  In this paper we simplify and otherwise improve the local resolution of
singularities algorithm of [G1]-[G3], providing a local resolution of
singularities method that works for functions with convergent power series over
an arbitrary local field of characteristic zero. The algorithm of this paper is
an entirely elementary classical analysis argument, using only the implicit
function theorem and elementary facts about power series and Newton polyhedra.
Several examples are given. The methods are quite different from traditional
resolution of singularities methods. In a separate paper arxiv:1104.4684, the
methods of this paper (but for the most part not the resolution of
singularities theorems themselves) are used to prove results concerning
oscillatory integrals, exponential sums, and related matters.
  Note: This paper should be viewed as a replacement for the resolution of
singularities portion of older versions of arxiv:1104.4684, as this paper is
quite a bit different from that algorithm, and should be a lot easier to read.
For one, standard blowups are used here, rather than the monomial maps of
arxiv:1104.4684 which forced one to deal with technical issues related to such
monomial maps not necessarily being one to one. Secondly, the complicated
iterated domain subdivisions using Newton polyhedra that were done in
arxiv:1104.4684 have been reduced to a short one-stage argument, given in
section 2. Thirdly, forming partitions of unity can be done more naturally in
the context of the newer algorithm (see Lemma 3.5).
