Masahiro Shiota (1983)
Annales de l'institut Fourier
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A semi-algebraic analytic manifold and a semi-algebraic analytic map are called a Nash manifold and a Nash map respectively. We clarify the category of Nash manifolds and Nash maps.
Jesús Escribano (2001)
Annales de l’institut Fourier
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We study triviality of Nash families of proper Nash submersions or, in a more general setting, the triviality of pairs of proper Nash submersions. We work with Nash manifolds and mappings defined over an arbitrary real closed field . To substitute the integration of vector fields, we study the fibers of such families on points of the real spectrum and we construct models of proper Nash submersions over smaller real closed fields. Finally we obtain results on finiteness of topological...
Michel Coste, Masahiro Shiota (2000)
Annales scientifiques de l'École Normale Supérieure
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Alberto Tancredi, Alberto Tognoli (1998)
Revista Matemática Complutense
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Masahito Shiota (1982)
Annales de l'institut Fourier
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We show under some assumptions that a differentiable function can be transformed globally to a polynomial or a rational function by some diffeomorphism. One of the assumptions is that the function is proper, the number of critical points is finite, and the Milnor number of the germ at each critical point is finite.
Michel Coste, Jesús M. Ruiz, Masahiro Shiota (2004)
Revista Matemática Complutense
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This is a survey on the history of and the solutions to the basic global problems on Nash functions, which have been only recently solved, namely: separation, extension, global equations, Artin-Mazur description and idempotency, also noetherianness. We discuss all of them in the various possible contexts, from manifolds over the reals to real spectra of arbitrary commutative rings.