2001/08/29 by Edward Bierstone, Bierstone, Edward, Pierre D. Milman +1
Mathematics · #26E10 #32S45 #58C25 #Advanced Topology and Set Theory #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics #Mathematical Dynamics and Fractals #Rings, Modules, and Algebras #math.AG #math.CV #msc:26E10 #msc:32S45 #msc:58C25
paper · pdf · doi:10.48550/arxiv.math/0108204
35 pages, AMSTEX
arxiv created 2001/08/29 · openalex publication_date 2001/08/29 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show that a version of the desingularization theorem of Hironaka holds for certain classes of infinitely differentiable functions (essentially, for subrings that exclude flat functions and are closed under differentiation and the solution of implicit equations). Examples are quasianalytic classes, introduced by E. Borel a century ago and characterized by the Denjoy-Carleman theorem. These classes have been poorly understood in dimension > 1. Resolution of singularities can be used to obtain many new results; for example, topological Noetherianity, Lojasiewicz inequalities, division properties.