Masahiro Shiota

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.

Nash functions on noncompact Nash manifolds

Abstract Several conjectures concerning Nash functions (including the conjecture that globally irreducible Nash sets are globally analytically irreducible) were proved in Coste et al. (1995) for compact affine Nash manifolds. We prove these conjectures for all affine Nash manifolds.

Piecewise linearization of real-valued subanalytic functions

We show that for a subanalytic function / on a locally compact subanalytic set X there exists a unique subanalytic triangulation (a simplicial complex K , a subanalytic homeomorphism n: K —► X) such that f o na , a 6 K , are linear. Let X be a subanalytic set contained and closed in a Euclidean space. A subanalytic triangulation of X is a pair (K , n) where K isa simplicial complex and n is a subanalytic homeomorphism from the underlying polyhedron K to X. Here we give K a subanalytic structure by realizing K in a Euclidean space so that K is closed in the Euclidean space. The existence of a subanalytic triangulation of X was shown by [2 and 4]. [2 and 4] use induction on the dimension of the ambient space like [5]. Moreover [10] proved the uniqueness up to PL homeomorphism, namely, that if there are two subanalytic triangulations (K ,it) and (K1 ,n) of X then K and K are PL homeomorphic. Our purpose is the following theorem of a unique subanalytic triangulation of a subanalytic function. Theorem 1. Let f be a subanalytic function on X. Then there exists a unique subanalytic triangulation (K ,n) of X such that for every simplex o in K, f o na is linear. Here the uniqueness means that for another subanalytic triangulation (K1, ri) of X with the same property as (K , it) there exists a PL homeomorphism r from K to K such that fonox = fon. I proved this in weaker forms in [9]. I constructed n using the integrations of vector fields. Hence triangulations of subanalytic functions in [9] were of class C only, and the uniqueness did not follow because of the failure of the topological Hauptvermutung [6]. As for this paper, we apply the projection method of [5] to the local proof of Theorem 1. The uniqueness theorem [10] pastes the local subanalytic triangulations and proves globally Theorem 1, which is similar to the proof of the theorem of C°° triangulation of C°° manifold (CairnsWhitehead, e.g. [7]). We use also the Alexander trick (see for its definition [8, 10] and the statement before Lemma 9). Teissier [11] also used the projection Received by the editors September 16, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 32B25; Secondary 32B20.