A differential Puiseux theorem in generalized series fields of finite rank
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 20 (2011) no. 2, pp. 247-293.

We study differential equations $F\left(y,...,{y}^{\left(n\right)}\right)=0$ where $F$ is a formal series in $y,{y}^{\prime },...,{y}^{\left(n\right)}$ with coefficients in some field of generalized power series ${𝕂}_{r}$ with finite rank $r\in {ℕ}^{*}$. Our purpose is to express the support $\mathrm{Supp}\phantom{\rule{4pt}{0ex}}{y}_{0}$, i.e. the set of exponents, of the elements ${y}_{0}\in {𝕂}_{r}$ that are solutions, in terms of the supports of the coefficients of the equation, namely $\mathrm{Supp}\phantom{\rule{4pt}{0ex}}F$.

Nous étudions des équations différentielles $F\left(y,...,{y}^{\left(n\right)}\right)=0$$F$ est une séries formelle en $y,{y}^{\prime },...,{y}^{\left(n\right)}$, à coefficients dans un corps de séries généralisées ${𝕂}_{r}$ de rang fini $r\in {ℕ}^{*}$. Notre objet est d’exprimer le support – c’est-à-dire l’ensemble $\mathrm{Supp}\phantom{\rule{4pt}{0ex}}{y}_{0}$ des exposants – des éléments ${y}_{0}\in {𝕂}_{r}$ solutions, en fonction des supports des coefficients de l’équation, dont l’union est notée $\mathrm{Supp}\phantom{\rule{4pt}{0ex}}F$.

DOI: 10.5802/afst.1293
Mickaël Matusinski 1

1 Universität Konstanz, Fachbereich Mathematik und Statistik, 78457 Konstanz, Allemagne.
@article{AFST_2011_6_20_2_247_0,
author = {Micka\"el Matusinski},
title = {A differential {Puiseux} theorem in generalized series fields of finite rank},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {247--293},
publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques},
volume = {Ser. 6, 20},
number = {2},
year = {2011},
doi = {10.5802/afst.1293},
mrnumber = {2847885},
zbl = {1222.34012},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1293/}
}
TY  - JOUR
AU  - Mickaël Matusinski
TI  - A differential Puiseux theorem in generalized series fields of finite rank
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2011
SP  - 247
EP  - 293
VL  - 20
IS  - 2
PB  - Université Paul Sabatier, Institut de Mathématiques
PP  - Toulouse
UR  - https://afst.centre-mersenne.org/articles/10.5802/afst.1293/
DO  - 10.5802/afst.1293
LA  - en
ID  - AFST_2011_6_20_2_247_0
ER  - 
%0 Journal Article
%A Mickaël Matusinski
%T A differential Puiseux theorem in generalized series fields of finite rank
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2011
%P 247-293
%V 20
%N 2
%I Université Paul Sabatier, Institut de Mathématiques
%C Toulouse
%U https://afst.centre-mersenne.org/articles/10.5802/afst.1293/
%R 10.5802/afst.1293
%G en
%F AFST_2011_6_20_2_247_0
Mickaël Matusinski. A differential Puiseux theorem in generalized series fields of finite rank. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 20 (2011) no. 2, pp. 247-293. doi : 10.5802/afst.1293. https://afst.centre-mersenne.org/articles/10.5802/afst.1293/

[AvdD05] Aschenbrenner (M.), van den Dries (L.).— Asymptotic differential algebra, Analyzable functions, applications, Contemp. Math., vol. 373, AMS, Providence, RI, p. 49-85, (2005). | MR | Zbl

[Bou70] Bourbaki (N.).— Éléments de mathématique. Théorie des ensembles, Hermann, Paris (1970). | Zbl

[Can93] Cano (J.).— On the series defined by differential equations, with an extension of the Puiseux polygon construction to these equations, Analysis 13, no. 1-2, 103-119 (1993). | MR | Zbl

[CMR05] Cano (F.), Moussu (R.), Rolin (J.-P.).— Non-oscillating integral curves, valuations, J. Reine Angew. Math. 582, 107-141 (2005). | MR | Zbl

[É92] Écalle (J.).— Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac, Actualités Mathématiques. [Current Mathematical Topics], Hermann, Paris (1992). | MR

[Fuc63] Fuchs (L.).— Partially ordered algebraic systems, Pergamon Press, Oxford (1963). | MR | Zbl

[GS91] Grigoriev (D. Y.), Singer (M. F.).— Solving ordinary differential equations in terms of series with real exponents, Transactions Amer. Math. Soc. 327, no. 1, p. 329-351 (1991). | MR | Zbl

[Hah07] Hahn (H.).— Über die nichtarchimedischen Größensystem, Sitzungsberichte der Kaiserlichen Akad. der Wissens., Math. - Naturwissens. Klasse (Wien) 116, no. Abteilung IIa, p. 601-655 (1907).

[Inc44] Ince (E. L.).— Ordinary Differential Equations, Dover Publications, New-York (1944). | MR | Zbl

[Fin89] Fine (H. B.).— On the functions defined by differential equations, with an extension of the Puiseux polygon construction to these equations, Amer. J. Math. 11, no. 4, p. 317-328 (1889). | MR

[vdH97] van der Hoeven (J.).— Asymptotique automatique, Thèse, Université Paris VII, Paris (1997). | MR | Zbl

[vdH06] van der Hoeven (J.).— Transseries, real differential algebra, Lecture Notes in Mathematics, vol. 1888, Springer-Verlag, Berlin (2006). | MR | Zbl

[KP02] Krantz (S. G.), Parks (H. R.).— A primer of real analytic functions, 2nd ed., Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Boston Inc., Boston, mA (2002). | MR | Zbl

[Kuh00] Kuhlmann (S.).— Ordered exponential fields, Fields Institute monographs, vol. 12, American mathematical Society, Providence, RI (2000). | MR | Zbl

[Mat07] Matusinski (M.).— Équations différentielles à coefficients dans des corps de séries généralisés, Thèse, Université de Bourgogne (2007).

[KM10] Kuhlmann (S.), Matusinski (M.).— Hardy type derivations in generalized series fields., preprint 22 pages (2010).

[MR06] Matusinski (M.), Rolin (J.-P.).— Generalized power series solutions of sub-analytic differential equations, C. R. Math. Acad. Sci. Paris 342, no. 2, p. 99-102 (2006). | MR | Zbl

[Ros80] Rosenlicht (M.).— Differential valuations, Pacific J. Math. 86, no. 1, p. 301-319 (1980). | MR | Zbl

[Ros81] Rosenlicht (M.).— On the value group of a differential valuation. II, Amer. J. Math. 103, no. 5, p. 977-996 (1981). | MR | Zbl

[Ros83] Rosenlicht (M.).— The rank of a Hardy field, Trans. Amer. Math. Soc. 280, no. 2, p. 659-671 (1983). | MR | Zbl

Cited by Sources: