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A Note on the Weierstrass Preparation Theorem in Quasianalytic Local Rings

Published online by Cambridge University Press:  20 November 2018

Adam Parusiński  and
Jean-Philippe Rolin
Adam Parusiński
Affiliation:
Univ. Nice Sophia Antipolis, CNRS, LJAD, UMR 7351, 06100 Nice, France e-mail: parus@unice.fr
Jean-Philippe Rolin
Affiliation:
Univ. de Bourgogne (Dijon), I.M.B., 9 av. A. Savary, BP47870, 21078 Dijon Cedex, France e-mail: rolin@u-bourgogne.fr
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Abstract

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Consider quasianalytic local rings of germs of smooth functions closed under composition, implicit equation, and monomial division. We show that if the Weierstrass Preparation Theoremholds in such a ring, then all elements of it are germs of analytic functions.

Keywords

26E1026E0514P15Weierstrass Preparation Theoremquasianalytic local rings
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 57 , Issue 3 , 01 September 2014 , pp. 614 - 620
Copyright
Copyright © Canadian Mathematical Society 2014

References

[1] Acquistapace, F., Broglia, F., Bronshtein, M., Nicoara, A., and Zobin, N., Failure of the Weierstrass preparation theorem in quasi-analytic Denjoy–Carleman rings. Eprint arXiv:1212.4265, 2012.Google Scholar
[2] Bianconi, R., Nondefinability results for expansions of the field of real numbers by the exponential function and by the restricted sine function. J. Symbolic Logic 62 (1997), 1173–1178. http://dx.doi.org/10.2307/2275634 Google Scholar
[3] Bierstone, E. and Milman, P. D., Resolution of singularities in Denjoy–Carleman classes. Selecta Math. (N.S.) 10 (2004), 1–28. http://dx.doi.org/10.1007/s00029-004-0327-0 Google Scholar
[4] Bochnak, J. and Siciak, J., Analytic functions in topological vector spaces. Studia Math. 39 (1971), 77–112.Google Scholar
[5] Borel, E., Sur la généralisation du prolongement analytique. C. R. Acad. Sci. 130 (1900), 1115–1118.Google Scholar
[6] Borel, E., Sur les séries de polynômes et de fractions rationnelles. Acta Math. 24 (1901), 309–387. http://dx.doi.org/10.1007/BF02403078 Google Scholar
[7] Carleman, T., Les fonctions quasi-analytiques. Gauthier Villars, 1926.Google Scholar
[8] Childress, C. L., Weierstrass division in quasianalytic local rings. Canad. J. Math. 28 (1976), 938–953. http://dx.doi.org/10.4153/CJM-1976-091-7 Google Scholar
[9] Denjoy, A., Sur les fonctions quasi-analytiques de la variable r´eelle. C. R. Acad. Sci. Paris 123 (1921), 1320–1322.Google Scholar
[10] van den Dries, L., On the elementary theory of restricted elementary functions. J. Symbolic Logic 53 (1988), 796–808. http://dx.doi.org/10.2307/2274572 Google Scholar
[11] van den Dries, L., Tame topology and o-minimal structures. Cambridge University Press, 1998.Google Scholar
[12] van den Dries, L. and Speissegger, P., The field of reals with multisummable series and the exponential function. Proc. London Math. Soc. (3) 81 (2000), 513–565. http://dx.doi.org/10.1112/S0024611500012648 Google Scholar
[13] Elkhadiri, A., Link between Noetherianity and the Weierstrass Division Theorem on some quasianalytic local rings. Proc. Amer. Math. Soc. 140 (2012), 3883–3892. http://dx.doi.org/10.1090/S0002-9939-2012-11243-2 Google Scholar
[14] Elkhadiri, A. and Sfouli, H., Weierstrass division theorem in definable C1 germs in a polynomially bounded o-minimal structure. Ann. Polon. Math. 89 (2006), 127–137. http://dx.doi.org/10.4064/ap89-2-2 Google Scholar
[15] Elkhadiri, A., Weierstrass division theorem in quasianalytic local rings. Studia Math. 185 (2008), 83–86. http://dx.doi.org/10.4064/sm185-1-5 Google Scholar
[16] Komatsu, H., The implicit function theorem for ultradifferentiable mappings. Proc. Japan Acad. Ser. A Math. Sci. 55 (1979), 69–72. http://dx.doi.org/10.3792/pjaa.55.69 Google Scholar
[17] Malgrange, B., Id´eaux de fonctions diff´erentiables et division des distributions. Distributions, Ed. Ec. Polytech., Palaiseau, 2003, 1–21.Google Scholar
[18] Roumieu, C., Ultra-distributions d´efinies sur Rn et sur certaines classes de vari´et´es diff´erentiables. J. Analyse Math. 10(1962/1963), 153–192. http://dx.doi.org/10.1007/BF02790307 Google Scholar
[19] Rolin, J.-P., Sanz, F., and Schäfke, R., Quasi-analytic solutions of analytic ordinary differential equations and o-minimal structures. Proc. London Math. Soc. 95 (2007), 413–442. http://dx.doi.org/10.1112/plms/pdm016 Google Scholar
[20] Rolin, J.-P., Speissegger, P., and Wilkie, A. J., Quasianalytic Denjoy–Carleman classes and o-minimality. J. Amer. Math. Soc. 16 (2003), 751–777. http://dx.doi.org/10.1090/S0894-0347-03-00427-2 Google Scholar
[21] Thilliez, V., On quasianalytic local rings. Expo. Math. 26 (2008), 1–23. http://dx.doi.org/10.1016/j.exmath.2007.04.001 Google Scholar