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Quasianalytic Denjoy-Carleman classes and o-minimality. (English) Zbl 1095.26018
The authors obtain a new method for constructing o-minimal structures (i.e., linearly ordered structures $$M = \langle | M| , <, \ldots 1\rangle$$ such that the definable sets $$X \subseteq | M|$$ are the union of finitely many points and open intervals with end points in $$| M| \cup \{ \pm \infty\})$$ based on a normalization algorithm inspired by E. Bierstone and P. D. Milman [Invent. Math. 128, No. 2, 207–302 (1997; Zbl 0896.14006)] and then apply this construction to some Denjoy-Carleman classes and finally answer negatively to the following questions:
(i) Does every o-minimal expansion of the real field admits analytic cell decomposition, i.e., each definable set $$X$$ can be written as a disjoint union of finitely many definable sets, $$X=\bigcup_{j = 1}^K X_j$$, each “cell” $$X_j$$ being an open set, or the embedding of the graph of a definable function $$\varphi_j : U_i \subseteq | M| ^{n_j}$$ with the $$U_j$$ are definable open sets or isolated points?
(ii) Does there exists a “larger” o-minimal expansion $$M$$ of the field?
Let $$M = (M_j)$$, with $$1 \leq M_0 \leq M_1 \leq \ldots$$ be a log convex sequence of real numbers, $$B = [a_1, b_1] \times \dots\times [a_n, b_n]$$, with $$a_i < b_i$$, $$(i = 1 \ldots n)$$ a cube and $$C_B^0(M)$$ be the collection of all $$f : B \to \mathbb{R}$$, such that there is an open neighborhood $$U$$ of $$B$$ and a $$g : U \to \mathbb{R}$$ such that $$g\mid_B = f$$ and $$| g^{(\alpha)} (x) | \leq A^{| \alpha| + 1} M_{| \alpha| }$$ for all $$x \in U$$ and $$\alpha \in \mathbb{N}^n$$, $$\mathcal{C}_B (M) = \bigcup_0^\infty C_B^0 (M^{(j)})$$ (where $$M^{(j)} = (M_j, M_{j + 1}, \ldots)$$). $$\mathcal{C}_B (M)$$ is closed under differentiation. To ensure that it is closed under composition, taking implicit junctions and division by monomial terms one asks $$M$$ to be strongly-convex (i.e., $$(M_i/i!)$$ to be log-convex). If $$\sum M_i/M_{i + 1} = \infty (QA)$$ then the class $$C_B(M)$$ is quasianalytic (Denjoy-Carleman).
For each $$n \in \mathbb{N}$$ and $$f \in \mathcal{C}_{[-1, 1]^n} (M)$$ define $$\widetilde{ f} : \mathbb{R}^n\to \mathbb{R}^n$$ by $$\widetilde{ f} (x) = f(x)$$ if $$x \in [-1,1]^n$$, and $$\widetilde{f}(x) = 0$$ otherwise. Let $$\mathbb{R}_{\mathcal{C}(M)}$$ be the expansion of the real field by all $$\widetilde{ f}$$, for $$f \in \mathcal{C}_{[-1, 1]^n}$$, $$n \in \mathbb{N}$$.
The authors prove the following:
(1) If $$M$$ is strongly log convex and satisfies $$(QA)$$ then $$\mathbb{R}_{\mathcal{C}(M)}$$ is o-minimal, polynomially bounded (i.e., for every definable function $$f : \mathbb{R} \to \mathbb{R}$$ there exists $$p \in \mathbb{N}$$ such that $$| f(t)| < t^p$$ for all large $$t$$) model complete (i.e., the complement of any $$\mathcal{C}(M)$$ subanalytic set is a $$\mathcal{C}^\infty$$ cell decomposition).
(2) Given any $$C^\infty$$ function $$f : U \to \mathbb{R}$$ with $$U$$ open neighborhood of $$[-1, 1]^n$$, $$n \in \mathbb{N}$$, then there exist strongly log convex sequences $$M$$ and $$N$$, each satisfying $$(QA)$$ and $$f_1 \in \mathcal{C}^0_{[-1, 1]^n} (M)$$, $$f_2 \in \mathcal{C}^0_{[-1, 1]^n} (N)$$ such that $$f(x) = f_1(x) + f_2 (x)$$ for all $$n \in [-1, 1]^n$$.
(3) There exists a strongly log-convex $$M$$, satisfying $$(QA)$$ and a function $$f \in \mathcal{C}^0_{[-1, 1]} (M)$$ such that $$f$$ is nowhere analytic.
The authors follow the construction of o-minimal structures [L. van der Dries and P. Speissegger, Trans. Am. Math. Soc. 350, No. 11, 4377–4421 (1998; Zbl 0905.03022) and Proc. Lond. Math. Soc., III. Ser. 81, No. 3, 513–565 (2000; Zbl 1062.03029)] but as in the present situation there is no Weierstrass preparation theorem available, the substitute is a normalization algorithm, inspired by Bierstone-Milman (loc. cit.). (In fact the authors could have applied directly the desingularization of Bierstone-Milman.)
The negative answer to the question stated above is obtained by using (3) and (1) to construct o-minimal expansions $$\mathcal{R}_1$$ and $$\mathcal{R}_2$$ of the real field whose almagamation defines the set $$\mathbb{Z}$$ of all integers.

##### MSC:
 26E10 $$C^\infty$$-functions, quasi-analytic functions 14P15 Real-analytic and semi-analytic sets 03C64 Model theory of ordered structures; o-minimality 32S45 Modifications; resolution of singularities (complex-analytic aspects)
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