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Quasianalytic Ilyashenko Algebras

Published online by Cambridge University Press:  20 November 2018

Patrick Speissegger*
Affiliation:
Department of Mathematics and Statistics, McMaster University, 1280 Main Street West, Hamilton, ON L8S 4K1, Canada e-mail: [email protected]
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Abstract

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We construct a quasianalytic field $\mathcal{F}$ of germs at $+\infty $ of real functions with logarithmic generalized power series as asymptotic expansions, such that $\mathcal{F}$ is closed under differentiation and log-composition; in particular, $\mathcal{F}$ is a Hardy field. Moreover, the field $\mathcal{F}\,\circ \,\left( -\text{log} \right)$ of germs at ${{0}^{+}}$ contains all transition maps of hyperbolic saddles of planar real analytic vector fields.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2018

References

[1] Borichev, A. A. and Volberg, A. L., The finiteness of limit cycles, and uniqueness theorems for asymptotically holomorphic functions. Algebra i Analiz, 7(1995), 43–75.Google Scholar
[2] Ilyashenko, Y. S., Finiteness theorems for limit cycles. Translations of Mathematical Monographs, 94. American Mathematical Society, Providence, RI, 1991. Translated from the Russian by H. H. McFaden.Google Scholar
[3] Ilyashenko, Yulij and Yakovenko, Sergei, Lectures on analytic differential equations. Graduate Studies in Mathematics, 86. American Mathematical Society, Providence, RI, 2008.Google Scholar
[4] Kaiser, Tobias, The Dirichlet problem in the plane with semianalytic raw data, quasi analyticity, and o-minimal structure. Duke Math. J. 147(2009), 285–314.http://dx.doi.Org/10.1215/00127094-2009-012 Google Scholar
[5] Kaiser, Tobias, The Riemann mapping theorem for semianalytic domains and o-minimality. Proc. Lond. Math. Soc. (3) 98(2009), 427–444.http://dx.doi.Org/10.1112/plms/pdn034 Google Scholar
[6] Kaiser, T., Rolin, J.-P., and Speissegger, P., Transition maps at non-resonant hyperbolic singularities are o-minimal. J. Reine Angew. Math. 636(2009), 1–45.http://dx.doi.Org/10.1515/CRELLE.2009.081 Google Scholar
[7] Mourtada, Abderaouf, Action de dérivations irréductibles sur les algèbres quasi-régulières d'Hilbert. arxiv:O912.156O Google Scholar
[8] Rudin, Walter, Real and complex analysis. Third edition. McGraw-Hill, New York, 1987.Google Scholar
[9] van den Dries, Lou, Macintyre, Angus, and Marker, David, The elementary theory of restricted analytic fields with exponentiation. Ann. of Math. (2) 140(1994), 183–205.http://dx.doi.Org/10.2307/2118545 Google Scholar
[10] van den Dries, Lou, Macintyre, Angus, and Marker, David, Logarithmic-exponential series. Ann. Pure Appl. Logic 111(2001), 61–113.http://dx.doi.org/10.1016/S0168-0072(01)00035-5 Google Scholar
[11] van den Dries, Lou and Miller, Chris, On the real exponential field with restricted analytic functions. Israel J. Math. 85(1994), 19–56.http://dx.doi.Org/10.1007/BF02758635 Google Scholar
[12] van den Dries, Lou and Speissegger, Patrick, The real field with convergent generalized power series. Trans. Amer. Math. Soc. 350(1998), 4377–4421.http://dx.doi.org/10.1090/S0002-9947-98-02105-9 Google Scholar
[13] van der Hoeven, J., Transseries and real differential algebra. Lecture Notes in Mathematics, 1888. Springer-Verlag, Berlin, 2006.http://dx.doi.org/10.1007/3-540-35590-1 Google Scholar
[14] Wilkie, A. J., Model completeness results for expansions of the ordered field of real numbers by restricted Vfaffian functions and the exponential function. J. Amer. Math. Soc. 9(1996), 1051–1094. http://dx.doi.org/10.1090/S0894-0347-96-00216-0Google Scholar