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Thom’s first isotopy lemma: A semialgebraic version, with uniform bound. (English) Zbl 0844.14025
Broglia, Fabrizio (ed.) et al., Real analytic and algebraic geometry. Proceedings of the international conference, Trento, Italy, September 21-25, 1992. Berlin: Walter de Gruyter. 83-101 (1995).
In this paper the authors prove the semialgebraic version of Thom’s first isotopy lemma, and provide a uniform recursive bound on the degree of the algebraic trivialization in terms of the complexity of the data. Thom’s well-known first isotopy lemma guarantees the existence of isotopy of a family of stratified sets, under the regularity condition of Whitney. Its usual proof, however, involves integrations of vector fields, and therefore, it is not applicable to constructing semi-algebraic isotopy for semi-algebraic stratifications. The authors overcome this difficulty using the Nash triviality theorem proved in another paper by the same authors [Invent. Math. 108, No. 2, 349-368 (1992; Zbl 0801.14017)]. This paper provides the fundamental method for the related area, real algebraic geometry and singularity theory: There are expected, after this paper, fruitful applications.
For the entire collection see [Zbl 0812.00016].

14P10 Semialgebraic sets and related spaces
14P20 Nash functions and manifolds
58A35 Stratified sets
32S60 Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects)