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 $A$.

The notion of a real semigroup was introduced in [8] to provide a framework for the investigation of the theory of (diagonal) quadratic forms over commutative, unitary, semi-real rings. In this paper we introduce and study an outstanding class of such structures, that we call spectral real semigroups (SRS). Our main results are: (i) The existence of a natural functorial duality between the category of SRSs and that of hereditarily normal spectral spaces; (ii) Characterization of the SRSs as the...

We show some advantages of splitting vector bundles by blowups.

We construct a global system of real analytic coordinates on the real Teichmüller space of a compact real algebraic curve X, using so-called strict uniformization of the real algebraic curve X. A global coordinate system is then obtained via real quasiconformal deformations of the Kleinian subgroup of PGL2(R) obtained as a group of covering transformations of a strict uniformization of X.

Un feuilletage $ℱ$ de codimension un sur une variété orientable $M$ est de Rolle s’il vérifie la propriété suivante : une courbe transverse à $ℱ$ coupe au plus une fois chaque feuille. Soit $Q$ une fonction tapissante sur $M$, i.e. propre et possédant un nombre fini de valeurs critiques. Nous montrons que si l’ensemble des singularités de la restriction de $Q$ aux feuilles de $F$ vérifie certaines propriétés de finitude, alors la restriction de $ℱ$ au complémentaire d’un nombre fini de feuilles possède une structure...

For any subanalytic $C^k$-Whitney field (k finite), we construct its subanalytic $C^k$-extension to ${ℝ}^n$. Our method also applies to other o-minimal structures; e.g., to semialgebraic Whitney fields.

Let V ⊂ ℝⁿ, n ≥ 2, be an unbounded algebraic set defined by a system of polynomial equations $h₁\left(x\right)=\cdots =h_r\left(x\right)=0$ and let f: ℝⁿ→ ℝ be a polynomial. It is known that if f is positive on V then ${f|}_V$ extends to a positive polynomial on the ambient space ℝⁿ, provided V is a variety. We give a constructive proof of this fact for an arbitrary algebraic set V. Precisely, if f is positive on V then there exists a polynomial $h\left(x\right)={\sum }_{i=1}^{r}h{²}_i\left(x\right){\sigma }_i\left(x\right)$, where ${\sigma }_i$ are sums of squares of polynomials of degree at most p, such that f(x) + h(x) > 0 for x...