Resolution of singularities in Denjoy-Carleman classes

By Edward Bierstone, Pierre D. Milman. 2001

99 citations recorded by the source corpus.

Abstract We show that a version of the desingularization theorem of Hironaka $${\cal C}^\infty $$ holds for certain classes of 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, curve selection, Łojasiewicz inequalities, division properties.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/c3174d93184bbda260098c3401f2b4b17b6c02c6

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In Applied Mathematics

1. Michael Greenblatt
2. A. Rainer
3. Peter W. Michor
4. Andreas Kriegl
5. B. Clarke

In Geometry and Topology

1. Michael Greenblatt
2. A. Rainer
3. Peter W. Michor
4. Andreas Kriegl
5. B. Clarke

Citations in the corpus

• Oscillatory integral decay, sublevel set growth, and the Newton polyhedron (2008, methodology): “In [PSt1] and [PSt2], the method of algebraic estimates is used for this purpose; in [PSt2] resolution of singularities algorithms of Bierstone and Milman such as [ BM ] are also used.”
• The convenient setting for non-quasianalytic Denjoy–Carleman differentiable mappings☆ (2008, background): “The category of convenient complex vector spaces and holomorphic mappings between them is cartesian closed, i.e.”