Academic literature on the topic 'Trichotomie de Zilber'

AbstractThis paper ties together much of the model theory of the last 50 years. Shelah's attempts to generalize the Morley theorem beyond first order logic led to the notion of excellence, which is a key to the structure theory of uncountable models. The notion of Abstract Elementary Class arose naturally in attempting to prove the categoricity theorem for Lω1,ω(Q). More recently, Zilber has attempted to identify canonical mathematical structures as those whose theory (in an appropriate logic) is categorical in all powers. Zilber's trichotomy conjecture for first order categorical structures was refuted by Hrushovski, by the introducion of a special kind of Abstract Elementary Class. Zilber uses a powerful and essentailly infinitary variant on these techniques to investigate complex exponentiation. This not only demonstrates the relevance of Shelah's model theoretic investigations to mainstream mathematics but produces new results and conjectures in algebraic geometry.

AbstractThis paper began as a generalization of a part of the author's PhD thesis about ACFA and ended up with a characterization of groups definable in TA. The thesis concerns minimal formulae of the form x Є A ∧ σ(x) = f(x) for an algebraic curve A and a dominant rational function f: A → σ(A). These are shown to be uniform in the Zilber trichotomy, and the pairs (A, f) that fall into each of the three cases are characterized. These characterizations are definable in families. This paper covers approximately half of the thesis, namely those parts of it which can be made purely model-theoretic by moving from ACFA, the model companion of the class of algebraically closed fields with an endomorphism, to TA, the model companion of the class of models of an arbitrary totally-transcendental theory T with an injective endomorphism, if this model-companion exists. A TA analog of the characterization of groups definable in ACFA is obtained in the process. The full characterization of the cases of the Zilber trichotomy in the thesis is obtained from these intermediate results with heavy use of algebraic geometry.

We classify all possible combinatorial geometries associated with one-dimensional difference equations, in any characteristic. The theory of difference fields admits a proper interpretation of itself, namely the reduct replacing the automorphism by its nth power. We show that these reducts admit a successively smoother theory as n becomes large; and we succeed in defining a limit structure to these reducts, or rather to the structure they induce on one-dimensional sets. This limit structure is shown to be a Zariski geometry in (roughly) the sense of Hrushovski and Zil'ber. The trichotomy is thus obtained for the limit structure as a consequence of a general theorem, and then shown to be inherited by the original theory. 2000 Mathematical Subject Classification: 03C60; (primary) 03C45, 03C98, 08A35, 12H10 (secondary)

AbstractIn this paper we prove a form of the Zilber's trichotomy conjecture for reducts of algebraically closed valued fields of characteristic 0 which are expansions of the valued vector space structure. We prove first that a non-modular reduct of a nontrivially valued algebraically closed field containing the valued vector space structure defines a non-semilinear curve. Then we show that the expansion of such a reduct by a non-semilinear curve defines multiplication on a nonempty open set.

L'objet de cette thèse est l'étude de certains aspects des structures géométriques C-minimales, i. E. Les structures généralisant les structures ultramétriques, dans lesquelles la clôture algébrique perme de définir une notion de dimension et dans lesquelles tout ensemble définissable en dimension 1 a une « forme simple ». L'idée principale derrière les chapitres 1-3 de cette thèse est d'essayer de construire des structures algébriques(groupes ou corps infinis) dans de telles structures en partant d'hypothèses modèle théoriques. Cette problématique a été introduite par Zilber à propos des structures fortement minimales, et des résultats de trichotomie de Zilber ont déjà été trouvés pour les géométries de Zariski et les structures o-minimales entre autres. Dans le premier chapitre, on classifie à équivalence élémentaire près tous les espaces vectoriels C-minimaux, qui sont les exemples standard de structures C-minimales dites localement modulaires. Dans le chapitre deux, on démontre qu'on peut construire un groupe type-définissable infini dans toute structure géométrique C-minimale localement modulaire non triviale et S\aleph_1S-saturée. . Dans le chapitre trois, on montre que pour tout corps value algébriquement clos (une structure C-minimale géométrique typique), dans toute expansion algébrique non modulaire de la structure d'espace vectoriel value on peut définir la multiplication sur un ouvert. Le dernier chapitre étudie les imaginaires dans les espaces vectoriels C-minimaux. On trouve des codes pour un certain type fonctions, mais l'objectif final, qui est l'élimination faible des imaginaires aux sortes des clusters, n'est pas atteint
The main object of this thesis is the study of certain aspects of geometric C-minimal structures, i. E. Structures generalizing ultrametric structures, in which algebraic closure defines a notion of dimension on definable sets, and in which, uniformly in parameters, every definable subset of 1-space has a « simple form ». The main idea behind chapters 1-3 of this thesis is trying to construct algebraic structures (such as infinite groups and fields) in such structures which verify moreover some model theoretic assumptions. This kind of problems has already been suggested by Zilber for strongly minimal structures, and many results of Zilber's trichotomy are proved notably for Zariski geometries and o-minimal structures. In the first chapter, we classify up to elementarly equivalence, ail C-minimal vector spaces, which are the typical examples of the so called C-minimal locally modular structures. In the second chapter, we show how to construct an infinite type-definable group in any geometric non trivial locally modular C-minimal structure which is S\aleph_1S-saturated. In the third chapter, we show that for any algebraically closed valued field (a typical example of a geometric C-minimal structure), we can define multiplication on an open set in any non modular algebraic expansion of the vector space structure. Chapter four studies imaginaries in C-minimal vector spaces. We show how to find codes for some definable functions, without succeeding in the reach the final objective, which is weak elimination of imaginaries to the sort of clusters

Le travail de cette thèse a pour objet les interactions entre deux approches d'étude des équations différentielles: la théorie des modèles des corps différentiellement clos d'une part et l'étude dynamique des équations différentielles réelles d'autre part. Dans le premier chapitre, on présente un formalisme d'algèbre différentielle, en termes de D-schémas à la Buium au-dessus du corps des nombres réels (muni de la dérivation triviale), qui permet de rendre compte de ces deux approches d'étude en même temps. Le résultat principal est un critère d'orthogonalité aux constantes pour le type générique d'une D-variétés réelle absolument irréductible, basé sur la dynamique topologique de son flot réel analytique associé. Le deuxième chapitre est consacré aux équations différentielles algébriques décrivant le flot géodésique de variétés algébriques réelles munies de 2-formes symétriques non-dégénérées. A l'aide du critère précédent, on démontre un théorème d'orthogonalité aux constantes "en courbure strictement négative'', s'appuyant sur les résultats d'Anosov et de ses successeurs concernant la dynamique topologique - la propriété de mélange topologique faible - du flot géodésique d'une variété riemannienne compacte à courbure strictement négative. En dimension 2, on conjecture en fait une description plus précise - son type générique est minimal de prégéométrie triviale - de la structure associée aux équations différentielles géodésiques unitaires. On présente, dans le troisième chapitre, des motivations et des résultats partiels concernant cette conjecture
This thesis is dedicated to studying the interactions between two different approaches regarding differential equations: the model-theory of differentially closed fields on the one side and the dynamical analysis of real differential equations, on the other side. In the first chapter, we present a formalism from differential algebra, in terms of D-varieties à la Buium over the field of real numbers (endowed with the trivial derivation), that allows one to realise both approaches at the same time. The main result is a criterion of orthogonality to the constants, based on the topological dynamic of its associated real analytic flow. The second chapter is dedicated to the algebraic differential equations describing the (unitary) geodesic flow of a real algebraic variety endowed with an algebraic, non-degenerated symmetric 2-form. Using the previous criterion, we prove a theorem of orthogonality to the constants "in negative curvature'', that relies on the results of Anosov and of his followers, regarding the topological dynamic - the weakly mixing topological property - for the geodesic flow of a compact Riemannian manifold with negative curvature. In dimension 2, we conjecture a more precise description - its generic type is minimal and has a trivial pregeometry- for the structure associated to the unitary geodesic equation. In the third chapter, we present some motivations and partial results on this conjecture

This thesis explores the geometric principles underlying many of the known Trichotomy Theorems. The main aims are to unify the field construction in non-linear o-minimal structures and generalizations of Zariski Geometries as well as to pave the road for completely new results in this direction. In the first part of this thesis we introduce a new axiomatic framework in which all the relevant structures can be studied uniformly and show that these axioms are preserved under elementary extensions. A particular focus is placed on the study of a smoothness condition which generalizes the presmoothness condition for Zariski Geometries. We also modify Zilber's notion of universal specializations to obtain a suitable notion of infinitesimals. In addition, families of curves and the combinatorial geometry of one-dimensional structures are studied to prove a weak trichotomy theorem based on very weak one-basedness. It is then shown that under suitable additional conditions groups and group actions can be constructed in canonical ways. This construction is based on a notion of ``geometric calculus'' and can be seen in close analogy with ordinary differentiation. If all conditions are met, a definable distributive action of one one-dimensional type-definable group on another are obtained. The main result of this thesis is that both o-minimal structures and generalizations of Zariski Geometries fit into this geometric framework and that the latter always satisfy the conditions required in the group constructions. We also exhibit known methods that allow us to extract fields from this. In addition to unifying the treatment of o-minimal structures and Zariski Geometries, this also gives a direct proof of the Trichotomy Theorem for "type-definable" Zariski Geometries as used, for example, in Hrushovski's proof of the relative Mordell-Lang conjecture.