# Nombre de composantes connexes d'une vari\'et\'e r\'eelle et R-places

## TL;DR

This paper explores R-places in real algebraic geometry, establishing their connections with orderings, valuations, and the topology of real varieties, and provides explicit formulas for counting connected components of smooth projective real varieties.

## Contribution

It introduces R-places and their relationship with orderings and valuations, and derives explicit formulas for the number of connected components of real algebraic varieties.

## Key findings

- Explicit formula for connected components of smooth projective real varieties
- Connections between R-places, orderings, and valuations
- Open problems in the theory of R-places

## Abstract

The purpose of this paper is to present results and open problems related to R-places. The first section recalls basic facts, the second introduces R-places and their relationship with orderings and valuations. The third part involves Real Algebraic Geometry and gives results proved using the space of R-places. Theorem 14 gives explicitly, in terms of the function field of the variety, the number of connected components of a non-empty smooth projective real variety. The fourth and fifth parts are devoted to the links with the real holomorphy rings and the valuation fans. Then we present an approach to abstract real places and conclude with some open questions.