### A local Bernstein inequality on real algebraic varieties.

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We characterize composition operators on spaces of real analytic functions which are open onto their images. We give an example of a semiproper map φ such that the associated composition operator is not open onto its image.

Some representations of Nash functions on continua in ℂ as integrals of rational functions of two complex variables are presented. As a simple consequence we get close relations between Nash functions and diagonal series of rational functions.

We prove that any divisor $Y$ of a global analytic set $X\subset {\mathbb{R}}^{n}$ has a generic equation, that is, there is an analytic function vanishing on $Y$ with multiplicity one along each irreducible component of $Y$. We also prove that there are functions with arbitrary multiplicities along $Y$. The main result states that if $X$ is pure dimensional, $Y$ is locally principal, $X/Y$ is not connected and $Y$ represents the zero class in ${H}_{q-1}^{\infty}(X,{\mathbb{Z}}_{2})$ then the divisor $Y$ is globally principal.

We study here several finiteness problems concerning affine Nash manifolds $M$ and Nash subsets $X$. Three main results are: (i) A Nash function on a semialgebraic subset $Z$ of $M$ has a Nash extension to an open semialgebraic neighborhood of $Z$ in $M$, (ii) A Nash set $X$ that has only normal crossings in $M$ can be covered by finitely many open semialgebraic sets $U$ equipped with Nash diffeomorphisms $({u}_{1},\cdots ,{u}_{m}):U\to {\mathbb{R}}^{m}$ such that $U\cap X=\{{u}_{1}\cdots {u}_{r}=0\}$, (iii) Every affine Nash manifold with corners $N$ is a closed subset of an affine Nash manifold...

This is a survey on the history of and the solutions to the basic global problems on Nash functions, which have been only recently solved, namely: separation, extension, global equations, Artin-Mazur description and idempotency, also noetherianness. We discuss all of them in the various possible contexts, from manifolds over the reals to real spectra of arbitrary commutative rings.

Let $M\subset {\mathbb{R}}^{n}$ be a real-analytic submanifold and H(M) the algebra of real analytic functions on M. If K ⊂ M is a compact subset we consider ${S}_{K}=f\in H\left(M\right)\left|f\right(x)\ne 0forallx\in K$; ${S}_{K}$ is a multiplicative subset of $H\left(M\right)$. Let ${S}_{K}^{-1}H\left(M\right)$ be the localization of H(M) with respect to ${S}_{K}$. In this paper we prove, first, that ${S}_{K}^{-1}H\left(M\right)$ is a regular ring (hence noetherian) and use this result in two situations: 1) For each open subset $\Omega \subset {\mathbb{R}}^{n}$, we denote by O(Ω) the subalgebra of H(Ω) defined as follows: f ∈ O(Ω) if and only if for all x ∈ Ω, the germ of f at x, ${f}_{x}$, is algebraic...

We discuss some conditions which guarantee that the Kuratowski limit of a sequence of analytic sets is a Nash set.

Let $X\subset {{\u2102}^{n}}^{n}$ be a closed real-analytic subset and put$$\mathcal{A}:=\{z\in X\mid \exists \phantom{\rule{4pt}{0ex}}A\subset X,\phantom{\rule{4.0pt}{0ex}}\text{germ}\phantom{\rule{4.0pt}{0ex}}\text{of}\phantom{\rule{4.0pt}{0ex}}\text{a}\phantom{\rule{4.0pt}{0ex}}\text{complex-analytic}\phantom{\rule{4.0pt}{0ex}}\text{set,}\phantom{\rule{4.0pt}{0ex}}z\in A,\phantom{\rule{0.166667em}{0ex}}{dim}_{z}A\>0\}$$This article deals with the question of the structure of $\mathcal{A}$. In the main result a natural proof is given for the fact, that $\mathcal{A}$ always is closed. As a main tool an interesting relation between complex analytic subsets of $X$ of positive dimension and the Segre varieties of $X$ is proved and exploited.

We deal with a reduction of power series convergent in a polydisc with respect to a Gröbner basis of a polynomial ideal. The results are applied to proving that a Nash function whose graph is algebraic in a "large enough" polydisc, must be a polynomial. Moreover, we give an effective method for finding this polydisc.