Real Algebraic and Analytic Geometry Preprint Server

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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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