Multivariable Newton-Puiseux Theorem for Generalised Quasianalytic Classes

TL;DR

This paper develops an explicit method for solving equations involving functions from broad quasianalytic classes, including generalized power series and multisummable series, extending classical Newton-Puiseux techniques.

Contribution

It introduces a multivariable Newton-Puiseux theorem applicable to a wide range of quasianalytic function classes, providing explicit solutions for associated equations.

Findings

Explicit solutions for equations in quasianalytic classes

Extension of Newton-Puiseux theorem to multivariable cases

Application to classes like Dulac maps and multisummable series

Abstract

We show how to solve explicitly an equation satisfied by a real function belonging to certain general quasianalytic classes. 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.