Jesús María Ruiz Sancho

Approximation in compact Nash manifolds

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 Nerons 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 reviewers opinion this article makes crucial progress in the theory of global Nash functions.

On sequentially compact T2 compactifications

Abstract It is known that a compact space can fail to be sequentially compact. In this paper we consider the following problem: when does a space admit a sequentially compact T 2 compactification? In the first section we develop a method to produce such compactifications, and we apply it in the second section to study the question using coverings. Moreover, we obtain solutions for locally compact T 2 spaces, and for metrizable spaces.