On the real spectrum of a ring of global analytic functions

Abstract

We present here some applications of the theory of real spectra of excellent rings to the ring of global analytic functions on a compact real analytic manifold. Section 1 contains the facts of the theory that shall be used in the sequel. Section 2 describes the good relationship between global semianalytic subsets of the manifold and constructible subsets of the real spectrum of the ring of global analytic functions. This leads to the solution of Hilbert's 17th problem, to the real Nullstellensatz and to the finiteness theorems, all in this global analytic setting. Finally, Section 3 gives a quick overview on several questions related to connectedness, either of constructible sets or of global semianalytic sets