Global problems of Nash functions
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 real numbers by an arbitrary real closed field. Again, these problems have been answered in the affirmative by \textit{M. Coste}, \textit{J. M. Ruiz} and \textit{M. Shiota} [J. Reine Angew. Math. 536, 209-235 (2001; Zbl 0981.14028)]. In that paper the authors, using the Tarski-Seidenberg theorem and proving uniform bounds on the complexities of Nash functions, obtain the solution to the already mentioned problems. Also, they prove the idempotency of the real spectrum and discuss conditions for the rings of abstract Nash functions to be noetherian.NEWLINENEWLINEFor the entire collection see [Zbl 0952.00066].
scientific article