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Zbl 0805.14027
Ruiz, Jesús M.; Shiota, Masahiro
On global Nash functions. (English)
[J] Ann. Sci. Éc. Norm. Supér. (4) 27, No. 1, 103-124 (1994). ISSN 0012-9593

There is a number of open problems concerning Nash functions, and a central one is the ``separation problem'', which asks whether Nash functions are sufficient to separate the global analytic components of real algebraic sets. The authors introduce a new problem (``equal complexities''): if a semialgebraic set is described by $s$ simultaneous global analytic inequalities, can it be described by $s$ Nash inequalities? The main result of the paper is that ``separation'' and ``equal complexities'' are equivalent for compact Nash manifolds. There are other interesting results, such as the fact that ``separation'' implies ``extension'' (a Nash function on a Nash subset may be extended to a Nash function on the ambient manifold). The introduction of the paper gives informations about related results in the literature about Nash functions. Since the writing of the paper under review, the authors and the reviewer (to appear in Am. J. Math.) have proven that the separation problem has a positive answer on a compact Nash manifold; hence ``equal complexities'' also hold.\par The problem of ``equal complexities'' is in the line of the works initiated by {\it L. Bröcker} about the number of inequalities needed to describe semialgebraic sets; the main tool in this activity is the notion of fan, which comes from the algebraic theory of quadratic forms. The main result of the paper is translated in terms of fans, and then the proof uses algebraic tools.
[M.Coste (Rennes)]

MSC 2000:: *14P20 Nash functions and manifolds
32C07 Real-analytic sets

Keywords: separation problem; problem of equal complexities; Nash functions; number of inequalities; fans

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