We show that in the class of compact sets K in ${ℝ}^n$ with an analytic parametrization of order m, the sets with Zariski dimension m are exactly those which admit a Bernstein (or a van der Corput-Schaake) type inequality for tangential derivatives of (the traces of) polynomials on K.

Applying general results on separation of semialgebraic sets and spaces of orderings, we produce a catalogue of all possible geometric obstructions for separation of 3-dimensional semialgebraic sets and give some hints on how separation can be made decidable.

We establish, for smooth projective real curves, an analogue of the classical Clifford inequality known for complex curves. We also study the cases when equality holds.

In 1988 it was proved by the first author that the closure of a partially semialgebraic set is partially semialgebraic. The essential tool used in that proof was the regular separation property. Here we give another proof without using this tool, based on the semianalytic L-cone theorem (Theorem 2), a semianalytic analog of the Cartan-Remmert-Stein lemma with parameters.

Let ${ℳ}_g$ be the moduli space of smooth complex projective curves of genus $g$. Here we prove that the subset of ${ℳ}_g$ formed by all curves for which some Brill-Noether locus has dimension larger than the expected one has codimension at least two in ${ℳ}_g$. As an application we show that if $X\in {ℳ}_g$ is defined over $ℝ$, then there exists a low degree pencil $uX\to {\mathbb{P}}^1$ defined over $ℝ$.

We provide a simple characterization of codimension two submanifolds of ${\mathbb{P}}^n\left(ℝ\right)$ that are of algebraic type, and use this criterion to provide examples of transcendental submanifolds when $n\ge 6$. If the codimension two submanifold is a nonsingular algebraic subset of ${\mathbb{P}}^n\left(ℝ\right)$ whose Zariski closure in ${\mathbb{P}}^n\left(ℂ\right)$ is a nonsingular complex algebraic set, then it must be an algebraic complete intersection in ${\mathbb{P}}^n\left(ℝ\right)$.

Soit $\Gamma$ un groupe de type fini non élémentaire. On note $D^n\left(\Gamma \right)$ l’ensemble des structures hyperboliques de dimension $n$ sur $\Gamma$. $D^n\left(\Gamma \right)$ peut se réaliser comme fermé dans un espace semi-algébrique qui admet une compactification naturelle par le spectre réel. On note ${\overline{D^n\left(\Gamma \right)}}^{\mathrm{sp}}$ le compactifié via le $\underset{\_}{spectre}$ réel de $D^n\left(\Gamma \right)$. L’objet de cet article est de décrire les points ajoutés dans ${\overline{D^n\left(\Gamma \right)}}^{\mathrm{sp}}$. La compactification obtenue de cette manière permet d’interpréter “les points frontières” comme des représentations de $\Gamma$ dans ${\mathrm{SO}}_F^{+}(n,1)$ où $F(\supset ℝ)$ est un corps réel...

On démontre la formule d’orientations complexes pour les $M$-courbes dans ${ℝP}^2$ de degré $4d+1$ ayant $4$ nids. Cette formule généralise celle pour les $M$-courbes à nid profond. C’est un pas vers la classification des $M$-courbes de degré $9$.

This paper is devoted to the computation of distance to set, called S, defined by polynomial equations. First we consider the case of quadratic systems. Then, application of results stated for quadratic systems to the quadratic equivalent of polynomial systems (see [5]), allows us to compute distance to semi-algebraic sets. Problem of computing distance can be viewed as non convex minimization problem: $d(u,S)={inf}_{x\in S}{\parallel x-u\parallel }^2$, where u is in ${ℛ}^n$. To have, at least, lower approximation of distance, we consider the dual...