Experience shows that in geometric situations the separating ideal associated with two orderings of a ring measures the degree of tangency of the corresponding ultrafilters of semialgebraic sets. A related notion of separating ideals is introduced for pairs of valuations of a ring. The comparison of both types of separating ideals helps to understand how a point on a surface is approached by different half-branches of curves.

Given global semianalytic sets A and B, we define a minimal analytic set N such that Ā∖N and B̅∖N can be separated by an analytic function. Our statement is very similar to the one proved by Bröcker for semialgebraic sets.

It is known that for $C^{\infty }$ determining sets Markov’s property is equivalent to Bernstein’s property. We are interested in finding a generalization of this fact for sets which are not $C^{\infty }$ determining. In this paper we give examples of sets which are not $C^{\infty }$ determining, but have the Bernstein and generalized Markov properties.

The Siciak extremal function establishes an important link between polynomial approximation in several variables and pluripotential theory. This yields its numerous applications in complex and real analysis. Some of them can be found on a rich list drawn up by Klimek in his well-known monograph "Pluripotential Theory". The purpose of this paper is to supplement it by applications in constructive function theory.

We define open book structures with singular bindings. Starting with an extension of Milnor’s results on local fibrations for germs with nonisolated singularity, we find classes of genuine real analytic mappings which yield such open book structures.

Soit $P\in ℝ[X_1,...,X_n]$ un polynôme. On appelle série de Dirichlet associée à $P$ la fonction : $s\mapsto Z(P;s)=\sum _{m\in {\mathbb{N}}^{*n}}P(m{)}^{-s}(s\in ℂ)$. Dans cet article nous étudions l’existence et les propriétés du prolongement méromorphe d’une telle série sous l’hypothèse qu’il existe $B\in ]0,1[$ tel que : i) $P\left(x\right)\to +\infty$ quand $\left|\right|x\left|\right|\to +\infty$ et $x\in [B,+\infty {[}^n$ et ii) $d\left(Z\right(P),[B,+\infty {[}^n)\>0$ où $Z\left(P\right)=\{z\in {ℂ}^n|P\left(z\right)=0\}$. Cette hypothèse est probablement optimale et en tout cas contient strictement toutes les classes de polynômes déjà traitées antérieurement. Sous cette hypothèse nos principaux résultats sont : l’existence du prolongement méromorphe au plan...

We give some approximation theorems in the Whitney topology for a general class of analytic fiber bundles. This leads to a classification theorem which generalizes the classical ones.

It is proved that the set of smooth points of a semialgebraic set is semialgebraic.

Define for a smooth compact hypersurface $M^n$ of ${\mathbf{R}}^{n+1}$ its crumpleness $\kappa \left(M^n\right)$ as the ratio ${diam}_{{\mathbf{R}}^{n+1}}(M^n)/r(M^n)$, where $r\left(M^n\right)$ is the distance from $M^n$ to its central set. (In other words, $r\left(M^n\right)$ is the maximal radius of an open non-selfintersecting tube around $M^n$ in ${\mathbf{R}}^{n+1}.)$We prove that any $n$-dimensional non-singular compact algebraic hypersurface of degree $d$ is rigidly isotopic to an algebraic hypersurface of degree $d$ and of crumpleness $\le exp\left(c\right(n\left)d^{\alpha \left(n\right)d^{n+1}}\right)$. Here $c\left(n\right)$, $\alpha \left(n\right)$ depend only on $n$, and rigid isotopy means an isotopy passing only through hypersurfaces of degree...

In this paper we deal with a best approximation of a vector with respect to a closed semi-algebraic set C in the space ℝⁿ endowed with a semi-algebraic norm ν. Under additional assumptions on ν we prove semi-algebraicity of the set of points of unique approximation and other sets associated with the distance to C. For C irreducible algebraic we study the critical point correspondence and introduce the ν-distance degree, generalizing the notion developed by other authors for the Euclidean norm. We...

Nel presente lavoro si studiano le applicazioni polinomiali proprie $$\phi :{\mathbb{R}}^n\to {\mathbb{R}}^q.$$ In particolare si prova: 1) se $\phi :{\mathbb{R}}^n\to \mathbb{R}$ è un'applicazione polinomiale tale che ${\phi }^{-1}\left(y\right)$ è compatto per ogni $y\in \mathbb{R}$, allora $\phi$ è propria; 2) se $\phi :{\mathbb{R}}^n\to {\mathbb{R}}^q$ è polinomiale a fibra compatta e $\phi \left({\mathbb{R}}^n\right)$ è chiuso in ${\mathbb{R}}^q$ allora $\phi$ è propria; 3) l'insieme delle applicazioni polinomiali proprie di ${\mathbb{R}}^n$ in ${\mathbb{R}}^q$ è denso, nella topologia $C^{\mathrm{\infty }}$, nello spazio delle applicazioni $C^{\mathrm{\infty }}$ di ${\mathbb{R}}^n$ in ${\mathbb{R}}^q$.

The spectrum of the Laplace operator on algebraic and semialgebraic subsets $A$ in ${\mathbf{R}}^N$ is studied and the number of small eigenvalues is estimated by the degree of $A$.