Global problems of Nash functions
type: Item
instance of: scholarly article
zbMATH Open document ID: 0986.14038
publication date: 2002-05-29T00:00:00Z
Mathematics Subject Classification ID: 14P20
Mathematics Subject Classification ID: 32C07
Mathematics Subject Classification ID: 58A07
zbMATH DE Number: 1574627
zbMATH Keywords: approximation theorem
zbMATH Keywords: Nash manifolds
zbMATH Keywords: separation problem
zbMATH Keywords: factorization problem
zbMATH Keywords: extension problem
zbMATH Keywords: global equations
zbMATH Keywords: complexity of Nash functions
author: Jesús M. Ruiz
MaRDI profile type: Publication
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title: Global problems of Nash functions
review text: Nash functions are analytic functions which are algebraic over the polynomials. The paper under review is an excellent survey on some problems concerning Nash functions and the methods for their solutions. NEWLINENEWLINENEWLINEThe problems consider here are: separation, factorization, global equations and extension. To be more precise, the separation problem asks whether a prime ideal of Nash functions on a Nash manifold \(M \subset \mathbb R^n\) has a prime extension to the ring of global analytic functions on \(M\). In the factorization problem the question is whether a Nash function \(f\) with a factorization \(f = f_1 f_2\) as a product of analytic functions has a similar factorization \(f = g_1 g_2\) as a product of Nash functions. Finally, whether a finite ideal sheaf is generated by its global sections and whether every Nash function on \(M\) is the restriction of a Nash function on \(\mathbb R^n\) to \(M\) are the global equation and the extension problems, respectively. NEWLINENEWLINENEWLINEIn the case of a compact Nash manifold \(M\), using the so-called Néron desingularization an important approximation theorem is proved and then these problems are solved in the affirmative [see the paper of \textit{M. Coste}, \textit{J. M. Ruiz} and \textit{M. Shiota}, Am. J. Math. 117, No. 4, 905-927 (1995; Zbl 0873.32007)]. NEWLINENEWLINENEWLINEWhen \(M\) is not necessarily compact \textit{M. Coste}, \textit{J. M. Ruiz} and \textit{M. Shiota} [Compos. Math. 103, No. 1, 31-62 (1996; Zbl 0885.14029)] showed that these problems are equivalent. After that \textit{M. Coste} and \textit{M. Shiota} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 33, No. 1, 139-149 (2000; Zbl 0981.14027)] solved the global equation problem. The proof of \textit{M. Coste} and \textit{M. Shiota} relies in a semialgebraic version of Thom's first isotopy lemma proved by these two authors. NEWLINENEWLINENEWLINEThe same problems have been considered in a more general setting, namely, replacing the field of re…
reviewed by: Antonio Diaz-Cano
description: scientific article