The notion of spaces of orderings was introduced by Murray Marshall in the 1970's and provides an abstract framework for studying orderings on fields and the reduced theory of quadratic forms over fields. The structure of a space of orderings (X, G) is completely determined by the group structure of G and the quaternary relation (a_1, a_2) = (a_3, a_4) on G -- the groups with additional structure arising in this way are called reduced special groups. The theory of reduced special groups, in turn, can be conveniently axiomatized in the first order language L_SG. Numerous important notions in this theory, such as isometry, isotropy, or being an element of a value set of a form, make an extensive use of, so called, positive primitive formulae in the language L_SG. Therefore, the following question, which can be viewed as a type of very general and highly abstract local-global principle, is of great importance:<p>Is it true that if a positive primitive formula holds in every finite subspace of a space of orderings, then it also holds in the whole space?<p>This problem is now known as the pp conjecture. The answer to this question is affirmative in many cases, although it has always seemed unlikely that the conjecture has a positive solution in general. In this thesis, we discuss, discovered by us, first counterexamples for which the pp conjecture fails. Namely, we classify spaces of orderings of function fields of rational conics with respect to the pp conjecture, and show for which of such spaces the conjecture fails, and then we disprove the pp conjecture for the space of orderings of the field R(x,y). Some other examples, which can be easily obtained from the developed theory, are also given. In addition, we provide a refinement of the result previously obtained by Vincent Astier and Markus Tressl, which shows that a pp formula fails on a finite subspace of a space of orderings, if and only if a certain family of formulae is verified.

The notion of spaces of orderings was introduced by Murray Marshall in the 1970's and provides an abstract framework for studying orderings on fields and the reduced theory of quadratic forms over fields. The structure of a space of orderings (X, G) is completely determined by the group structure of G and the quaternary relation (a_1, a_2) = (a_3, a_4) on G -- the groups with additional structure arising in this way are called reduced special groups. The theory of reduced special groups, in turn, can be conveniently axiomatized in the first order language L_SG. Numerous important notions in this theory, such as isometry, isotropy, or being an element of a value set of a form, make an extensive use of, so called, positive primitive formulae in the language L_SG. Therefore, the following question, which can be viewed as a type of very general and highly abstract local-global principle, is of great importance:<p>Is it true that if a positive primitive formula holds in every finite subspace of a space of orderings, then it also holds in the whole space?<p>This problem is now known as the pp conjecture. The answer to this question is affirmative in many cases, although it has always seemed unlikely that the conjecture has a positive solution in general. In this thesis, we discuss, discovered by us, first counterexamples for which the pp conjecture fails. Namely, we classify spaces of orderings of function fields of rational conics with respect to the pp conjecture, and show for which of such spaces the conjecture fails, and then we disprove the pp conjecture for the space of orderings of the field R(x,y). Some other examples, which can be easily obtained from the developed theory, are also given. In addition, we provide a refinement of the result previously obtained by Vincent Astier and Markus Tressl, which shows that a pp formula fails on a finite subspace of a space of orderings, if and only if a certain family of formulae is verified.

Les espaces d'ordres abstraits sont introduits par M. Marshall dans les années 70, dans la perspective d'offrir un cadre abstrait à l'étude des formes quadratiques. Vers le début des années 90, les travaux de M. Dickmann, L. de Lima et de F. Miraglia, ont donné naissance à la version duale des groupes spéciaux. Le premier thème que nous traiterons est la caractérisation des points d'un espace d'ordres du corps de fonctions d'une variété réelle, nous reprendrons un résultat de Brumfiel affirmant l'existence d'une correspondance entre ces ordres et des ultrafiltres de semi-algébriques. Nous appliquerons ceci au corps R(x,y). Suivra la caractérisation des ordres de ce corps à travers la notion de demi-branche de Bézout. Le second thème traite des principes locaux-globaux généralisés (ou Conjecture pp). Le premier résultat de la thèse porte sur la séparation des constructibles et sur la principalité des basiques. Nous montrerons que le langage des groupes spéciaux nous offre une vision claire du fait que ces principes découlent trivialement du principe de l'isotropie étendu. Le second résultat traite des contre-exemples à la conjecture dans le cas de la conique rationnelle donnée par l'équation x2+y2=3. Le dernier résultat (le plus important), aborde la conjecture pp dans le cadre du corps R(x,y). Nous nous intéresserons à des familles de polynômes vérifiant certaines conditions géométriques et montrerons que toute formule pp, ayant ses paramètres dans cette famille, vérifie un principe local-global. Nous les baptiserons formules V-universelles. Nous clorons le dernier chapitre par deux méthodes de construction. / Spaces of orderings were introduced in the last of 70s by M. Marshall, in order to provide an abstract framework to the study of a generalized theory of quadratic forms. In the 90s, the work of M. Dickmann , joined by his student L. Lima and his collaborator F. Miraglia , gave rise to the dual version of this theory, the special groups. First, we focus on the characterization of points of a space of orderings in special the case of function fields of real varieties, we review a Brumfiel?s result proving the existence of a one-to-one correspondence between points of such a space and some family of ultrafilters of semi-algebraic sets of the variety. We apply this to the case where the field is R(x,y). Another characterization uses the concept of Bezout?s half-branch. The second topic deals with a generalized local-global principle, known as the pp-conjecture. The first new result of this thesis focuses on the separation of constructibles and the principality of basics. We show that the first order language traduces well these properties and offers a clear vision that they derive trivially from extended isotropy theorem. The second result presents a generalized counter-example to the pp-conjecture in the case of the rational conic defined by the equation x2 + y2 = 3. The last result (most important), addresses the pp-conjecture in the context of the space of orderings of the field R(x,y). We will focus on families of polynomials satisfying certain geometric conditions and show that any pp-formula, with its parameters in this family, verifies a local-global principle. We baptize them V-universal formulas. We close the last chapter by two construction methods.