An o-minimal structure which does not admit $C^{\infty }$ cellular decomposition

Le Gal, Olivier; Rolin, Jean-Philippe

Annales de l'Institut Fourier, Tome 59 (2009) no. 2, pp. 543-562.

Nous présentons un exemple de structure o-minimale n’admettant pas la propriété de décomposition cellulaire $C^{\infty }$. Pour ce faire, nous construisons une fonction $H$ dont le germe en $0$ admet un représentant $C^k$ pour tout entier $k$, mais n’admet aucun représentant $C^{\infty }$. Une condition de transcendance sur les coefficients de la série de Taylor de $H$ assure alors la quasi-analyticité de certaines algèbres différentielles ${𝒜}_n\left(H\right)$ engendrées par $H$. La o-minimalité de la structure engendrée par $H$ est enfin déduite de cette quasi-analyticité.

We present an example of an o-minimal structure which does not admit $C^{\infty }$ cellular decomposition. To this end, we construct a function $H$ whose germ at the origin admits a $C^k$ representative for each integer $k$, but no $C^{\infty }$ representative. A number theoretic condition on the coefficients of the Taylor series of $H$ then insures the quasianalyticity of some differential algebras ${𝒜}_n\left(H\right)$ induced by $H$. The o-minimality of the structure generated by $H$ is deduced from this quasianalyticity property.

[1] Bierstone, Edward; Milman, Pierre D. Semianalytic and subanalytic sets, Inst. Hautes Études Sci. Publ. Math. (1988) no. 67, pp. 5-42 | DOI | Numdam | MR | Zbl

[2] Denef, J.; van den Dries, L. $p$-adic and real subanalytic sets, Ann. of Math. (2), Volume 128 (1988) no. 1, pp. 79-138 | DOI | MR | Zbl

[3] van den Dries, Lou Tame topology and o-minimal structures, London Mathematical Society Lecture Note Series, 248, Cambridge University Press, Cambridge, 1998 | MR | Zbl

[4] van den Dries, Lou; Speissegger, Patrick The real field with convergent generalized power series, Trans. Amer. Math. Soc., Volume 350 (1998) no. 11, pp. 4377-4421 | DOI | MR | Zbl

[5] van den Dries, Lou; Speissegger, Patrick The field of reals with multisummable series and the exponential function, Proc. London Math. Soc. (3), Volume 81 (2000) no. 3, pp. 513-565 | DOI | MR | Zbl

[6] Gabrielov, Andrei Complements of subanalytic sets and existential formulas for analytic functions, Invent. Math., Volume 125 (1996) no. 1, pp. 1-12 | DOI | MR | Zbl

[7] Malgrange, Bernard Idéaux de fonctions différentiables et division des distributions, Distributions, Ed. Éc. Polytech., Palaiseau, 2003, pp. 1-21 With an Appendix: “Stanisław Łojasiewicz (1926–2002)” | MR

[8] Mandelbrojt, S. Sur les fonctions indéfiniment dérivables, Acta Math., Volume 72 (1940), pp. 15-29 | DOI | MR

[9] Rolin, J.-P.; Speissegger, P.; Wilkie, A. J. Quasianalytic Denjoy-Carleman classes and o-minimality, J. Amer. Math. Soc., Volume 16 (2003) no. 4, p. 751-777 (electronic) | DOI | MR | Zbl

[10] Wilkie, A. J. A theorem of the complement and some new o-minimal structures, Selecta Math. (N.S.), Volume 5 (1999) no. 4, pp. 397-421 | DOI | MR | Zbl