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Approximation in compact Nash manifolds

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Coste, M. and Ruiz Sancho, Jesús María and Shiota, Masahiro (1995) Approximation in compact Nash manifolds. American Journal of Mathematics, 117 (4). pp. 905-927. ISSN 0002-9327

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Official URL: http://www.jstor.org/stable/10.2307/2374953


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Abstract

Let 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.


Item Type:Article
Uncontrolled Keywords:Real nullstellensatz; extension theorem; sets; Nash manifold; global analytic functions; Nash functions; Néron desingularization; approximation theorem
Subjects:Sciences > Mathematics > Algebraic geometry
Sciences > Mathematics > Set theory
ID Code:19997
Deposited On:20 Feb 2013 16:54
Last Modified:12 Dec 2018 15:13

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