An o-minimal structure without mild parameterization

Annals of Pure and Applied Logic 162 (2011) 409–418

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An o-minimal structure without mild parameterization Margaret E.M. Thomas ∗ School of Mathematics, The University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom

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Article history: Received 13 April 2010 Accepted 5 November 2010 Available online 22 January 2011 Communicated by T. Scanlon MSC: 03C64 26A99 Keywords: Parameterization O-minimal

abstract We prove, by explicit construction, that not all sets definable in polynomially bounded ominimal structures have mild parameterization. Our methods do not depend on the bounds particular to the definition of mildness and therefore our construction is also valid for a generalized form of parameterization, which we call G-mild. Moreover, we present a cell decomposition result for certain o-minimal structures which may be of independent interest. This allows us to show how our construction can produce polynomially bounded, model complete expansions of the real ordered field which, in addition to lacking G-mild parameterization, nonetheless still have analytic cell decomposition. © 2011 Elsevier B.V. All rights reserved.

1. Introduction Parameterizations are geometric coverings of sets by the images of certain functions. These parameterizing functions are always differentiable up to a prescribed order and have particular bounds imposed on their derivatives. Our focus here is on the parameterization of sets definable in o-minimal expansions of real closed fields, especially on those which are definable in expansions of the real ordered field R := ⟨R, +, −, ·, 0, 1, <⟩, and on their parameterization by functions definable within the same structure. Let us make precise a general definition of parameterization, before considering particular variants. First, we let M be an o-minimal structure over a real closed field M. Definition 1.1. Let X be a definable set in M of dimension k. A parameterization of X is a finite collection S of functions φ : (0, 1)k −→ X definable in M such that X =



φ((0, 1)k ).

φ∈S

We may then refine this definition by placing certain constraints on the derivatives of the functions in a parameterization S . The first parameterization result was given by Gromov [6] who, in refining work of Yomdin [19], stated an algebraic lemma for compact semialgebraic sets, i.e. for closed and bounded sets definable in R (see also [2]). This result was then generalized by Pila and Wilkie to certain sets definable in o-minimal expansions of real closed fields (see [16]). The statement of the latter involves the form of parameterization given by the next definition. From now on we shall use the following multiindex notation: for any α = (α1 , . . . , αk ) ∈ Nk , we define the modulus |α| := α1 + · · · + αk , the factorial α! := α1 ! · · · αk ! and the differential operator Dα :=

∗

∂ |α| α . ∂ x1 . . . ∂ xk k α1

Corresponding address: Fachbereich Mathematik und Statistik, Zukunftskolleg, Universität Konstanz, Box 216, D-78457 Konstanz, Germany. E-mail address: [email protected].

0168-0072/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.apal.2010.11.004

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Definition 1.2. Let X be a subset of (0, 1)n , for some n ∈ N, definable in M , with dim(X ) = k. An r-parameterization of X , for r ∈ N, is a parameterization Sr satisfying, for each function φ ∈ Sr , (i) φ ∈ C (r ) ((0, 1)k ); (ii) |Dα φ(¯x)| ≤ 1, for each α ∈ Nk with |α| ≤ r and for all x¯ ∈ (0, 1)k . Pila and Wilkie’s o-minimal Reparameterization Theorem may then be stated as follows. Theorem 1.3 (Reparameterization Theorem; [16], Theorem 2.3). Let X be a subset of (0, 1)n definable in M , for some n ∈ N. For any r ∈ N, X has an r-parameterization. By rescaling, if necessary, this theorem applies to any set X definable in M which can be contained in a ball whose radius lies in N (where Q is identified with the prime submodel of M ). The primary application of this theorem is in bounding the density of rational points lying on certain subsets of Rn . For further details, we refer the reader to some of the extensive literature on this subject (for example, [1,12,16,13,14]). Briefly, it is shown in [16] that all of the rational points of interest lying on the image of an r-parameterizing function are contained in the union of few zero sets of polynomials of bounded degree. Combining this result with the above Reparameterization Theorem gives a general bound on the density of rational points for o-minimal subsets of Rn . It is observed in [16] that the bound arising in this way is the best possible for o-minimal expansions of R in general, as any improvement has a counterexample definable in the structure Ran , the real ordered field expanded by all restricted analytic functions (see [12], Section 7.5, for more details). However, an improvement in this bound is conjectured by Wilkie ([16], 1.11) for sets definable in Rexp := ⟨R, x → ex ⟩. One approach to this conjecture is to consider specific examples of definable sets, especially those which are independently known to have few intersections with the zero sets of polynomials. At the same time, new parameterizations of such sets are sought, where different constraints are placed on the derivatives of the parameterizing functions. This amendment is to allow for better bounds on the number of zero sets of polynomials needed to contain the rational points of interest. Mild parameterization is just such a modification, introduced by Pila as part of an analysis of Pfaffian functions in [13] and later made use of in [15,7,8]. It is a kind of parameterization by smooth functions, i.e. those which are C ∞ . The definition we present here is slightly more general, being a natural extension to any o-minimal structure M over a real closed field M, with the parameterizing functions definable in the same structure. Definition 1.4. A smooth function φ : (0, 1)k −→ (0, 1) is said to be (A, C )-mild if

|Dα f (¯x)| ≤ α!(A |α|C )|α| , for all α ∈ Nk and for all x¯ ∈ (0, 1)k . We say that a definable map Φ : (0, 1)k −→ (0, 1)n is (A, C )-mild if each of its coordinate functions is (A, C )-mild. A mild parameterization of a set X ⊆ (0, 1)n definable in M is a parameterization S of X for which all φ ∈ S are (A, C )-mild, for some A, C ≥ 0. Note that, over R, if a function f : (0, 1) −→ R is analytic on a neighbourhood of [0, 1], then f is (A, 0)-mild, for some A ≥ 0. It has already been established (see [7]) that any set definable in a reduct of Ran has a mild parameterization by functions which are (A, 0)-mild, for some A ≥ 0. Here we address the question of whether or not there is a general result, analogous to the Pila–Wilkie Reparameterization Theorem, for mild parameterization of sets definable in o-minimal expansions of real closed fields. We shall answer this question negatively, by presenting the explicit construction of an o-minimal structure and a definable set within it which does not have a mild parameterization. In fact, the resulting structure will be an o-minimal expansion of R, thereby demonstrating that there is no analogous mild parameterization theorem even for o-minimal expansions of R. The definable set which we construct to witness a lack of mild parameterization will be the graph of a one variable function. The method of constructing this function will not depend on the bounds on derivatives particular to the definition of mildness, namely, in the case of the parameterization of a one-dimensional set, the function m → m!(AmC )m , for some A, C ≥ 0. So the result we obtain will be more general. For each function of N, we may define a corresponding smooth parameterization of one-dimensional sets. We may then construct a counterexample to the supposition that all onedimensional sets definable in o-minimal expansions of R have such a parameterization. Let us make this statement more precise by generalizing our previous definition of mildness as follows. Definition 1.5. Let G : Nk → (0, ∞) be a function. We say that a smooth map f : (0, 1)k −→ (0, 1) is G-mild if |Dα f (¯x)| ≤ G(α), for all α ∈ Nk and for all x¯ ∈ (0, 1)k . We say that a definable map Φ : (0, 1)k −→ (0, 1)n is G-mild if each of its coordinate functions is G-mild. Let X be a subset of (0, 1)n definable in M of dimension k. For a given function G : Nk → (0, ∞), we say that S is a G-mild parameterization of X if S is a parameterization of X for which all φ ∈ S are G-mild. Our previous definition of mildness is the special case of this definition for which G(α) = α!(A |α|C )|α| , for some A, C ≥ 0 and all α ∈ Nk . The first theorem we shall prove may now be stated as follows.

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411

Theorem 1.6. For any function G : N → (0, ∞), there exist both an o-minimal, polynomially bounded, model complete expansion  R of R with smooth cell decomposition and a one-dimensional set X ⊆ (0, 1)2 definable in  R such that X does not have a G-mild parameterization. We shall establish the existence of such structures by explicit construction. These methods will demonstrate that the witness set X may in fact be chosen to be the graph of an analytic function. Consequently, we may strengthen this result in the following manner. Theorem 1.7. For any function G : N → (0, ∞), there exist both an o-minimal, polynomially bounded, model complete expansion  R of R with analytic cell decomposition and a one-dimensional set X ⊆ (0, 1)2 definable in  R such that X does not have a G-mild parameterization. In order to prove this result, we shall prove a cell decomposition result for functions definable in locally polynomially bounded structures which may be of independent interest. Locally polynomially bounded structures are a particular class of o-minimal structures introduced by Jones and Wilkie [9], which include both Rexp and certain model complete, polynomially bounded o-minimal structures amongst their number. The outline of the remainder of this paper is as follows. In Section 2, we identify a sufficient condition for a one variable function, definable in a polynomially bounded, o-minimal structure, to satisfy if its graph is to fail to have a Gmild parameterization. Roughly, a function H : (0, 1) −→ (0, 1) will satisfy this condition if H has suitably large derivatives at a sequence of points approaching the origin, so as to confound the bounds of G-mild parameterization with sufficiently wild behaviour. In Section 3, we shall demonstrate the existence of functions which satisfy a more general version of this condition on derivatives. We shall present two different methods for this, one of which produces functions which are also analytic on (0, 1). In Section 4, we shall use a result of Le Gal from [11] to find functions not only satisfying our condition, but which are also definable in an o-minimal, polynomially bounded, model complete expansion of R with smooth cell decomposition, thus proving Theorem 1.6. Finally, in Section 5, we shall combine a general cell decomposition result for o-minimal structures with a representation result for functions definable in locally polynomially bounded structures. This will directly prove Theorem 1.7. 2. A condition on derivatives In this section, let us fix a function G : N → R and an o-minimal, polynomially bounded expansion  R of R. Let H : (0, 1) → R be a one variable function definable in  R whose graph does have a G-mild parameterization S . Note that, given any function f on R and any parameterization of its graph, it is possible to derive another parameterization of the same set in which the graph of f over some interval (0, η) is parameterized by a single function ⟨φ, ψ⟩ : (0, 1) −→ R2 , with φ and ψ G-mild if the original parameterization were G-mild. So, without loss, we may assume that we have a pair of G-mild functions ⟨φ, ψ⟩ : (0, 1) −→ R2 in S parameterizing the graph of H(0,η) , for some η > 0. We may also simplify the argument by assuming, without loss, that φ(t ) → 0 as t → 0+ , with φ(t ) ̸= 0 for t close to 0. By o-minimality, there exists δ > 0 such that φ(0,δ) is bijective onto some interval (0, η′ ) ⊆ (0, η), for η′ > 0, and we may also assume that φ ′ (t ) ̸= 0 for t ∈ (0, δ). Let us consider the relationship between H and ⟨φ, ψ⟩. For all t ∈ (0, 1), we have that H (φ(t )) = ψ(t ). From this expression, by the product rule and the Faà di Bruno formula (see [3], Theorem 2.1), we obtain the following identity:

− H (n) (φ(t )) =

(φ ′ (t ))l1 · · · (φ (n) (t ))ln ψ (2n−1−

∑n

i=1 ili )

(t )

¯l∈Ln

(φ ′ (t ))2n−1

,

for all t ∈ (0, δ) and for all n ≥ 1, where the set Ln is defined, for each n ≥ 1, to be

 Ln :=

⟨l1 , . . . , ln ⟩ ∈ N | n − 1 ≤ n

n − i=1

ili ≤ 2(n − 1) and

n −

 li = n − 1 .

i=1

We know that φ and ψ are G-mild; by substituting the G-mild bounds into the above expression and rearranging, we establish the following inequality, for all n ≥ 1 and for all t ∈ (0, δ).

   n −  −   l1 ln G(1) · · · G(n) G 2n − 1 − ili      (n)  i=1 ¯  H (φ(t )) ≤  l∈Ln    (φ ′ (t ))2n−1          max{G(1), . . . , G(n)}n |Ln |   ≤   (φ ′ (t ))2n−1   n  (n · max{G(1), . . . , G(n)})  . ≤   (φ ′ (t ))2n−1

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Define a sequence of real numbers (Gn )n≥1 by Gn := (n · max{G(1), . . . , G(n)})n  ,





so that, for all n ≥ 1, Gn

 (n)  H (φ(t )) ≤

|φ ′ (t )|2n−1

.

Since φ ′ (t ) ≤ G(1), for all t ∈ (0, 1), we have that limt →0+ φ ′ (t ) ∈ R, by o-minimality. There are now two cases to consider: φ ′ (t ) → 0 as t → 0+ or φ ′ (t ) → a ̸= 0 as t → 0+ . Case 1 Suppose first that φ ′ (t ) → 0 as t → 0+ . Since we are assuming that φ is definable in a polynomially bounded structure,  we know that there exists some M ∈ N and some δ ∗ ∈ (0, δ) such that, for all t ∈ (0, δ ∗ ), φ ′ (t ) ≥ t n , for all n ≥ M. More-





over, by the Mean Value Theorem and the convergence of φ ′ , we can choose δ ∗ so that we also have Therefore, for all t ∈ (0, δ ∗ ) and for all n ≥ M, Gn

 (n)  H (φ(t )) ≤

|φ(t )|n(2n−1)

|φ(t )| t

≤ 1 on (0, δ ∗ ).

.

Case 2 Suppose that φ ′ (t) → a ̸= 0 as t → 0+ . Let ϵ ∈ R be such that 0 < ϵ < |a|. There is some δ ′ > 0 such that, whenever t ∈ (0, δ ′ ), we have φ ′ (t ) − a < ϵ . Moreover, we can clearly choose δ ′ such that |φ(t )| < min{|a| − ϵ, 1} on (0, δ ′ ). In this case we have that, for all t ∈ (0, δ ′ ) and all n ≥ 1,

 (n)  H (φ(t )) ≤ ≤ ≤

Gn

(|a| − ϵ)2n−1 Gn

|φ(t )|2n−1 Gn

|φ(t )|n(2n−1)

.

In both cases, for sufficiently small x ∈ (0, η) and sufficiently large n ∈ N, we have that

 (n)  H (x) ≤

Gn xn(2n−1)

.

This gives us the condition that we are looking for: if we can find a smooth function H : (0, 1) → R which is definable in a polynomially bounded, o-minimal structure and, for each n ≥ 1, we can find a point un ∈ (0, 1) at which the reverse inequality holds, i.e.

 (n)  H (un ) >

Gn n(2n−1) un

,

(2.1)

and, further, un tends to 0 as n → ∞, then the graph of H cannot possibly have a G-mild parameterization. In the next section we shall describe some different methods for constructing functions which have arbitrarily large derivatives, with each successive derivative taking its high value at one of a sequence of points approaching the origin. Then, in Section 4, we shall explain how these constructions can be used to find functions H which not only satisfy the above inequality, together with an appropriate choice of sequence (un )n∈N in (0, 1), but are also each definable in an o-minimal, polynomially bounded structure which is, moreover, model complete. 3. Functions with large derivatives Our aim in this section is to give two methods for proving the following theorem. Theorem 3.1. Let (Kn : RMn → R)n∈N be a sequence of real-valued functions. There exists a smooth function h : (0, 1) → R, with extension to R, and a strictly decreasing sequence (un )n∈N of real numbers in (0, 1) converging to 0 such that  (n) a smooth  h (un ) > Kn (u1 , . . . , uM ), for all n ≥ 1. n 3.1. The Whitney Extension Theorem The first method makes use of the Whitney Extension Theorem, which we state here. Theorem 3.2 (Whitney Extension Theorem; [18], Theorem I). Let A be a closed subset of R and let f : R −→ R be of class C m (for some m ∈ N ∪ {∞}) in A, in the sense that, for all x ∈ A and for each n ≤ m, there exists fx,n ∈ R such that

M.E.M. Thomas / Annals of Pure and Applied Logic 162 (2011) 409–418

f ( x + h) =

413

− fx,k+n hk + Rn (x, h), k! k≤m−n

whenever x + h ∈ A and where Rn (x, h) is a remainder term such that, given ϵ > 0, there exists δ > 0 such that if |h| < δ , then |Rn (x, h)| ≤ ϵ |h|m−n . Then there is a function F : R −→ R of class C m in R such that F (n) (x) = fx,n , for all x in A and for all n ≤ m, and F is analytic in R \ A. This construction allows us to form a smooth function with arbitrarily chosen derivatives on any closed set (which is, moreover, analytic off that closed set). Let us fix a strictly decreasing sequence (un )n∈N of real numbers in (0, 1) converging to 0 and define the closed set A := {un | n ∈ N} ∪ {0}. For each n ∈ N, set

 f x ,n =

2Kn (u1 , . . . , uMn ) if x = un , if either x = um for some m ̸= n, or x = 0.

1

 (nThen it is fairly clear to see that, by the Whitney Extension Theorem, there exists a smooth function h : R → R satisfying h ) (un ) > Kn (u1 , . . . , uM ), for all n ∈ N, and we can take the restricted function h(0,1) as satisfying the requirements of n

Theorem 3.1. 3.2. An analytic construction The function constructed using the Whitney Extension Theorem is only analytic on (0, 1) \ {un | n ∈ N} ∪ {0}. The values it takes outside [0, 1] are not of interest, since our ultimate objective is to define a suitable function only on (0, 1), but we may still look to construct a function which is also analytic on all of (0, 1). This property will enable us to draw some deeper conclusions later about the structures which we are building (see Section 5). In order to prove this property of analyticity more efficiently, we shall, in fact, first build a complex analytic function on a domain U containing (0, 1) which has the required property and which we may later restrict to (0, 1). Fix (un )n≥1 , a strictly decreasing sequence of reals lying in (0, 1), which satisfies u1 = 21 and, for all n ≥ 1, un+1 < u4n . Set u0 = 1. We shall also have another strictly decreasing sequence of reals, (vn )n≥1 , lying in (0, 1) which converges to 0. This sequence will be fixed later in terms of (un )n≥1 and will satisfy certain conditions which will be identified during the course of the following argument. From the outset we shall, for simplicity, insist that vn < un , for all n ≥ 1. Define the open complex domain



 U :=





n∈N

r ∈[un+1 ,un )

B vn+1 (r ) 4

∩ B1 (0).

Note that (0, 1) ⊆ U. We consider the following complex formal series, for all z ∈ U:

− [(z − u1 ) · · · (z − un )]n n ≥1

(z − un )2 + vn2

.

Our aim is to show that this series converges to a complex analytic function on U. We shall do so by showing that it is uniformly Cauchy on compact subsets of U, which is sufficient by combining the Cauchy criterion for uniform convergence of series (see, for example, [5], Definition 3.2.8) with the following classical theorem (see, for example, [17]). Theorem 3.3 ([17], Theorem 2.4.1). If (fn )n∈N is a sequence of complex analytic functions on an open subset U of C, which is uniformly convergent on each compact subset of U, then the limit exists and is analytic on U. Let K be a compact subset of our domain U. By compactness, there are finitely many r1 , . . . , rk ∈ (0, 1), with each rj lying in some [unj +1 , unj ), such that the sets B vnj +1 (rj ), for j = 1, . . . , k, cover K . Define 4



µ := min rj −

vnj +1 4



| j ∈ {1, . . . , k} ,

vnj +1

vn +1

> unj +1 − j4 > 0, for all j ∈ {1, . . . , k}. Moreover, for all j ∈ {1, . . . , k}, we have µ ≤ unj , vn +1 else, for some i ∈ {1, . . . , k}, µ > uni > ri > ri − i4 , which contradicts the minimality of µ. Set N ∈ N sufficiently large that uN < µ (so N > nj , for all j ∈ {1, . . . , k}). We shall consider n > N + 1. which is positive as rj −

4

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M.E.M. Thomas / Annals of Pure and Applied Logic 162 (2011) 409–418

Fix z ∈ K . There is some j ∈ {1, . . . , k} such that z ∈ B vnj +1 (rj ). If nj ≥ 1, then, as n > nj + 1, we have that 4

|z − un | ≤ (|z − rj | + |rj − un |)     vnj +1    n  +  un − un  <  j 4  n  1 + |u1 | ≤ 4  n 3 . = n

n

4

Otherwise, if nj = 0, so rj ∈ [ 12 , 1), then we have that

|z − u1 |n ≤ (|z − rj | + |rj − u1 |)n   n  v1  1 <  + 4 2 n  1 1 + < 4 2  n 3 = . 4

So, regardless of where z lies in K , we have that, for all n > N + 1,

(|z − u1 | · · · |z − un |) < n

 n 3 4

,

since, moreover, |z − um |n < 1, for  all m ∈ N. Now consider (z − un )2 + vn2 . Since vn → 0 as n → ∞, there exists N0 ≥ N + 1 such that, if n > N0 , then vn < µ − uN . So, if z is written as x + iy, then, for all n > N0 , we have (x − un )2 > (µ − uN )2 , since rj − x < Thus, for all n > N0 ,

vnj +1 4

and µ is a minimum.

2

|(z − un )2 + vn2 | − ((x − un )2 + vn2 )2 = y2 (y2 + 2((x − un )2 − vn2 )) ≥ 0. Therefore, we have that, for all n > N0 ,

|(z − un )2 + vn2 | ≥ (x − un )2 + vn2 ≥ (x − un )2 ≥ (µ − uN )2 , which gives us a lower bound for the nth denominator which is not dependent on n. Now let ϵ > 0 and choose NK ∈ N sufficiently large that both NK ≥ N0 and, for all n > NK , n1 > n2 > NK . Denote the partial sums by Sm (z ) :=

m − [(z − u1 ) · · · (z − un )]n n=1

(z − un )2 + vn2

,

for all m ≥ 1. We see that

  n1  −    [( z − u1 ) · · · (z − un )]n   Sn (z ) − Sn (z ) =   1 2 n=n +1  (z − un )2 + vn2 2   n n1 3 − 4 ≤ (µ − uN )2 n=n2 +1  n2 +1   n1 −n2  4 34 1 − 34 = (µ − uN )2 < ϵ.

 3 n 4

<

ϵ(µ−uN )2 4

. Let

M.E.M. Thomas / Annals of Pure and Applied Logic 162 (2011) 409–418

415

So the sequence of partial sums (Sm )m∈N is uniformly Cauchy on every compact subset K of U and therefore h(z ) :=

− [(z − u1 ) · · · (z − un )]n (z − un )2 + vn2

n ≥1

defines an analytic function on U. Consequently, the function h (0,1) is real analytic, which we shall henceforth relabel as h for convenience. Note that, even though h has a smooth extension to R, that extension may not be analytic at the origin, since the radius of convergence Rt of h about each t ∈ (0, 1) is such that Rt → 0+ as t → 0+ . Now let us consider the derivatives of h. Denote each term in the series by hn (t ) :=

[(t − u1 ) · · · (t − un )]n . (t − un )2 + vn2

Fix m > 1. If n ≥ m, then d m hn dt m

m   − m (t ) = ((t − um )n )(k) k=0

k



((t − u1 ) · · · (t − um ) · · · (t − un ))n (t − un )2 + vn2

(m−k)

and so we have that d m hn dt m

 m  m!((um − u1 ) · · · (um − um−1 )) , if n = m; 2 (um ) = vm  0, if n > m.

It is similarly easy to see that dh1 dt

(u1 ) =

1

v12

.

Therefore, we see that

 m!((um − u1 ) · · · (um − um−1 ))m   , if m > 1; Σm−1 +  m d h vm2 ( um ) = 1 m  dt  , if m = 1,  v12 ∑m−1 dm hn dm hn where Σm−1 := n=1 dt m (um ). Whenever n < m, the expression for dt m (um ) involves only u1 , . . . , un , um and v1 , . . . , vn ; in particular, it does not involve vm , and so nor does the expression for Σm−1 . Consequently, once the sequence (un )n≥1 has been fixed, we are free to choose the sequence (vn )n≥1 inductively so as to ensure that the nth derivative of h at un is as large as we should require, as long as (vn )n≥1 is a strictly decreasing sequence in (0, 1) converging to 0, with vn < un , for all n ≥ 1. 4. Analytic perturbation It remains for us to show how we may find functions, whose graphs do not have a G-mild parameterization, definable in polynomially bounded, o-minimal structures. As indicated in Section 2, we shall do so by first establishing the following proposition. Proposition 4.1. For any sequence of real numbers (Gn )n≥1 , there exist both a smooth function H : (0, 1) → R and a sequence (un )n∈N in (0, 1), with un → 0 as n → ∞, such that H is definable in a polynomially bounded, o-minimal structure (which, moreover, has smooth cell decomposition) and, at every n ≥ 1,

 (n)  H (un ) >

Gn n(2n−1)

un

.

(2.1)

The idea is to use a very powerful result of Le Gal in [11] indicating that any smooth function is only an analytic perturbation away from a smooth function which is definable in an o-minimal structure. In order to state this result we must first make the following definition. Definition 4.2. Let U ⊇ [0, 1] be a bounded, open domain. For any R ∈ (0, ∞), let AR (U) denote the following collection of real analytic functions on U:

AR (U) := {f ∈ C ω (U) | ∃S ∈ [0, ∞) ∀n ∈ N ∀x ∈ U f (n) (x) ≤ n! SRn }.





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The result we shall use is then the following. Theorem 4.3 ([11], Proposition 2.2 and Theorem 1.2). Let U ⊇ [0, 1] be a bounded, open domain and let h ∈ C ∞ (U). For any R ∈ (0, ∞) there exists f ∈ AR (U) such that the structure Rf +h := ⟨R, (f + h)[0,1] ⟩ is o-minimal, polynomially bounded and admits smooth cell decomposition. Proof of Proposition 4.1. Suppose that h ∈ C ∞ (U), for some U ⊇ [0, 1], and that f ∈ A1 (U) is such that Rf +h is ominimal, as given to us by this theorem. Let us consider the derivatives of f + h. There exists some S ∈ [0, ∞) such that, for all x ∈ U and all n ∈ N,

      (f + h)(n) (x) ≥ h(n) (x) − f (n) (x)   ≥ h(n) (x) − Sn! + n · n!, for all n ∈ N. By Theorem 3.1, there is a smooth function h : (0, 1) → R and a sequence   (un )n∈N in (0, 1), with un → 0 as n → ∞, such that h(n) (un ) > Kn (un ), for all n ≥ 1. Whichever construction is used, h has a smooth extension g to an open domain V ⊇ [0, 1]. Thus Theorem 4.3 can be applied to find f ∈ A1 (V ) such that, if we define H := f + g, then the structure RH := ⟨R, H [0,1] ⟩ is o-minimal, polynomially bounded and has smooth cell Now let Kn (x) =

Gn x2n−1

decomposition.  By construction, using the arguments of Section 2, the graph of any function H given by Proposition 4.1 does not have a G-mild parameterization. Finally, let us consider the following definition and the ensuing theorem, taken from [10]. Definition 4.4. A global differential algebra is a family of functions F = n∈N Fn , where Fn is a subalgebra of C ∞ (Rn ), which contains all n-variable polynomials, is closed under differentiation and is such that F is closed under composition on the right. These conditions must also be verified at infinity: for all f ∈ F , if σ (x) = √ x , the function (x1 , . . . , xn ) →



1 +x 2

f (σ (x1 ), . . . , σ (xn )) extends to a function g which is smooth on [−1, 1]n and the function

(x1 , . . . , xn−1 ) → g (σ (x1 ), . . . , σ (xn−1 ), 1) lies in Fn−1 . Theorem 4.5 ([10], Theorem 1.2). If F is a global differential algebra consisting of functions definable in an o-minimal, polynomially bounded structure, then the structure ⟨R, F ⟩ is model complete. This theorem tells us that, since RH is o-minimal and polynomially bounded, and H can be extended to a smooth function on an open neighbourhood V ⊇ [0, 1], the structure  R := ⟨R, {H (n) }n∈N ⟩ is o-minimal, polynomially bounded and model complete. It defines H, whose graph does not have a G-mild parameterization, hence Theorem 1.6 is proved, taking X to be the graph of H(0,1) . 5. Cell decomposition In this section we present a general result, which may be of independent interest, for cell decomposition in locally polynomially bounded structures. These are a particular class of o-minimal structures whose 0-definable functions have an especially nice implicit representation. Definition 5.1 ([9], Section 3). Fix M = ⟨M , F ⟩, an o-minimal, model complete structure expanding a real closed field M, where F is a collection of total, smooth functions f : M n −→ M for various n. Define

Fres := {f B | f ∈ F , B a bounded, open box in dom(f )}. We say that M is locally polynomially bounded if ⟨M ; Fres ⟩ is polynomially bounded. Observe that, for any of the functions H : (0, 1) −→ R given by Proposition 4.1, the structure ⟨R, {(H ◦ τ )(n) }n∈N ⟩, where

τ : R → [0, 1] is given by τ (x) =

x2 , 1+x2

is locally polynomially bounded.

 denote the smallest collection Definition 5.2 ([9], Section 4). In a locally polynomially bounded structure ⟨M , F ⟩, let F of functions, containing both F and all polynomials over Q, that is closed under Q-algebra operations and partial . A 0-definable function f : U −→ M differentiation. For every n ≥ 1, let Rn denote the Q-algebra of n-ary functions in F on an open set U ⊆ M n is implicitly F -defined if there exist m ≥ 1, functions g1 , . . . , gm ∈ Rn+m and 0-definable maps φ1 , . . . , φm : U −→ M such that • f = φi , for some i ∈ {1, . . . , m}; • ⟨φ1 (¯x), . . . , φm (¯x)⟩ is a regular zero of the system given by g1 (¯x, ·), . . . gm (¯x, ·), for all x¯ ∈ U.

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417

The implicit representation of 0-definable functions is then given by the following theorem. Theorem 5.3 ([9], Corollary 4.5). Let M = ⟨M , F ⟩ be a locally polynomially bounded structure and let f : U −→ M be a k 0-definable function on an open set U ⊆ M n . There are 0-definable open sets U1 , . . . , Uk ⊆ U with dim(U \ i=1 Ui ) < n such that each map f Ui , for i ∈ {1, . . . , k}, is implicitly F -defined. It is a consequence of this theorem that all locally polynomially bounded structures have smooth cell decomposition ([9], Corollary 4.4). We shall prove a cell decomposition theorem for those locally polynomially bounded structures M := ⟨M , F ⟩ where certain further conditions are placed on the functions in F . Let P be a (not necessarily definable) local property of functions, i.e. one that is attributable (or not) to each of the germs of a function. Analyticity, for functions over R, is one example of such a property. One can define the notion of P cell decomposition in the expected manner, by requiring that every germ of every function featuring in the definition of a cell decomposition has the local property P. By induction on dimension and the repeated application of various standard properties of o-minimal structures (see, for example, [4]), it is straightforward to establish the following theorem for P cell decomposition. Theorem 5.4. Let M be an o-minimal structure with the property that, for every definable function f : U −→ M on any open set U ⊆ M n , for n ∈ N, the germs of f at all x¯ ∈ U have the local property P, except perhaps for those x¯ lying in a definable subset D ⊆ U with dim(D) < n. Then M has P cell decomposition. We shall use this to prove the following P cell decomposition theorem for locally polynomially bounded structures. Theorem 5.5. Let P be a local property of functions which is preserved under the implicit function theorem, polynomial operations and partial differentiation. Let M := ⟨M , F ⟩ be a locally polynomially bounded structure. Suppose that, for all f ∈ F , P holds ¯ Then ⟨M , F ⟩ has P cell decomposition. for the germ of f at every x¯ ∈ M n , for appropriate n ≥ 1, except perhaps at x¯ = 0. This theorem has the following immediate corollary, which will prove Theorem 1.7. Corollary 5.6. Let ⟨R, F ⟩ be a locally polynomially bounded structure such that F consists of smooth, one variable functions which are analytic, except perhaps at 0. Then ⟨R, F ⟩ has analytic cell decomposition. The proof of Theorem 5.5 follows the pattern indicated in [9] for proving Corollaries 4.4–4.6. We will give an exposition of the analogous results. We start by fixing a locally polynomially bounded structure ⟨M , F ⟩ such that F is a collection of smooth, one variable functions for which the germ of every f ∈ F at every x¯ ∈ M n , for appropriate n ≥ 1, has property P, except perhaps the ¯ germ at x¯ = 0. Lemma 5.7. Suppose that f : U −→ M is a 0-definable function on an open set U ⊆ M n containing a point a¯ which is generic in M n (i.e. for every open, 0-definable W ⊆ M n with dim(W ) < n, a¯ is not in W ). Then there is an open, 0-definable neighbourhood V of a¯ , with V ⊆ U, such that f V is either identically zero or is implicitly F -defined from functions defined on an open set not intersecting any coordinate hyperplane. Proof. Let f : U −→ M be such a function on a neighbourhood of a generic point a¯ ∈ M n . By genericity of a¯ , it is easy to see that if f (¯a) = 0, then there is an open, 0-definable neighbourhood V of a¯ within U on which f is identically zero. So let us suppose f (¯a) ̸= 0. Corollary 4.4 of [9] provides us with a neighbourhood V ′ ⊆ U of a¯ such that f V ′ is implicitly F -defined, i.e. there are 0-definable maps φ1 , . . . , φm : V ′ −→ M, with f V ′ = φi for some i ∈ {1, . . . , m}, and functions g1 , . . . , gm ∈ Rn+m such that ⟨φ1 (¯x), . . . , φm (¯x)⟩ is a regular zero of the system given by g1 (¯x, ·), . . . , gm (¯x, ·), for all x¯ ∈ V ′ . Again by genericity of a¯ we have that, for any j ∈ {1, . . . , m}, if φj (¯a) = 0, then there is some open, n-dimensional neighbourhood Vj of a¯ on which φj ≡ 0. m So first let us shrink V ′ to V ′′ := j=1 Vj (where Vj = V ′ if φj (¯a) ̸= 0). Then the functions φ1 V ′′ , . . . , φm V ′′ , including f V ′′ , are implicitly defined by the same functions g1 , . . . , gm . For a moment, suppose that φl is a function such that φl ≡ 0 on V ′′ . By considering functions g˜1 , . . . , g˜m ∈ Rn+m−1 defined by the following: g˜j (¯x, y¯ ) =



gj (¯x, y1 , . . . , yl−1 , 0, yl , . . . , ym−1 )

if 1 ≤ j ≤ l − 1

gj+1 (¯x, y1 , . . . , yl−1 , 0, yl , . . . , ym−1 )

if l ≤ j ≤ m − 1,

we can observe that, since ⟨φ1 (¯x), . . . , φl−1 (¯x), φl+1 (¯x), . . . , φm (¯x)⟩ is a regular zero of the system given by g˜1 (¯x, ·), . . . , g˜m−1 (¯x, ·), for all x¯ ∈ V ′′ , the collection of functions φ1V ′′ , . . . , φl−1V ′′ , φl+1V ′′ , . . . , φmV ′′ is implicitly defined from g˜1 , . . . , g˜m−1 about a¯ ∈ M n . Consequently, by making as many similar reductions as necessary, we may assume that, in fact, none of the functions φ1V ′′ , . . . , φmV ′′ is identically zero, or, in other words, φj (¯a) ̸= 0 for all j ∈ {1, . . . , m}. So we shall complete the proof assuming this fact, i.e. referring to all of g1 , . . . , gm , but what follows will apply to any such reduction as it is necessary to make. It only remains to show that we may avoid coordinate hyperplanes. But this is straightforward, since no coordinate of (¯a, φ1 (¯a), . . . , φm (¯a)) is 0, so there is an open neighbourhood W of (¯a, φ1 (¯a), . . . , φm (¯a)) in M n+m such that W does not intersect any set {¯x | xj = 0}, for j ∈ {1, . . . , m + n}.

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Then g1 W , . . . , gm W will define the functions φ1 , . . . , φm implicitly on some open neighbourhood W ′ ⊆ M n of a¯ with W ⊆ Πnn+m (W ). By shrinking W ′ if necessary so that it is contained in V ′′ (= V ′ , with the above assumption), we have a new neighbourhood V of a¯ on which all φ1 , . . . , φm , including f , can be implicitly defined from functions defined on an open set not intersecting any coordinate hyperplane.  ′

The lemma above is presented in the form required for subsequent lemmas. However, we may conclude from it the following immediate corollary. Corollary 5.8. Suppose that f : U −→ M is as in the above Lemma, with a¯ ∈ U generic in M n . Then there is an open, 0-definable neighbourhood V of a¯ , with V ⊆ U, such that f V has local property P throughout V . Now we can prove an analogy to Corollary 4.5 of [9]. Lemma 5.9. Suppose that f : U −→ M is a 0-definable function on an open set U ⊆ M n . Then there are 0-definable open sets k U1 , . . . , Uk ⊆ U, with dim(U \ i=1 Ui )< n, such that, for each i ∈ {1, . . . , k}, f Ui is implicitly F -defined from functions which are themselves defined on an open set Wi which avoids all coordinate hyperplanes. Proof. We may explicitly define a type p(¯x) whose satisfaction in a locally polynomially bounded structure would contradict Lemma 5.7. Since the property of being locally polynomially bounded is preserved under elementary equivalence ([9], Theorem 3.1), p(¯x) must therefore contain a finite inconsistent set of formulae. Since U is open, we must have a statement holding in M which expresses precisely the required conclusion. Corollary 5.10. Suppose that f : U −→ M is a 0-definable function on an open set U ⊆ M n . Then there are 0-definable open k sets U1 , . . . , Uk ⊆ U with dim(U \ i=1 Ui ) < n such that, for each i ∈ {1, . . . , k}, f Ui has the local property P. Since this corollary gives us that all 0-definable functions f : U −→ M, on any open set U ⊆ M n , have the local property P, except perhaps on a definable set of lower dimension, Theorem 5.5 then follows via the P cell decomposition Theorem 5.4. Hence Theorem 1.7 is also proved. Remarks 5.11. (1) The existence of definable functions whose graphs do not have a G-mild parameterization is established here by appealing to the analytic perturbation result of Le Gal (Theorem 4.3). However, this would not be required if the functions constructed according to the methods of Section 3 were already known to be definable in a polynomially bounded o-minimal structure. (2) Although Theorem 1.6 shows that no o-minimal mild parameterization result can be obtained for o-minimal fields in general, mild parameterization still appears to be a useful tool in working towards Wilkie’s Conjecture of [16]. For example, if X is a surface which is definable in Rexp and has a mild parameterization, then it satisfies the conclusion of the conjecture (see [8]). Moreover, it might also be possible to use the methods employed in obtaining this latter result to address further questions in transcendence theory. Therefore, it still remains of interest to try to establish mild parameterization for particular subsets of Rn , beyond those definable in a reduct of Ran . Acknowledgements Thanks are due to A. J. Wilkie, under whose supervision this work was completed as part of the author’s D.Phil. thesis at the University of Oxford. Thanks are also due to Gareth O. Jones, Olivier Le Gal and Jean-Philippe Rolin for useful and illuminating discussions. This work was funded in part by the EPSRC and The Fields Institute for Research in Mathematical Sciences. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]

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