Ordinary Differential Equations
Generalised power series solutions of sub-analytic differential equations
Comptes Rendus. Mathématique, Volume 342 (2006) no. 2, pp. 99-102.

We show that if a solution y(x) of a sub-analytic differential equation admits an asymptotic expansion i=1cixμi, μiR+, then the exponents μi belong to a finitely generated semi-group of R+. We deduce a similar result for the components of non-oscillating trajectories of real analytic vector fields in dimension n.

Nous montrons que si une solution y(x) d'une équation différentielle sous-analytique admet un développement asymptotique de la forme i=1cixμi, μiR+, alors les exposants μi appartiennent à un semi-groupe finiment engendré de R+. Nous en déduisons un résultat analogue pour les composantes des trajectoires non oscillantes de champs de vecteurs analytiques réels en dimension n.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2005.11.005
Matusinski, Mickaël 1; Rolin, Jean-Philippe 1

1 I.M.B., université de Bourgogne, 9, avenue Savary, B.P. 47870, 21078 Dijon cedex, France
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Matusinski, Mickaël; Rolin, Jean-Philippe. Generalised power series solutions of sub-analytic differential equations. Comptes Rendus. Mathématique, Volume 342 (2006) no. 2, pp. 99-102. doi : 10.1016/j.crma.2005.11.005. http://www.numdam.org/articles/10.1016/j.crma.2005.11.005/

[1] Cano, F.; Moussu, R.; Rolin, J.-P. Non-oscillating integral curves and valuations, J. Reine Angew. Math., Volume 582 (2005), pp. 107-141

[2] Cano, J. On the series defined by differential equations, with an extension of the Puiseux polygon construction to these equations, Analysis, Volume 13 (1993) no. 1–2, pp. 103-119

[3] Grigoriev, D.Y.; Singer, M.F. Solving ordinary differential equations in terms of series with real exponents, Trans. Amer. Math. Soc., Volume 327 (1991) no. 1, pp. 329-351

[4] Kaup, L.; Kaup, B. Holomorphic Functions of Several Variables, de Gruyter Stud. in Math., vol. 3, Walter de Gruyter & Co., Berlin, 1983 (An introduction to the fundamental theory, With the assistance of Gottfried Barthel. Translated from the German by Michael Bridgland)

[5] Lion, J.-M.; Rolin, J.-P. Théorème de préparation pour les fonctions logarithmico-exponentielles, Ann. Inst. Fourier (Grenoble), Volume 47 (1997) no. 3, pp. 859-884

[6] van den Dries, L.; Macintyre, A.; Marker, D. The elementary theory of restricted analytic fields with exponentiation, Ann. of Math. (2), Volume 140 (1994) no. 1, pp. 183-205

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