Multivariable Newton-Puiseux Theorem for Generalised Quasianalytic Classes

Servi, Tamara

Annales de l'Institut Fourier, Tome 65 (2015) no. 1, pp. 349-368.

Nous montrons comment résoudre explicitement une équation satisfaite par une fonction réelle appartenant à certaines classes quasianalytiques générales. Plus précisément, nous montrons que si $f(x_1,...,x_m,y)$ appartient à une telle classe, alors les solutions $y=\varphi \left(x_1,...,x_m\right)$ de l’équation $f=0$ au voisinage de l’origine peuvent être exprimées par morceaux comme des compositions finies de fonctions dans la classe, de racines $n$-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.

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 $f\left(x_1,...,x_m,y\right)$ belongs to such a class, then the solutions $y=\varphi \left(x_1,...,x_m\right)$ of the equation $f=0$ in a neighbourhood of the origin can be expressed, piecewise, as finite compositions of functions in the class, taking $n^{\text{th}}$ 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.

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