Original paper

Selecta Mathematica

, Volume 10, Issue 1, pp 1-28

First online:

Resolution of singularities in Denjoy-Carleman classes

  • Edward BierstoneAffiliated withDepartment of Mathematics, University of Toronto Email author 
  • , Pierre D. MilmanAffiliated withDepartment of Mathematics, University of Toronto

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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 26E10 32S45 58C25 Secondary 14E15 14P15 30D60

Key words.

Quasianalytic Denjoy-Carleman class resolution of singularities