A Coordinate-Dependent Local Resolution of Singularities with Applications

Details A Coordinate-Dependent Local Resolution of Singularities with Applications A Coordinate-Dependent Local Resolution of Singularities with Applications

2006-09-07

arxiv.org/abs/math/0609217v4

Mathematics

Classical Analysis and ODEs

40 pages, final published version

Scientific paper

In this paper, a geometric resolution of singularities algorithm is developed. This method is elementary in its statement and proof, using explicit coordinate systems as much as possible. Each coordinate change used in the resolution procedure is one-to-one on its domain, and is of one of a few explicit canonical forms. As applications of these methods to classical analysis, two theorems are proven. First and foremost, a general theorem regarding the existence of critical integrability exponents is established. Secondly, a new proof of a well-known inequality of Lojasiewicz is given.

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