We show how to solve explicitly an equation satisfied by a real function belonging to certain general quasianalytic classes. More precisely, we show that if belongs to such a class, then the solutions of the equation in a neighbourhood of the origin can be expressed, piecewise, as finite compositions of functions in the class, taking roots and quotients. Examples of the classes under consideration are the collection of convergent generalised power series, a class of functions which contains some Dulac Transition Maps of real analytic planar vector fields, quasianalytic Denjoy-Carleman classes and the collection of multisummable series.
Nous montrons comment résoudre explicitement une équation satisfaite par une fonction réelle appartenant à certaines classes quasianalytiques générales. Plus précisément, nous montrons que si appartient à une telle classe, alors les solutions de l’équation au voisinage de l’origine peuvent être exprimées par morceaux comme des compositions finies de fonctions dans la classe, de racines -ièmes et de quotients. Parmi les exemples de telles classes figurent les séries généralisées convergentes, une classe de fonctions qui contient certaines applications de transition de Dulac de champs de vecteurs analytiques du plan réel, les classes quasianalytiques de Denjoy-Carleman et la collection des séries multisommables.
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.2933
Classification: 30D60, 32B20, 32S45, 03C64
Keywords: Newton-Puiseux, quasianalytic classes, monomialisation, o-minimality
@article{AIF_2015__65_1_349_0, author = {Servi, Tamara}, title = {Multivariable {Newton-Puiseux} {Theorem} for {Generalised} {Quasianalytic} {Classes}}, journal = {Annales de l'Institut Fourier}, pages = {349--368}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {65}, number = {1}, year = {2015}, doi = {10.5802/aif.2933}, zbl = {1326.30032}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2933/} }
TY - JOUR TI - Multivariable Newton-Puiseux Theorem for Generalised Quasianalytic Classes JO - Annales de l'Institut Fourier PY - 2015 DA - 2015/// SP - 349 EP - 368 VL - 65 IS - 1 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2933/ UR - https://zbmath.org/?q=an%3A1326.30032 UR - https://doi.org/10.5802/aif.2933 DO - 10.5802/aif.2933 LA - en ID - AIF_2015__65_1_349_0 ER -
Servi, Tamara. Multivariable Newton-Puiseux Theorem for Generalised Quasianalytic Classes. Annales de l'Institut Fourier, Volume 65 (2015) no. 1, pp. 349-368. doi : 10.5802/aif.2933. https://aif.centre-mersenne.org/articles/10.5802/aif.2933/
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