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The purpose of this note is to correct a mistake in the article “A curve selection lemma in spaces of arcs and the image of the Nash map” Compositio Math. 142 (2006), 119–130. It is due to an overlooked hypothesis in the definition of generically stable subset of the space of arcs X∞ of a variety X defined over a perfect field k.
We prove two main results on Denjoy–Carleman classes: (1) a composite function theorem which asserts that a function
in a quasianalytic Denjoy–Carleman class
, which is formally composite with a generically submersive mapping
, at a single given point in the source (or in the target) of
can be written locally as
belongs to a shifted Denjoy–Carleman class
; (2) a statement on a similar loss of regularity for functions definable in the
-minimal structure given by expansion of the real field by restricted functions of quasianalytic class
. Both results depend on an estimate for the regularity of a
of the equation
as above. The composite function result depends also on a quasianalytic continuation theorem, which shows that the formal assumption at a given point in (1) propagates to a formal composition condition at every point in a neighbourhood.
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