Functions de Nash sur les variétés affines. (Nash functions on affine varieties) (Q1089398)

The real spectrum of a ring is equipped with a structure sheaf which, if the ring is \({\mathbb{R}}[X_ 1,...,X_ n]\) coincides with the sheaf of Nash functions over \({\mathbb{R}}^ n\). If V is an algebraic set over an arbitrary real closed field and A is its coordinate ring, we observe that the structure sheaf over \(Spec_ rA\) does not necessarily coincide with the common sheaf of Nash functions over V. The study of the relationship between them was proposed by \textit{M.-F. Roy} in Géométrie algébrique réelle et formes quadratiques, Journ. Soc. Math. Fr., Univ. Rennes 1981, Lect. Notes Math. 959, 406-432 (1982; Zbl 0497.14009)], as well as the conjecture that the set of points of V for which the stalks of both sheaves are isomorphic is the set of quasi- regular points of V, i.e. the set of points whose complex branches are the complexification of the real ones. We give a surjective sheaf morphism between these sheaves and a necessary and sufficient condition, for a semialgebraic open subset U of V, in order to have a surjective ring morphism on the global sections over the constructible Ũ associated to U. We also solve the conjecture about the quasi-regular points. Moreover the semialgebraicity of the set of quasi- regular points of V is proved.