×

zbMATH — the first resource for mathematics

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)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Shreeram S. Abhyankar and Tzuong Tsieng Moh, Newton-Puiseux expansion and generalized Tschirnhausen transformation. I, II, J. Reine Angew. Math. 260 (1973), 47 – 83; ibid. 261 (1973), 29 – 54. · Zbl 0272.12102 · doi:10.1515/crll.1973.260.47 · doi.org
[2] Shreeram S. Abhyankar and Tzuong Tsieng Moh, Newton-Puiseux expansion and generalized Tschirnhausen transformation. I, II, J. Reine Angew. Math. 260 (1973), 47 – 83; ibid. 261 (1973), 29 – 54. · Zbl 0272.12102 · doi:10.1515/crll.1973.260.47 · doi.org
[3] Edward Bierstone and Pierre D. Milman, Semianalytic and subanalytic sets, Inst. Hautes Études Sci. Publ. Math. 67 (1988), 5 – 42. · Zbl 0674.32002
[4] Edward Bierstone and Pierre D. Milman, Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant, Invent. Math. 128 (1997), no. 2, 207 – 302. · Zbl 0896.14006 · doi:10.1007/s002220050141 · doi.org
[5] C. L. Childress, Weierstrass division in quasianalytic local rings, Canad. J. Math. 28 (1976), no. 5, 938 – 953. · Zbl 0355.32009 · doi:10.4153/CJM-1976-091-7 · doi.org
[6] Lou van den Dries, o-minimal structures and real analytic geometry, Current developments in mathematics, 1998 (Cambridge, MA), Int. Press, Somerville, MA, 1999, pp. 105 – 152. · Zbl 0980.03043
[7] Lou van den Dries and Chris Miller, Geometric categories and o-minimal structures, Duke Math. J. 84 (1996), no. 2, 497 – 540. · Zbl 0889.03025 · doi:10.1215/S0012-7094-96-08416-1 · doi.org
[8] Lou van den Dries and Patrick Speissegger, The real field with convergent generalized power series, Trans. Amer. Math. Soc. 350 (1998), no. 11, 4377 – 4421. · Zbl 0905.03022
[9] Lou van den Dries and Patrick Speissegger, The field of reals with multisummable series and the exponential function, Proc. London Math. Soc. (3) 81 (2000), no. 3, 513 – 565. · Zbl 1062.03029 · doi:10.1112/S0024611500012648 · doi.org
[10] Andrei Gabrielov, Complements of subanalytic sets and existential formulas for analytic functions, Invent. Math. 125 (1996), no. 1, 1 – 12. · Zbl 0851.32009 · doi:10.1007/s002220050066 · doi.org
[11] Yitzhak Katznelson, An introduction to harmonic analysis, John Wiley & Sons, Inc., New York-London-Sydney, 1968. · Zbl 1055.43001
[12] Hikosaburo Komatsu, The implicit function theorem for ultradifferentiable mappings, Proc. Japan Acad. Ser. A Math. Sci. 55 (1979), no. 3, 69 – 72. · Zbl 0467.26004
[13] J.-M. Lion and J.-P. Rolin, Théorème de préparation pour les fonctions logarithmico-exponentielles, Ann. Inst. Fourier (Grenoble) 47 (1997), no. 3, 859 – 884 (French, with English and French summaries). · Zbl 0873.32004
[14] S. Mandelbrojt, Sur les fonctions indéfiniment dérivables, Acta Math., 72 (1940), pp. 15-29. · Zbl 0023.05505
[15] Chris Miller, Expansions of the real field with power functions, Ann. Pure Appl. Logic 68 (1994), no. 1, 79 – 94. · Zbl 0823.03018 · doi:10.1016/0168-0072(94)90048-5 · doi.org
[16] Chris Miller, Infinite differentiability in polynomially bounded o-minimal structures, Proc. Amer. Math. Soc. 123 (1995), no. 8, 2551 – 2555. · Zbl 0823.03019
[17] Adam Parusiński, Lipschitz stratification of subanalytic sets, Ann. Sci. École Norm. Sup. (4) 27 (1994), no. 6, 661 – 696. · Zbl 0819.32007
[18] Charles Roumieu, Ultra-distributions définies sur \?\(^{n}\) et sur certaines classes de variétés différentiables, J. Analyse Math. 10 (1962/1963), 153 – 192 (French). · Zbl 0122.34802 · doi:10.1007/BF02790307 · doi.org
[19] Walter Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1987. · Zbl 0925.00005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.