Real Algebraic and Analytic Geometry |

(r) does not imply (n) or (npf) for definable sets in non polynomially bounded o-minimal structures.

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Submission: 2007, February 2.

*Abstract:
It is known that if two subanalytic strata
satisfy Kuo's ratio test, then the normal cone of
the larger stratum Y along the smaller X
satisfies two nice properties: the fiber of the
normal cone at any point of X is the
tangent cone to the fiber of Y over that point
and the projection of the normal cone to X is
open ("normal pseudo-flatness"). We present an
example with X a line and Y a surface which is
definable in any non polynomially bounded
o-minimal structure, such that the pair satisfies
Kuo's ratio test, but neither of the above
properties hold for the normal cone. Also (r*)
fails, showing also that a 1981 theorem of
Navarro Aznar and Trotman is only true for
polynomially bounded o-minimal structures.*

Mathematics Subject Classification (2000): 03C64, 14P15, 32B20, 58A35.

Keywords and Phrases: Stratified sets, o-minimal structures, normal cone.

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