An o-minimal structure which does not admit C ∞ cellular decomposition

Olivier Le Gal[1]; Jean-Philippe Rolin[2]

  • [1] University of Toronto Department of Mathematics Toronto, Ontario M5S 2E4 (Canada)
  • [2] Université de Bourgogne IMB, UFR Sciences et Techniques 9, Avenue Alain Savary BP 47870 21078 Dijon (France)

Annales de l’institut Fourier (2009)

  • Volume: 59, Issue: 2, page 543-562
  • ISSN: 0373-0956

Abstract

top
We present an example of an o-minimal structure which does not admit C ∞ 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 ∞ representative. A number theoretic condition on the coefficients of the Taylor series of H then insures the quasianalyticity of some differential algebras 𝒜 n ( H ) induced by H . The o-minimality of the structure generated by H is deduced from this quasianalyticity property.

How to cite

top

Le Gal, Olivier, and Rolin, Jean-Philippe. "An o-minimal structure which does not admit $C^{\infty }$ cellular decomposition." Annales de l’institut Fourier 59.2 (2009): 543-562. <http://eudml.org/doc/10403>.

@article{LeGal2009,
abstract = {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 $\mathcal\{A\}_n(H)$ induced by $H$. The o-minimality of the structure generated by $H$ is deduced from this quasianalyticity property.},
affiliation = {University of Toronto Department of Mathematics Toronto, Ontario M5S 2E4 (Canada); Université de Bourgogne IMB, UFR Sciences et Techniques 9, Avenue Alain Savary BP 47870 21078 Dijon (France)},
author = {Le Gal, Olivier, Rolin, Jean-Philippe},
journal = {Annales de l’institut Fourier},
keywords = {o-minimal; smooth cell decomposition},
language = {eng},
number = {2},
pages = {543-562},
publisher = {Association des Annales de l’institut Fourier},
title = {An o-minimal structure which does not admit $C^\{\infty \}$ cellular decomposition},
url = {http://eudml.org/doc/10403},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Le Gal, Olivier
AU - Rolin, Jean-Philippe
TI - An o-minimal structure which does not admit $C^{\infty }$ cellular decomposition
JO - Annales de l’institut Fourier
PY - 2009
PB - Association des Annales de l’institut Fourier
VL - 59
IS - 2
SP - 543
EP - 562
AB - 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 $\mathcal{A}_n(H)$ induced by $H$. The o-minimality of the structure generated by $H$ is deduced from this quasianalyticity property.
LA - eng
KW - o-minimal; smooth cell decomposition
UR - http://eudml.org/doc/10403
ER -

References

top
  1. Edward Bierstone, Pierre D. Milman, Semianalytic and subanalytic sets, Inst. Hautes Études Sci. Publ. Math. (1988), 5-42 Zbl0674.32002MR972342
  2. J. Denef, L. van den Dries, p -adic and real subanalytic sets, Ann. of Math. (2) 128 (1988), 79-138 Zbl0693.14012MR951508
  3. Lou van den Dries, Tame topology and o-minimal structures, 248 (1998), Cambridge University Press, Cambridge Zbl0953.03045MR1633348
  4. Lou van den Dries, Patrick Speissegger, The real field with convergent generalized power series, Trans. Amer. Math. Soc. 350 (1998), 4377-4421 Zbl0905.03022MR1458313
  5. Lou van den Dries, Patrick Speissegger, The field of reals with multisummable series and the exponential function, Proc. London Math. Soc. (3) 81 (2000), 513-565 Zbl1062.03029MR1781147
  6. Andrei Gabrielov, Complements of subanalytic sets and existential formulas for analytic functions, Invent. Math. 125 (1996), 1-12 Zbl0851.32009MR1389958
  7. Bernard Malgrange, Idéaux de fonctions différentiables et division des distributions, Distributions (2003), 1-21, Ed. Éc. Polytech., Palaiseau MR2065138
  8. S. Mandelbrojt, Sur les fonctions indéfiniment dérivables, Acta Math. 72 (1940), 15-29 Zbl66.0245.01MR1783
  9. J.-P. Rolin, P. Speissegger, A. J. Wilkie, Quasianalytic Denjoy-Carleman classes and o-minimality, J. Amer. Math. Soc. 16 (2003), 751-777 (electronic) Zbl1095.26018MR1992825
  10. A. J. Wilkie, A theorem of the complement and some new o-minimal structures, Selecta Math. (N.S.) 5 (1999), 397-421 Zbl0948.03037MR1740677

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.