A constructive elementary method for local resolution of singularities

TL;DR

This paper introduces a simplified, elementary method for local resolution of singularities over arbitrary local fields of characteristic zero, using classical analysis tools and Newton polyhedra, improving on previous algorithms.

Contribution

It provides a new elementary resolution of singularities algorithm that avoids complex blowups and iterative subdivisions, making the process more accessible and easier to implement.

Findings

The method works for functions with convergent power series over any local field of characteristic zero.

The algorithm is elementary, relying only on the implicit function theorem and basic properties of power series.

Examples demonstrate the effectiveness and simplicity of the new approach.

Abstract

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).