  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.
