On $\mathbb{R}$-places and related topics

Abstract

In this survey $K$ will be a formally real field, which means that $-1$ is not a finite sum of squares of elements of $K,$ hence $K$ has characteristic $0$. As often in the literature, we shall write real field insteadof formally real fieldIt is well known from Artin-Schreier theory that such fields are exactly those admitting at least one total order compatible with the field structure. After some background in Real Algebra, we introduce and study the space of $\mathbb{R}$-places. Thereafter, we present other mathematical notions, such as valuation fans, orderings of higher level and the real holomorphy ring. By use of these tools we obtain an outstanding result in Real Algebraic Geometry. Finally we provide some steps towards an abstract theory of $\mathbb{R}$-places.