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Resumen:
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Let ??Rn be a compact Nash manifold; A,B the rings of Nash, analytic global functions on ?. The main result of this paper is the following: Theorem 1. Let ?,?? be a pair of Nash submanifolds of some Rn ,Rq and let us suppose ? is compact. Let F1,?,Fq:?×???R be Nash functions. Then every analytic solution y=f(x) of the system F1(x,y)=?=Fq(x,y)=0 can be approximated, in the Whitney topology, by the global Nash solutions y=g(x). The main tool used to prove the above results is this version of Néron's desingularisation theorem: Any homomorphism of A-algebras C?B, with C finitely generated over A, factorizes through a finitely generated A-algebra D such that A?D is regular. Using Theorem 1 the authors are able to solve several interesting problems that have been open for many years. For example they prove: (I) Every analytic factorization of a global Nash function, defined over ?, is equivalent to a Nash factorization. (II) Every semialgebraic subset of ? which is a global analytic subset is also a global Nash subset. (III) Every prime ideal of A generates a prime ideal in B. (IV) Every coherent ideal subsheaf of the sheaf N(?) of Nash functions on ? is generated by its global sections. The case where ? is noncompact is only partially studied in this paper. In the reviewer's opinion this article makes crucial progress in the theory of global Nash functions.
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