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

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.