Nous étudions l'anneau des fonctions rationnelles qui se prolongent par continuité sur R n . Nous établissons plusieurs propriétés algébriques de cet anneau dont un Nullstellensatz fort. Nous étudions les propriétés schématiques associées et montrons une version régulue des théorèmes A et B de Cartan. Nous caractérisons géométriquement les idéaux premiers de cet anneau à travers leurs lieux d'annulation et montrons que les fermés régulus coïn-cident avec les fermés algébriquement constructibles.We study the ring of rational functions admitting a continuous extension to the real affine space. We establish several properties of this ring. In particular, we prove a strong Nullstellensatz. We study the scheme theoretic properties and prove regulous versions of Theorems A and B of Cartan. We also give a geometrical characterization of prime ideals of this ring in terms of their zero-locus and relate them to euclidean closed Zariski-constructible sets.

Abstract. We study the spaces of stable real and quaternionic vector bundles on a real algebraic curve. The basic relationship is established with unitary representations of an extension of Z/2 by the fundamental group. By comparison with the space of real or quaternionic connections, some of the basic topological invariants of these spaces are calculated.

Let X be a rational nonsingular compact connected real algebraic surface. Denote by Aut(X) the group of real algebraic automorphisms of X. We show that the group Aut(X) acts n‐transitively on X, for all natural integers n. As an application we give a new and simpler proof of the fact that two rational nonsingular compact connected real algebraic surfaces are isomorphic if and only if they are homeomorphic as topological surfaces.