Coste, M.Ruiz Sancho, Jesús MaríaShiota, Masahiro2023-06-202023-06-201995-080002-932710.2307/2374953https://hdl.handle.net/20.500.14352/57924Let Ω⊂Rn be a compact Nash manifold; A,B the rings of Nash, analytic global functions on Ω. The main result of this paper is the following: Theorem 1. Let Ω,Ω′ be a pair of Nash submanifolds of some Rn ,Rq and let us suppose Ω is compact. Let F1,⋯,Fq:Ω×Ω′→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 Ω, is equivalent to a Nash factorization. (II) Every semialgebraic subset of Ω 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(Ω) of Nash functions on Ω is generated by its global sections. The case where Ω 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.engjournal articlehttp://www.jstor.org/stable/10.2307/2374953http://www.jstor.orgrestricted access512.7510.22515.171.5Real nullstellensatzextension theoremsetsNash manifoldglobal analytic functionsNash functionsNéron desingularizationapproximation theoremGeometria algebraicaTeoría de conjuntos1201.01 Geometría Algebraica1201.02 Teoría Axiomática de Conjuntos