < x < b wix = 0)] of the theory of the real numbers, the algorithm ∃ ∧ iX=0 obtained is far from feasible. The best-known al- gorithm for eliminating quantifiers is doubly expresses that the sign change property holds exponential in the number of quantifier blocks for polynomials of degree n. One sees that all [17], and it is known that even deciding first- of the axioms for real closed ordered fields (see order sentences for the ordered group (R, +, <) [21]) are contained in T h(R). On the other hand, is exponentially hard [14]. there is no first-order sentence that expresses It is also worth noting that for any formula the completeness of the ordering of R. Indeed, φ(v1, . . . , vn), the formula (v1,...,vn) ob- we cannot hope to characterize the real numbers tained by quantifier elimination is actually equiv- up to isomorphism by T h(R), as the Löwenheim- alent to φ(v1, . . . , vn) in any other real closed Skolem theorem (see [3]) insures there will be field as well. Since the decision made in Step 2 models of T h(R) of every infinite cardinality. depends only on Z, it follows that a sentence φ Tarski’s Theorem. The first-order theory of the is true in R if and only if it is true in every real closed field if and only if it is true in some real field of real numbers is decidable. That is, there closed field. This means that the axioms for real is an algorithm that will decide on input φ closed fields completely axiomatize T h(R). whether or not φ is a sentence true of R. In [35] Tarski asked if his results could be ex- The proof of Tarski’s theorem breaks into tended to include the exponential function. x two steps. Namely, let exp = e . Is the first-order L L ∪ { }

754 NOTICES OF THE AMS VOLUME 43, NUMBER 7 exp-theory of R decidable?1 We will give a par- to define the integers. Since we can define Z, L tial answer to Tarski’s problem in Section 4. Gödel’s results show that this theory is far from One might hope to generalize the arguments decidable. used for the ordered field, but serious difficul- ties arise. First, quantifier elimination fails. In [5], Definable Sets and o-Minimality We say that X Rn is definable using if there van den Dries showed that the formula ⊆ L is an -formula φ(x1, . . . , xn, y1, . . . , ym) and y > 0 w (wy = x z = yew ) L b1, . . . , bm R such that ∧ ∃ ∧ ∈ is not equivalent to a quantifier-free exp-for- n L X = (a1, . . . , an) R : mula. { ∈ φ(a1, . . . , an, b1, . . . , bm) holds . There are rather silly ways to extend the lan- } guage so that we do have elimination of quan- For example, let φ(x1, x2, y) be the formula tifiers, but the trick is to extend the language so 2 2 2 2 u (x1 u) + (x2 u ) < y . that we have quantifier elimination and the quan- ∃ − − tifier-free formulas are still natural and mean- Substituting the real number π for the variable ingful. y, we get the definable set 2 Even if we could eliminate quantifiers, there (x1, x2) : the distance from (x1, x2) to the is no obvious uniform algorithm for deciding { π quantifier-free sentences like parabola y = x2 is less than . 2 } e2 2 5 3+e 3 n m e − e = e − , We say that a function f : R R is definable − if its graph (x, y) Rn Rm→: f (x) = y is de- since, as far as we know, there may be some per- finable. { ∈ × } verse exponential-algebraic relations that hold By quantifier elimination every definable sub- in Z. One expects that there are no surprising set can be defined by a quantifier-free formula. exponential algebraic relations holding between This simple idea was given an important geo- integers, but the only known proof of this relies metric reformulation by Seidenberg [31]. We say on a conjecture in transcendental number the- that a subset of Rn is semialgebraic if it is a fi- ory. nite Boolean combination of sets of the form x = (x1, . . . , xn) : f (x) > 0 and x : g(x) = 0 Schanuel’s Conjecture. Let λ1, . . . , λn C be { } { } ∈ where f , g R[X1, . . . , Xn]. It is easy to see that linearly independent over Q. Then the semialgebraic∈ sets are exactly the sets defined λ λ Q(λ1,...,λn,e 1,...,e n) has transcendence de- by quantifier-free formulas. Thus the semialge- gree at least n over Q. braic sets are exactly the definable sets. If X is a semialgebraic subset of Rn+1, then the pro- Schanuel’s conjecture generalizes the Linde- jection of X to Rn is x Rn : y (x, y) X . mann-Weierstrass theorem, viz., if α , . . . , α 1 n Since this is definable, it{ is∈ also semialgebraic∃ ∈ }2 are algebraic numbers that are linearly inde- These are very useful ideas. For example, let α1 αn pendent over Q, then e ,...,e are alge- A Rn be definable. Then the closure of A is braically independent over Q. Schanuel’s con- ⊆ jecture has many nice consequences, such as x Rn : { ∈ the algebraic independence of e and π. In [22] n 2 Macintyre proved that if Schanuel’s conjecture > 0 y y A (xi yi) < , ∀ ∃ ∈ ∧ − } holds, one does not obtain any unexpected ex- iX=1 ponential-algebraic relations on Z. This leads a definable set. Thus the closure of a semialge- to a decision procedure for the quantifier-free braic set is semialgebraic. exp-sentences. L Semialgebraic subsets of R are particularly Tarski also considered the theory of the field simple. For any polynomial f(X) R[X], of complex numbers. Here too one can prove x R : f(x) > 0 is a finite union of open∈ in- quantifier elimination and decidability. The the- tervals.{ ∈ Thus any} semialgebraic subset of R is a ory of the complex field is just the theory of al- finite union of points and intervals. This simple gebraically closed fields of characteristic zero. fact is the starting point of the modern model- x On the other hand, the theory of (C, +, , , e ) theoretic approach to R. Let ∗ be a language − · L will behave very badly, since one can use the for- extending , and let R denote the reals as an L ∗ mula z, w (z2 = 1 ewz = 1 exwz = 1) 2Similarly for the field of complex numbers the defin- ∀ − ∧ → able sets are exactly the constructible sets of algebraic geometry; quantifier elimination is equivalent to Cheval- 1Of course, even if this theory is decidable, decision com- ley’s theorem that the projection of a constructible set putations will, in general, be unfeasible. is constructible.

JULY 1996 NOTICES OF THE AMS 755 ∗ -structure. For example, = exp and One is led to the question: Is the real field with L x L L R = Rexp = (R, +, , , e , <, 0, 1) . We say that exponentiation o-minimal? We will see Wilkie’s ∗ − · R∗ is o-minimal if every subset of R definable positive answer to this question in the section using -formulas is a finite union of points and “Wilkie’s Theorem”. L∗ intervals. Pillay and Steinhorn [27] introduced o- minimality, generalizing an earlier idea of van Ran and Subanalytic Sets den Dries [5]. Remarkably, o-minimality has Most of the results on o-minimal structures men- strong consequences for definable subsets of Rn tioned above were proved before we knew of any for n > 1. interesting o-minimal structures other than the We inductively define the notion of cells as fol- real field. In [4] van den Dries gave the first new lows example of an o-minimal theory. X R is a cell if and only if it is either a n • Let an = f : for some open U [0, 1] , ⊆ L L ∪ { ⊃ point or an interval. f : U R is analytic . • If X Rn is a cell and f : X R is a con- → bn } ⊆ → We define f : R R by tinuous definable function, then the graph → n of f is a cell. b f (x) x [0, 1] f (x) = ∈ • If X Rn is a cell and f , g : X R are con- ⊆ → ( 0 otherwise. tinuous definable functions and f (x) > g(x) b for all x X, then (x, y) : x X and ∈ { ∈ f(x) > y > g(x) is a cell, as are We let Ran be the resulting an -structure. Denef } L (x, y) : x X and f(x) > y and and van den Dries [7] proved that Ran is o-min- { ∈ } (x, y) : x X and y > f (x) . imal and that Ran has quantifier elimination if { ∈ } we add x 1 to the language. Quantifier elim- 7→ x Theorem. (van den Dries [5]/Knight-Pillay-Stein- ination is proven by using the Weierstrass prepa- horn[20]) ration theorem to replace arbitrary analytic func-

i) If R∗ is o-minimal, then every definable set X tions of several variables by analytic functions can be partitioned into finitely many disjoint that are polynomial in one of the variables. cells. Tarski’s elimination procedure is then used to ii) If f : X R is a definable function, then there eliminate this variable. → is a partition of X into finitely many cells Denef and van den Dries also showed that if such that f is continuous on each cell. f : R R is definable in Ran, then f is asymp- → This is just the beginning. In any o-minimal totic to axq for some a R and q Q. In par- ∈ ∈ structure, definable sets have many of the good ticular, although we can define the restriction of topological and geometric properties of the semi- the exponential function to bounded intervals, algebraic sets (see, for example, [6] and [11]). For we cannot define the exponential function glob- example: ally. It is also impossible to define the sine func- • Any definable set has finitely many con- tion globally, for its zero set would violate o-min- nected components. imality. Definable bounded sets can be definably tri- • Although Ran may seem unnatural, the de- angulated. finable sets form an interesting class. n+m m • Suppose X R is definable. For a R We say that X Rn is semi-analytic if for all ⊆ ∈ ⊆ let x in Rn there is an open neighborhood U of x n Xa = x = (x1, . . . , xn) R : (x, a) X . such that X U is a finite Boolean combination { ∈ ∈ } ∩ of sets x U : f (x) = 0 and x U : g(x) > 0 { ∈ } { ∈ } There are only finitely many definable home- where f, g : U R are analytic. We say that omorphism types for the sets X . → a X Rn is subanalytic if for all x in Rn there is • (Curve selection) If X Rn is definable and ⊆ ⊆ an open U and Y Rn+m a bounded semiana- a is in the closure of X, then there is a con- ⊂ lytic set such that X U is the projection of Y tinuous definable f : (0, 1) X such that ∩ → into U. It is well known (see [1] for an excellent lim f (x) = a. survey) that subanalytic sets share many of the x 1 → nice properties of semialgebraic sets. If X Rn is bounded, then X is definable in • (Pillay [26]) If G is a definable group, then ⊂ G is definably isomorphic to a Lie group. Ran if and only if X is subanalytic. Indeed n • If we assume in addition that all definable Y R is definable in Ran if and only if it is the ⊆ functions are majorized by polynomials, image of a bounded subanalytic set under a then many of the metric properties of semi- semialgebraic map. Most of the known proper- algebraic sets and asymptotic properties of ties of subanalytic sets generalize to sets defined semialgebraic functions also generalize. in any polynomial bounded o-minimal theory.

756 NOTICES OF THE AMS VOLUME 43, NUMBER 7 Wilkie’s Theorem m wi w1, . . . , wm (( e = xi) The big breakthrough in the subject came in ∃ i^=1 1991. While quantifier elimination for Rexp is im- m n possible, Wilkie [37] proved the next best thing. wi ri,j ai e = 0). ∧ i=1 j=1 Wilkie’s Theorem. Let φ(x1, . . . , xm) be an exp X Y L formula. Then there is n m and We see that expresses x x ≥ f1,...,fs Z[x1,...,xn,e 1,...,e n] such that ∈ m n φ(x ,...,x ) is equivalent to ri,j 1 n Φ ai xj = 0. iX=1 jY=1 x1 xn xm+1 . . . xn f1(x1, . . . , xn, e , . . . , e ) = . . . n ∃ ∃ Let Xr,a denote the set of x R such that = f (x , . . . , x , ex1 , . . . , exn ) = 0. ∈ s 1 n (x, r, a) holds. By o-minimality, mn m Thus every formula is equivalent to an exis- Xr,a : r R ,a R represents only fi- { ∈ ∈ } tential formula (this property is called model nitelyΦ many homeomorphism types. completeness), and every definable set is the pro- In addition to answering the question of o- jection of an exponential variety. minimality, some headway has been made on the Wilkie’s proof depends heavily on the fol- problem of decidability. Making heavy use of Wilkie’s methods and Khovanski’s theorem, Mac- lowing special case of a theorem of Khovanski intyre and Wilkie [23] have shown that if [19]. Before Wilkie’s theorem, Khovanski’s re- Schanuel’s conjecture is true, then the first-order sult was the best evidence that Rexp is o-mini- theory of Rexp is decidable. One would expect mal; indeed Khovanski’s theorem is also the cru- that the computational complexity of this the- cial tool needed to deduce o-minimality from ory would be horrific, but the only known lower model completeness. bounds are those imposed by the theory of the n additive ordered group. Macintyre and Wilkie Khovanski’s Theorem. If f1, . . . , fm : R R are n → also gave an axiomatization of Th(Rexp) (as- exponential polynomials, then x R : f1(x) = { ∈ suming Schanuel’s conjecture); unfortunately . . . fn(x) = 0 has finitely many connected com- } the axioms are quite complicated and ugly. ponents. Miller [25] provided an interesting counter- If X R is definable in Rexp, then by Wilkie’s point to Wilkie’s theorem. Using ideas of Rosen- ⊆ theorem there is an exponential variety V Rn licht [30], he showed that if R∗ is any o-minimal ⊆ such that X is the projection of V. By Khovan- expansion of the real field that contains a func- ski’s theorem, V has finitely many connected tion that is not majorized by a polynomial, then components and X is a finite union of points and exponentiation is definable in R∗. intervals. Thus Rexp is o-minimal. Ran, exp Using the o-minimality of Rexp, one can im- x Let an,exp be an e , and let Ran, exp be the prove some of Khovanski’s results on “fewno- L L ∪ { } real numbers with both exponentiation and re- mials”. From algebraic geometry we know that stricted analytic functions. In [8] van den Dries, we can bound the number of connected com- Macintyre, and I examined this structure. Using n ponents of a hypersurface in R uniformly in the the Denef-van den Dries quantifier elimination degree of the defining polynomial. Khovanski for Ran and a mixture of model-theoretic and val- [19] showed that it is also possible to bound the uation theoretic ideas, we were able to show number of connected components uniformly in that Ran, exp has quantifier elimination if we add the number of monomials in the defining poly- log to the language. Using quantifier elimination nomial. We will sketch the simplest case of this. and Hardy field style arguments (but avoiding Let n,m be the collection of polynomials in the geometric type of arguments used by Kho- F R[X1, . . . , Xn] with at most m monomials. For vanski), we were able to show that Ran, exp is o- p n,m let minimal. Using arguments similar to Wilkie’s, o- ∈ F minimality was also proved in [10]. + n Since the language an,exp has size 2ℵ0 , one V (p) = x = (x1, . . . , xn) R : L { ∈ would not expect to give a simple axiomatiza- n tion of the first-order theory of Ran, exp. Ressayre xi 0 p(x) = 0 . ≥ ∧ } noticed that the model-theoretic analysis of i^=1 Ran, exp uses very little global information about We claim that there are only finitely many home- exponentiation. This observation leads to a “rel- + omorphism types of V (p) for p n,m. Let ative” axiomatization. The theory T h(R ) ∈ F an, exp m,n(x1, . . . , xn, r1,1, . . . , r1,n, . . . , rm,1, . . . , is axiomatized by the theory of Ran and by ax- rm,n, a1, . . . , am) be the formula ioms asserting that exponentiation is an in- Φ

JULY 1996 NOTICES OF THE AMS 757 creasing homomorphism from the additive group nite, so we want exp(t 1) > tr for all r R. To − ∈ onto the multiplicative group of positive ele- remedy this, we extend 0 to r ments that majorizes every polynomial. 1 = art : ar = 0 for all r > 0 R which we Using this axiomatization and quantifier elim- { } ⊕ order Plexiographically. We then define anΓ expo- ination, one can show that any definable func- nentialΓ exp : R((t 0 )) R((t 1 )) by tion is piecewise given by a composition of poly- → Γ Γ n nomials, exp, log, and restricted analytic α r exp(α + r + ) = t− e functions on [0, 1]n. For example, the definable n! x 2 X function f (x) = ee ex 3x is eventually in- − − where r R, is infinitesimal and all elements creasing and unbounded. Thus for some large of the support∈ of α are negative. We will still not enough r R there is a function g : (r, + ) R 1 ∈ ∞ → be able to define an exponential of exp(t− ), so such that f (g(x)) = x for x > r. The graph of g we must repeat this process infinitely many is the definable set (x, y) : x > r and y 2 { times. The resulting field will have an exponen- ee ey 3y = x . Thus g is a definable func- − − } tia, but not every positive element will have a log- tion, and there is some way to express g ex- arithm. So a further infinite construction needs plicitly as a composition of rational functions, to be done to add logs (for details see [9]). exp, log, and restricted analytic functions. In The resulting model, which we call R((t))LE, is most cases it is in no way clear how to get these very useful for examining the asymptotic be- explicit representations of an implicitly defined havior of definable functions, as the formal se- function. One important corollary is that every ries representing the function reflects its as- definable function is majorized by an iterated ymptotic expansion. Consider the function exponential. f(x) = (log x)(log log x), and let g be a composi- tional inverse to f defined on (r, + ) for some Logarithmic-Exponential Series r R. In [16] Hardy conjectured that∞ g is not ∈ The axiomatization of the theory of Ran, exp leads asymptotic to a composition of exp, log and to a relatively natural algebraic construction of semialgebraic functions. Shackell [32] came close nonstandard models of Ran, exp. These models to verifying this conjecture by proving that the can be used to study asymptotic properties of inverse to (log log x)(log log log x) is not asymp- definable functions. For example, we use them totic to such a composition.3 Using the series ex- to answer a problem of Hardy. pansion for g in R((t))LE, we are able to show that We first show how to build many nonstandard Hardy’s conjecture is correct (see [9]). LE models of Ran. We also use asymptotic expansion in R((t)) 2 Suppose ( , +) is an abelian group and < is to show, for example, that (0, + ), x et dt | ∞ 0 an ordering of compatible with addition. For and the bold Riemann zeta function restrictedR formal sums Γ to (1, + ) are not definable inΓ Ran, exp. Γ ∞ γ f = aγ t And Beyond γ The most pressing current problem in the sub- X∈ ject is to find o-minimal expansions of R al- Γ we define the support of f to be the set lowing richer collections of functions. Recently γ : aγ = 0 and let R((t )) be the set of for- van den Dries and Speissegger [12], building on { ∈ 6 } γ mal sums f = aγ t such thatΓ the support of work of Tougeron [36] and Gabrielov [15], have f is wellΓ ordered.P We can make R((t )) an ordered shown how to prove model completeness and o- field by defining addition and multiplicationΓ in minimality for expansions containing functions γ the usual way and f = aγt > 0 iff aγ0 > 0 with divergent Taylor series coming from certain where γ0 is least such that aγ0 = 0. If r R and quasi-analytic classes. For example, they have 0 P 6 ∈ r > 0, then t < r = rt . Thus t is an infinitesi- shown that one can add a function with domain mal element. If is also a Q-vector space, then (0, ) and asymptotic expansion R((t )) is a real closed field and hence a non- ∞ standardΓ model Γof the theory of the real field. xlog n. It is possible to view all of the restricted func- nX=1 tions as acting on R((t )). For example, if x [0, 1], then we can write x = r + where There is some hope that in a sufficiently rich ∈ Γ r R [0, 1] and is infinitesimal. Then we de- o-minimal expansion of R one would have all of ∈ x∩ r n the Poincaré return maps for polynomial vector fine e as e n∞=0 n! . Using the Denef-van den fields in R2 . If this is true, it is likely that one Dries quantifierP elimination, one shows that R((t )) is a model of the theory of Ran. could give a new proof that such a vector field ToΓ build a nonstandard model of exponenti- ation, we start with 0 = R. We cannot define a 3On the other hand, the inverse to x log x is asymp- R 1 total exponential on R((t )) because t− is infi- totic to x/ log x. Γ

758 NOTICES OF THE AMS VOLUME 43, NUMBER 7 has only a finite number of limit cycles.4 More- [19] A. Hovanskii, On a class of systems of transcen- over, one expects that such a proof would yield dental equations, Soviet Math. Dokl. 22 (1980), a bound on the number of limit cycles uniform 762–765. in the degrees of the polynomials defining the [20] J. Knight, A. Pillay, and C. Steinhorn, Definable sets in ordered structures II, Trans. Amer. Math. vector field. We are far from answering this, Soc. 295 (1986), 593–605. and undoubtedly this problem will fuel the sub- [21] S. Lang, Algebra, Addison-Wesley, 1971. ject for some time to come. [22] A. Macintyre, Schanuel’s conjecture and free ex- ponential rings, Ann. Pure Appl. Logic 51 (1991), References 241–246. [1] E. Bierstone and P. Milman, Semialgebraic and [23] A. Macintyre and A. Wilkie, On the decidability subanalytic sets, IHES Publ. Math. 67 (1988), 5–42. of the real exponential field (to appear). [2] J. Bochnak, M. Coste, and M. F. Roy, Géométrie [24] D. Marker, Introduction to the model theory of algébrique réelle, Springer-Verlag, 1986. fields, Model Theory of Fields (D. Marker, M. Mess- [3] C. C. Chang and H. J. Keisler, Model theory, mer, and A. Pillay, eds.), Lecture Notes in Logic, North-Holland, 1977. vol. 5, Springer-Verlag, 1996. [4] J. Denef and L. van den Dries, p-Adic and real [25] C. Miller, Exponentiation is hard to avoid, Proc. subanalytic sets, Ann. Math. 128 (1988), 79–138. Amer. Math. Soc. 122 (1994), 257–259. [5] L. van den Dries, Remarks on Tarski’s problem [26] A. Pillay, Groups and fields definable in o-mini- concerning (R, +, , exp), Logic Colloquium ‘82 mal structures, J. Pure Appl. Algebra 53 (1988). · (G. Longi, G. Longo, and A. Marcja, eds.), North- [27] A. Pillay and C. Steinhorn, Definable sets in or- Holland, 1984. dered structures I, Trans. Amer. Math. Soc. 295 [6] ———, Tame topology and o-minimal structures, (1986), 565–592. monograph in preparation. [28] J. P. Ressayre, Integer parts of real closed expo- [7] ———, A generalization of the Tarski-Seidenberg nential fields, Arithmetic, Proof Theory, and Com- theorem, and some nondefinability results, Bull. putational Complexity (P. Clote and J. Krajicek, Amer. Math. Soc. (N. S.) 15 (1986), 189–193. eds.), Oxford Univ. Press, 1993, pp. 278–288. [8] L. van den Dries, A. Macintyre, and D. Marker, [29] A. Robinson, Model theory as a framework for al- The elementary theory of restricted analytic fields gebra, Studies in Model Theory (M. Morley, ed.), with exponentiation, Ann. of Math. 140 (1994), Math. Assoc. Amer., 1973. 183–205. [30] M. Rosenlicht, The rank of a Hardy field, Trans. [9] ———, Logarithmic-exponential power series, J. Amer. Math. Soc. 280 (1983), 659–671. London Math. Soc. (to appear). [31] A. Seidenberg, A new decision method for ele- [10] L. van den Dries and C. Miller, On the real ex- mentary algebra, Ann. of Math. 60 (1954), ponential field with restricted analytic functions, 365–374. Israel J. Math. 85 (1994), 19–56. [32] J. Shackell, Inverses of Hardy’s L-functions, Bull. [11] ———, Geometric categories and o-minimal struc- London Math. Soc. 25 (1993), 150–156. ture, Duke Math. J. (to appear). [33] A. Tarski, Sur les ensembles définissables de nom- [12] L. van den Dries and P. Speissegger, The real field bres réels, Fund. Math. 17 (1931), 210–239. with convergent generalized power series is model [34] ———, A decision method for elementary algebra complete and o-minimal, preprint. and geometry (prepared for publication by J. C.C. [13] J. Ecalle, Finitude des cycles-limites et accéléro- McKinsey), RAND Corp. monograph, 1948. sommation de l’application de retour, Bifurca- [35] ———, The completeness of elementary algebra tions of Planar Vector Fields (J. P. Françoise and and geometry, Institut Blaise Pascal, Paris, 1967 R. Roussarie, eds.), Lecture Notes in Math., vol. (a reprint of page proofs of a 1940 work). 1455, Springer-Verlag, 1990, pp. 74–159. [36] J.-Cl. Tougeron, Sur les ensembles semi-analy- [14] M. J. Fischer and M. O. Rabin, Super-exponential tiques avec conditions Gevrey au bord, Ann. Sci. complexity of Pressburger arithmetic, Complexity École Norm. Sup. (4) 27 (1994), 173–208. of Computation (R. Karp, ed.), Proc. SIAM-AMS [37] A. J. Wilkie, Some model completeness results for Sympos. Appl. Math., Amer. Math. Soc., 1974. expansions of the ordered field of reals by Pfaffian [15] A. Gabrielov, Existential formulas for analytic functions and exponentiation, J. Amer. Math. Soc. functions, preprint. (to appear). [16] G.H. Hardy, Properties of logarithmico-exponen- tial functions, Proc. London Math. Soc. 10 (1912), 54–90. [17] J. Heintz, M-F. Roy, and P. Solerno, Complexité du principe de Tarski-Seidenberg, C. R. Acad. Sci. Paris Sér. I Math. 309 (1989), 825–830. [18] Y. Il iashenko, Finiteness theorems for limit cy- cles, Transl. Math. Monographs, vol. 94, Amer. Math. Soc., 1991.

4This is part of Hilbert's 16th problem that was solved by Ecalle [13] and Il'iashenko [18].

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