Adapted coordinates in two dimensions and a proof of Puiseux's theorem

TL;DR

This paper presents an elementary proof of Puiseux series convergence and the existence of smooth adapted coordinates in two dimensions, using Newton's method and ideas from resolution of singularities, simplifying previous complex proofs.

Contribution

Provides a quick, elementary proof of Puiseux series convergence and smooth adapted coordinates in two dimensions, extending prior results to the smooth case.

Findings

Elementary proof of Puiseux series convergence

Existence of smooth adapted coordinates in two dimensions

Simplification of previous complex proofs

Abstract

A method for finding Puiseux series goes back to Isaac Newton, which gives the terms of Puiseux series through an infinite recursive process; an additional argument is then used to show that the resulting Puiseux series are convergent. This paper provides an argument based on Newton's method and some ideas from resolution of singularities that gives a quick proof of both the existence and convergence of Puiseux series. It is then shown that similar ideas can be used to give a short proof of the existence of smooth adapted coordinates for oscillatory integrals in two dimensions, a result first proved in the real-analytic case by Varchenko [V] and then recently for the general smooth case by Ikromov-Muller [IM]. The arguments of this paper are entirely elementary.