Selecta Mathematica

, 10:1

Resolution of singularities in Denjoy-Carleman classes

Original paper

DOI: 10.1007/s00029-004-0327-0

Cite this article as:
Bierstone, E. & Milman, P.D. Sel. math., New ser. (2004) 10: 1. doi:10.1007/s00029-004-0327-0


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.

Mathematics Subject Classification (2000)

Primary 26E1032S4558C25Secondary 14E1514P1530D60

Key words.

QuasianalyticDenjoy-Carleman classresolution of singularities

Copyright information

© Birkhäuser-Verlag 2004

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TorontoTorontoCanada
  2. 2.Department of MathematicsUniversity of TorontoTorontoCanada