A counter-example concerning quantifier elimination in quasianalytic structures

Krzysztof Jan Nowak

IMUJ PREPRINT 2012/04

Abstract

This paper provides an example of a quasianalytic structure which, unlike the classical analytic structure, does not admit quantifier elimination in the language of restricted quasianalytic functions augmented by the reciprocal function 1/x. Our construction applies rectilinearization of terms as well as some theorems on power substitution for Denjoy–Carleman classes and on non-extendability of quasianalytic germs.

1. Introduction. In our earlier papers [15, 17], we established, for a quasianalytic structure, quantifier elimination as well as rectilinearization and description of definable functions by terms, in the language augmented by rational powers. Full generality was achieved in the latter paper, which constitutes a basis for a new, self-contained article being currently prepared.

The main purpose of this paper is to provide a negative answer to the problem, posed in [17], whether a quasianalytic structure admits quantifier elimination in the language augmented merely by the reciprocal function 1/x. In the case of the classical structure ℝ_{an}, the affirmative answer was given by J. Denef and L. van den Dries [6]. The construction of a counter-example, given in Section 4, is based on the rectilinearization of terms as well as on two function-theoretic results concerning Denjoy–Carleman classes. The first, presented in Section 2, refers to power substitution for those classes. The other is a non-extendability theorem, which is a refinement of a theorem by V. Thilliez [22].

2010 Mathematics Subject Classification. Primary: 03C10, 26E10; Secondary: 14P15, 03C64.

Key words: quasianalytic structures, quantifier elimination, Denjoy–Carleman classes.

Let us recall (cf. [14, 15, 18]) that a quasianalytic structure is the expansion of the real field by restricted quasianalytic functions determined by a quasianalytic system Q = (Q_{m})_{m∈ℕ} of sheaves of local ℝ-algebras of smooth functions on ℝ^{m}, submitted to the following six conditions:

1. each algebra Q(U) contains the restrictions of polynomials;

2. Q is closed under composition, i.e. the composition of Q-mappings is a Q-mapping (whenever it is well defined);

3. Q is closed under inverse, i.e. if φ : U ⟶ V is a Q-mapping between open subsets U, V ⊂ ℝ^{n}, a ∈ U, b ∈ V and if ∂φ/∂x(a) ≠ 0, then there are neighbourhoods U_{a} and V_{b} of a and b, respectively, and a Qdiffeomorphism ψ : V_{b} ⟶ U_{a} such that φ ∘ ψ is the identity mapping on V_{b};

4. Q is closed under differentiation;

5. Q is closed under division by a coordinate, i.e. if f ∈ Q(U) and f(x_{1}, …, x_{i−1}, a_{i}, x_{i+1}, …, x_{m}) = 0 as a function in the variables x_{j}, j ≠ i, then f(x) = (x_{i} − a_{i})g(x) with some g ∈ Q(U);

6. Q is quasianalytic, i.e. if f ∈ Q(U) and the Taylor series f̂_{a} = of f at a point a ∈ U vanishes, then f vanishes in the vicinity of a.

Q-mappings give rise, in the ordinary manner, to the category Q of Qmanifolds, which is a subcategory of that of smooth manifolds and smooth mappings. Similarly, Q-analytic, Q-semianalytic and Q-subanalytic sets can be defined. The above conditions ensure some (limited) resolution of singularities in the category Q, including transformation to normal crossings by blowing up (cf. [2, 19]), upon which the geometry of quasianalytic structures relies; especially, in the absence of their good algebraic properties (cf. [19, 14, 15]).

The examples of such categories are provided by quasianalytic Denjoy– Carleman classes Q_{M′}, where M′ = (M′_{n})_{n∈ℕ} are log-convex sequences. The class Q_{M′} consists of smooth functions f(x) = f(x_{1}, …, x_{m}) in m variables, m ∈ ℕ, which are locally submitted to the following growth condition for their derivatives:

|f^{(n)}(x)| ≤ AB^{n}n!M′_{n}

with some constants A, B > 0 depending only on the vicinity of a given point. Often this growth condition is formulated in a slightly different way:

|f^{(n)}(x)| ≤ AB^{n}M_{n},

where M_{n} = n!M′_{n}. Obviously, the class Q_{M} contains the real analytic functions. It is quasianalytic iff

∑_{n=0}^{∞} \frac{M′_{n}}{(n + 1)M′_{n+1}} = ∞

(the Denjoy–Carleman theorem). It is closed under composition (Roumieu [20]) and under inverse (Komatsu [8]); furthermore, it is closed under differentiation and under division by a coordinate iff

sup_{n} \sqrt[n]{(\frac{M′_{n+1}}{M′_{n}} )} < ∞

(cf. [12, 21]). On the other hand, every polynomially bounded, o-minimal structure R determines a quasianalytic system of sheaves of germs of smooth functions that are locally definable in R.

It is well-known (cf. [4, 5, 21]) that, given two log-convex sequences M′ and N′, the inclusion E′(M′) ⊂ E′(N′) holds iff there is a constant C > 0 such that M′_{n} ≤ C^{n}N′_{n} for all n ∈ ℕ or, equivalently,

sup \{\sqrt[n]{\frac{M′_{n}}{N′_{n}}} : n ∈ ℕ\} < ∞.

2. Power substitution for Denjoy–Carleman classes. Let M′ = (M′_{n}) be an increasing sequence of real numbers with M′_{0} = 1. Let I be an interval (open or closed) contained in ℝ. By E′(I, M′) (resp. E(I, M)) we denote the class of functions on I that satisfy the above conditions. Denote by E′(M′) (resp. E(M)) the set of germs at zero of smooth functions from the Denjoy-Carleman class corresponding to these sequences. In order to ensure some important algebraic and analytic properties of E(M), it suffices to assume that the sequence M or M′ is log-convex. The former implies that the set E(M) is a ring; it is closed under multiplication by the Leibniz formula. The latter assumption is stronger, and implies, moreover, that E′(M′) is closed under composition (as proven by Roumieu [20] and Cartan [4] in particular cases; see also [2]).

The main purpose of this section is Theorem 1, which is valid for an arbitrary sequence M.

Theorem 1. Let p > 1 be an integer and I the interval [0, 1] or [−1, 1] according as p is even or odd. Consider power substitution x = ξ^{p}, which is a bijection of I onto itself. Let f : I ⟶ ℝ be a smooth function. If

F (ξ) := (f ∘ φ)(ξ) = f(ξ^{p}) ∈ E(I, M),

then f(x) ∈ E(I, M^{(p)}), where the sequence M^{(p)} is defined by putting M_{n}^{(p)} := 1/n^{(p−1)n} · M_{pn}.

In terms of the corresponding sequences M′, one can rephrase the theorem as follows. If

F (ξ) := (f ∘ φ)(ξ) = f(ξ^{p}) ∈ E′(I, M′),

then f(x) ∈ E′(I, M^{′(p)}), where the sequence M^{′(p)} is defined by putting M_{n}^{′(p)} := M′_{pn}.

Remark 1. Using a function constructed by Bang, we shall show at the end of this section that, in the case where p = 2 and the sequence M is logconvex, E(I, M^{(2)}) is the smallest class containing all those functions f(x). As communicated to us in written form by E. Bierstone and F.V. Pacheco [3], application of a suitable variation of Bang’s function yields the result for an arbitrary positive integer p.

Remark 2. The case p = 2 of Theorem 1 may be related to the following problem posed and investigated by Mandelbrojt [11] and [12] (Chap. VI):

Consider a smooth function f(x) on the interval [−1, 1] and suppose that F (ξ) := f(cos ξ) belongs to a class E(ℝ, M). To which class on the interval [−1, 1] does f belong?

A complete solution for sequences M such that lim_{n→∞} \sqrt[n]{M_{n}} = ∞ was given by Lalaguë [9], Chap. III. He proved that those functions f must belong to the class E(I, N^{(2)}), N_{n}^{(2)} = 1/n^{n} · N_{2n}, where N := M^{c} is the log-convex regularization of the initial sequence M, and gave a formula for the smallest class containing all those functions f. Moreover, the smallest class coincides with E(I, N^{(2)}), if

lim inf_{n→∞} \frac{1}{n} \sqrt[n]{M_{n}} > 0.

The last condition is equivalent to the existence of a constant C > 0 such that M_{n} ≥ n!C^{n} for all n ∈ ℕ, and thus implies that the Denjoy-Carleman class corresponding to the sequence M contains analytic functions.

Whenever the sequence M′ (corresponding to M) is increasing, it is not difficult to draw Lalaguë’s result from Theorem 1 for the case p = 2. Then those functions f must belong to the class E′(I, M^{′(2)}). The proof relies on the observation that the class E′(I, N′) is closed under analytic substitutions if a sequence N′ is increasing. This, in turn, follows immediately from Cauchy’s inequalities and the formula for the derivatives of a composite function, recalled below:

(f ∘ g)^{(n)}(x) = ∑_{k=1}^{n} f^{(k)}(g(x)) · \frac{1}{k!} · \frac{d^{n}}{dX^{n}} (g(X) − g(x))^{k} |_{X=x}.

Before establishing Theorem 1, we need two lemmas stated below.

Lemma 1. Consider the Taylor expansions

∑_{i=1}^{∞} \frac{x^{i}}{i} = − log (1 − x) and (∑_{i=1}^{∞} \frac{x^{i}}{i} )^{k} = ∑_{n=1}^{∞} c_{n}x^{n}.

Then we have the estimate

c_{n} ≤ (2e)^{n} · \frac{k!}{n^{k}} for all k, n ∈ ℕ.

Indeed, it is easy to verify the estimate:

| log(1 − z)| ≤ |log 2 + \frac{π}{6} \sqrt{−1}| ≤ 1 for all z ∈ ℂ, |z| ≤ 1/2.

Hence and by Cauchy’s inequalities, we get |c_{n}| ≤ 2^{n}. Since e^{n} > n^{k}/k! for all n, k ∈ ℕ, we have

c_{n} ≤ 2^{n} < 2^{n} · e^{n} · \frac{k!}{n^{k}} = (2e)^{n} · \frac{k!}{n^{k}} ,

as asserted.

As an immediate consequence, we obtain the

Corollary.

∑_{i_{1}+...+i_{k}=n} \frac{1}{i_{1}} · . . . · \frac{1}{i_{k}} = c_{n} ≤ (2e)^{n} · \frac{k!}{n^{k}} for all k, n ∈ ℕ.

Lemma 2. Let p, k ∈ ℕ with p > 1, k ≥ 1, and

α_{k}(X, x) := \frac{1}{k!} (X^{1/p} − x^{1/p}) for X, x > 0,

where α_{k} is regarded as a function in one variable X and parameter x. Then we have the estimate

|α_{k}^{(n)}(x, x)| ≤ (2e)^{n} · n^{n−k} · x^{− \frac{pn−k}{p}} for all n, k ∈ ℕ, x > 0.

Consider first the case k = 1, α_{1}(X, x) = X^{1/p} − x^{1/p}. Then

α_{1}^{(n)}(x, x) = ±\frac{(p − 1)(2p − 1) · . . . · ((n − 1)p − 1)}{p^{n}} · x^{− \frac{pn−1}{p}} ,

whence

|α_{1}^{(n)}(x, x)| ≤ (n − 1)! ≤ n^{n−1} · x^{− \frac{pn−1}{p}} ,

as asserted. The Taylor expansion of α_{1}(X, x) at X = x is

α_{1}(X, x) = ∑_{i=1}^{∞} a_{i} · x^{− \frac{pi−1}{p}} (X − x)^{i},

where

a_{i} := ±\frac{1}{i!} · \frac{(p − 1)(2p − 1) · . . . · ((i − 1)p − 1)}{p^{i}};

obviously, |a_{i}| ≤ (i − 1)!/i! = 1/i. We get

α_{k}(X, x) = (∑_{i=1}^{∞} a_{i}x^{− \frac{pi−1}{p}} (X − x)^{i})^{k} = ∑_{j=1}^{∞} b_{j} · x^{− \frac{pj−k}{p}} (X − x)^{j},

where

b_{j} := \frac{1}{k!} ∑_{i_{1}+...+i_{k}=j} a_{i_{1}} · . . . · a_{i_{k}}.

Then

|b_{n}| ≤ \frac{1}{k!} ∑_{i_{1}+...+i_{k}=n} |a_{i_{1}}| · . . . · |a_{i_{k}}| ≤ \frac{1}{k!} ∑_{i_{1}+...+i_{k}=n} \frac{1}{i_{1}} · . . . · \frac{1}{i_{k}} =

= \frac{c_{n}}{k!} ≤ \frac{(2e)^{n}}{n^{k}} for all n, k ∈ ℕ;

the last inequality follows from the corollary to Lemma 1. Hence

|α_{k}^{(n)}(x, x)| ≤ n! · |b_{n}| · x^{− \frac{pn−k}{p}} ≤ (2e)^{n} · \frac{n!}{n^{k}} · x^{− \frac{pn−k}{p}} ≤ (2e)^{n} · n^{n−k} · x^{− \frac{pn−k}{p}}

for all n, k ∈ ℕ, x > 0, as asserted.

Now we can readily pass to the proof of Theorem 1. So suppose that

|F^{(n)}(ξ)| ≤ A^{n}M_{n}

for all n ∈ ℕ, ξ ∈ I and some constant A > 0. We are going to estimate the growth of the n-th derivative f^{(n)}. Fix n ∈ ℕ and put:

p_{n}(x) := T_{0}^{n}(x) = ∑_{k=0}^{n−1} f^{k}(0)\frac{x^{k}}{k!} , r_{n}(x) := f(x) − p_{n}(x),

P_{n}(ξ) := p_{n}(ξ^{p}) and R_{n}(ξ) := r_{n}(ξ^{p}).

Obviously,

P_{n}^{(pn)} ≡ 0 and R_{n}^{(pn)} ≡ F^{(pn)}.

From the Taylor formula, we therefore obtain the estimate

|R_{n}^{(pn)}(ξ)| ≤ A^{pn}M_{pn} · \frac{ξ^{pn−k}}{(pn − k)!}

for all k < pn, ξ ∈ I. We still need an elementary inequality

\frac{1}{(pn − k)!} ≤ \frac{e^{pn}}{n^{pn−k}} for all k < pn,  (∗)

which can be proven via the following well-known Stirling formula

\sqrt{2πn} · \frac{n^{n}}{e^{n}} < n! < e\sqrt{n} · \frac{n^{n}}{e^{n}} .

Indeed, it suffices to show that

\frac{e^{pn−k}}{(pn − k)^{pn−k}} ≤ \frac{e^{pn}}{n^{pn−k}} .

When pn−k ≥ n or, equivalently, k ≤ (p−1)n, the last inequality is evident. Suppose thus that pn − k < n or, equivalently, k > (p − 1)n. This inequality is, of course, equivalent to

(\frac{n}{pn − k} )^{pn−k} ≤ e^{k},

which holds as shown below:

(\frac{n}{pn − k} )^{pn−k} = (1 + \frac{k − (p − 1)n}{pn − k})^{pn−k} =

= [(1 + \frac{k − (p − 1)n}{pn − k})^{\frac{pn−k}{k−(p−1)n}} ]^{k−(p−1)n} < e^{k}.

Now, the foregoing estimate along with inequality (*) yield

|R_{n}^{(pn)}(ξ)| ≤ A^{pn}M_{pn} · \frac{e^{pn}}{n^{pn−k}} · ξ^{pn−k}.

Applying the formula for the derivatives of a composite function, we obtain

r_{n}^{(n)}(x) = ∑_{k=1}^{n} R_{n}^{(k)}(ξ) · α_{k}^{(n)}(x, x).

Hence and by Lemma 2, we get

|f^{(n)}(x)| = |r_{n}^{(n)}(x)| ≤ ∑_{k=1}^{n} A^{pn}M_{pn} · \frac{e^{pn}}{n^{pn−k}} · |ξ|^{pn−k} · (2e)^{n}n^{n−k} · |ξ|^{−(pn−k)} =

= n · (2e)^{n} · (eA)^{pn} · \frac{M_{pn}}{n^{(p−1)n}} ,

which completes the proof of Theorem 1.

Finally, we show that, whenever the sequence M is log-convex, E(I, M^{(2)}) is the smallest class containing all smooth functions f(x) on the interval I = [0, 1] such that F (ξ) = f(ξ^{2}) ∈ E(I, M). We make use of a classical function constructed by Bang [1], applied in his proof that the classes determined by log-convex sequences contain functions with sufficiently large derivatives (the result due to Cartan [4] and Mandelbrojt [5]; see also [21], Section 1, Theorem 1).

The logarithmic convexity of the sequence M yields for every j, k ∈ ℕ the inequality

(\frac{1}{m_{k}} )^{k−j} ≤ \frac{M_{j}}{M_{k}} where m_{k} := \frac{M_{k+1}}{M_{k}} .

Consequently,

F (ξ) := ∑_{k=0}^{∞} \frac{M_{k}}{(2m_{k})^{k}} cos(2m_{k}ξ)

is an even smooth function on ℝ such that

F (ξ) ∈ E(ℝ, M) and |F^{(2n)}(0)| ≥ M_{2n}

for all n ∈ ℕ. Therefore F (ξ) = f(ξ^{2}) for some smooth function f on ℝ, and we get

f^{(n)}(0) = \frac{n!}{(2n)!}F^{(2n)}(0) and |f^{(n)}(0)| ≥ \frac{n!M_{2n}}{(2n)!} ,

which is the desired result.

It was communicated to us by E. Bierstone and F.V. Pacheco [3] that, in order to obtain this result for an arbitrary positive integer p, one can replace cos x by the function

C_{p}(x) := ∑_{j=0}^{∞} \frac{x^{jp}}{(jp)!}

with the properties C_{p}^{(pn)}(x) = C_{p}(x), C_{p}^{(pn)}(0) = 1 and |C_{p}^{(n)}(x)| ≤ e for all n ∈ ℕ, x ∈ [−1, 1]. Then

F (ξ) := ∑_{k=0}^{∞} \frac{M_{k}}{(2m_{k})^{k}} C_{p}(2m_{k}ξ)

is a smooth function on ℝ such that

F (ξ) ∈ E([−1, 1], M) and |F^{(pn)}(0)| ≥ M_{pn}

for all n ∈ ℕ. As before, there exists a smooth function f on ℝ such that

F (ξ) = f(ξ^{p}), f^{(n)}(0) = \frac{n!}{(pn)!}F^{(pn)}(0) and |f^{(n)}(0)| ≥ \frac{n!M_{pn}}{(pn)!} .

We conclude this section with some examples, one of which (namely, for k = 2) will be applied to the construction of our counter-example in the last section.

Example. Fix an integer k ∈ ℕ, k ≥ 1, and put

log^{(k)} := \underbrace{log ∘ … ∘ log}_{k times}, and e ↑↑ k := (\underbrace{exp ∘ … ∘ exp}_{k times})(1);

let n_{k} be the smallest integer greater than e ↑↑ k. Then the sequence

(log^{(k)} n)^{n} for n ≥ n_{k}

is log-convex. Further, the shifted sequence:

M′ = (M′_{n}), M′_{n} := \frac{1}{(log^{(k)} n_{k})^{n_{k}}} · (log^{(k)}(n_{k} + n))^{(n_{k}+n)} ,

determines a quasianalytic class closed under derivatives; the former follows from Cauchy’s condensation criterion. It is easy to check that the sequences M^{′(p)}, p > 1, are quasianalytic when k > 1, but are not quasianalytic when k = 1.

3. Non-extendability of quasianalytic germs. In this section we are concerned with a result by V. Thilliez [22] on the extension of quasianalytic function germs in one variable, recalled below. Consider two log-convex sequences M′ and N′ such that E′(M′) ⊂ E′(N′). Denote by E′(M′)^{+} the local ring of right-hand side germs at zero (i.e. germs of functions from E′([0, ε], M′) for some ε > 0).

Theorem 2. If E′(N′) is a quasianalytic local ring, then

O ⊊ E′(M′) ⊂ E′(N′) ⟹ E′(M′)^{+} \ E′(N′) ≠ ∅,

i.e. there exist right-hand side germs from E′(M′)^{+} which do not extend to germs from E′(N′).

Remark 3. Theorem 2 may be related to the research by M. Langenbruch [10] on the extension of ultradifferentiable functions in several variables, which is principally focused on the non-quasianalytic case, which seems to be more difficult in this context. His extension problem is, roughly speaking, as follows:

Given two compact convex subsets K, K_{1} of ℝ^{m} such that int (K) ≠ ∅ or K = {0} and K ⊂ int (K_{1}), characterize the sequences of positive numbers M and N such that every function from the class E(K, M) extends to a function from E(K_{1}, N).

M. Langenbruch applies, however, different methods and techniques in comparison with V. Thilliez. In particular, his approach is based on the theory of Fourier transform and plurisubharmonic functions.

On the other hand, Thilliez’s approach relies on Grothendieck’s version of the open mapping theorem (cf. [7], Chap. 4, Part I, Theorem 2 or [13], Part IV, Chap. 24) and Runge approximation. It also enables the formulation of the non-extendability theorem for quasianalytic function germs on a compact convex subset K ⊂ ℝ^{m} with 0 ∈ K.

Nevertheless, in order to construct our counter-example in the next section, we need a refinement of the above non-extendability theorem, stated below. Thilliez’s proof can be adapted mutatis mutandis. We shall outline it for the reader’s convenience. Consider an increasing countable family M^{′(p)}, p ∈ ℕ, of log-convex sequences, i.e.

1 = M_{0}^{′(p)} ≤ M_{1}^{′(p)} ≤ M_{2}^{′(p)} ≤ M_{3}^{′(p)} ≤ … for all p ∈ ℕ.

Then we receive an ascending sequence of local rings

E′(M^{′(1)}) ⊂ E′(M^{′(2)}) ⊂ E′(M^{′(3)}) ⊂ …

such that E′(M^{′(p)}) is dominated by E′(M^{′(q)}) for all p, q ∈ ℕ with p ≤ q.

Theorem 2^{∗}. If every local ring E′(M^{′(p)}) is quasianalytic, then

O ⊊ E′(M′) ⊂ ⋃_{p∈ℕ} E′(M^{′(p)}) ⟹ E′(M′)^{+} \ ⋃_{p∈ℕ} E′(M^{′(p)}) ≠ ∅.

Before proving Theorem 2^{∗}, we adopt the following notation. For a smooth function f on an interval I ⊂ ℝ and r > 0, put

∥f∥_{M′,I,r} := \{sup \frac{|f^{(n)}(x)|}{r^{n}n!M_{n}} : n ∈ ℕ, x ∈ I\} .

For k ∈ ℕ, k > 0, let B_{k}(M′) or B_{k}(M′)^{+}, respectively, denote the Banach space with norm

∥ · ∥_{M′,[−1/k,1/k],k} or ∥ · ∥_{M′,[0,1/k],k},

of those smooth functions on the interval [−1/k, 1/k] or [0, 1/k]) such that

∥f∥_{M′,[−1/k,1/k],k} < ∞ or ∥f∥_{M′,[0,1/k],k} < ∞, respectively.

Because the canonical embeddings

B_{k}(M′) ↪ B_{l}(M′) and B_{k}(M′)^{+} ↪ B_{l}(M′)^{+}, k, l ∈ ℕ, k ≤ l,

are compact linear operators, one can endow the local rings E′(M′) and E′(M′)^{+} with the inductive topologies. Similarly, the countable union of local rings ⋃_{p∈ℕ} E′(M^{′(p)}) is the inductive limit of the sequence B_{k}(M^{′(k)}), k ∈ ℕ, k > 0, of Banach algebras. Further, the local ring

⋃_{p∈ℕ} E′(M^{′(p)}) ∩ E′(M′)^{+}

is the inductive limit of the sequence

B_{k}(M^{′(k)}) ∩ B_{k}(M′)^{+}, k ∈ ℕ, k > 0,

of Banach algebras with norms

∥f∥_{k} := max {∥f∥_{M^{′(k)},[−1/k,1/k],k} , ∥f∥_{M′,[0,1/k],k}}.

Clearly, the restriction operator

R : ⋃_{p∈ℕ} E′(M^{′(p)}) ∩ E′(M′)^{+} ⟶ E′(M′)^{+}

is continuous and injective by quasianalyticity.

To proceed with reduction ad absurdum, suppose that R is surjective. By Grothendieck’s version of the open mapping theorem (cf. [7], Chap. 4, Part I or [13], Part IV, Chap. 24), R is a homeomorphism. Next, by Grothendieck’s factorization theorem (loc.cit.), for each k ∈ ℕ there is an l ∈ ℕ and a constant C > 0 such that

R (B_{l}(M^{′(l)})) ⊃ R (B_{l}(M^{′(l)}) ∩ B_{l}(M′)^{+}) ⊃ B_{k}(M′)^{+},

and

∥f∥_{M^{′(l)},[−1/l,1/l],l} ≤ ∥R^{−1}f∥_{l} ≤ C∥f∥_{M′,[0,1/k],k}

for all f ∈ B_{k}(M′)^{+}. In particular, there is an l ∈ ℕ and a constant A > 0 such that

R (B_{l}(M^{′(l)})) ⊃ B_{1}(M′)^{+},

and

∥f∥_{M^{′(l)},[−1/l,1/l],l} ≤ A∥f∥_{M′,[0,1],1}

for all f ∈ B_{1}(M′)^{+}. In particular,

|P (−\frac{1}{l})| ≤ A∥P ∥_{M′,[0,1],1}

for every polynomial P ∈ ℂ[x]. Put

W := \{z ∈ ℂ : dist (z, [0, 1]) ≤ \frac{1}{2l}\} and B := sup \{\frac{(2l)^{n}}{M′_{n}} : n ∈ ℕ\} < ∞;

the last inequality holds because O ⊊ E′(M′) whence

sup \{\sqrt[n]{M′_{n}} : n ∈ ℕ\} = ∞.

It follows from Cauchy’s inequalities that

sup {|P^{(n)}(x)| : x ∈ [0, 1]} ≤ n!(2l)^{n} sup {|P (x)| : x ∈ W },

and hence

∥P ∥_{M′,[0,1],1} ≤ B sup {|P (x)| : x ∈ W }.

Consequently,

|P (−\frac{1}{l})| ≤ AB sup \{|P (x)| : x ∈ W \}

for every polynomial P ∈ ℂ[x]. But, by virtue of Runge approximation, there exists a sequence of polynomials P_{ν} ∈ ℂ[x] which converges uniformly to 0 on W, and to 1 for x = −1/l. This contradicts the above estimate, and thus the theorem follows.

4. Construction of a counter-example. Now, we can readily give a counter-example indicating that quasianalytic structures, unlike the classical structure ℝ_{an}, may not admit quantifier elimination in the language augmented merely by the reciprocal function 1/x. The example we construct is a plane curve through 0 ∈ ℝ^{2} which is definable in the quasianalytic structure corresponding to the log-convex sequence

M′ = (M′_{n}), M′_{n} := \frac{1}{(log log 3)^{3}} · (log log (n + 3))^{(n+3)} ;

this sequence determines a quasianalytic class closed under derivatives (cf. the example from the end of Section 2). By Theorem 2^{∗}, we can take a function germ

f ∈ E′(M′)^{+} \ ⋃_{p odd} E′(M^{′(p)}).

Let V ⊂ ℝ^{2} be the graph of a representative of this germ in a right-hand side neighbourhood [0, ε].

To proceed with reductio ad absurdum, suppose V is given by a term in the language augmented by the reciprocal function 1/x. Then there would exist a rectilinearization of this term via a finite sequence of blowings-up at points (see [16] and [17], Section 4, Remark 5). Consequently, the germ of V at zero would be contained in the image φ([−δ, δ]), where

φ = (φ_{1}, φ_{2}) : [−δ, δ] ⟶ ℝ^{2}, φ(0) = 0,

is a Q-analytic homeomorphism. But then the order of φ_{1} at zero must be odd, and thus the set V would have a parametrization near zero of the form (ξ^{p}, g(ξ)), where p is odd and g is a Q-analytic function at zero. Hence and by Theorem 1,

f(x) = g(x^{1/p}) ∈ E′(I, M^{′(p)}),

which is a contradiction.

Remark. In view of Puiseux’s theorem for definable functions (cf. [18], Section 2), every smooth definable function germ in one variable belongs to E′(M^{′(p)}) for some positive integer p. Therefore the structure under study will not admit quantifier elimination, even considered with the richer language of restricted definable quasianalytic functions augmented by the reciprocal function 1/x.

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Institute of Mathematics Faculty of Mathematics and Computer Science Jagiellonian University ul. Profesora Łojasiewicza 6 30-348 Kraków, Poland e-mail address: nowak@im.uj.edu.pl