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Model Theory of Fields and Heights Haydar Göral To cite this version: Haydar Göral. Model Theory of Fields and Heights. General Mathematics [math.GM]. Université Claude Bernard - Lyon I, 2015. English. NNT : 2015LYO10087. tel-01184906v3 HAL Id: tel-01184906 https://hal.archives-ouvertes.fr/tel-01184906v3 Submitted on 13 Nov 2015 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Universit´e Claude Bernard Lyon 1 Ecole´ doctorale InfoMath, ED 512 Sp´ecialit´e:Math´ematiques N. d’ordre 87-2015 Model Theory of Fields and Heights La Th´eorie des Mod`eles des Corps et des Hauteurs Th`ese de doctorat Soutenue publiquement le 3 Juillet 2015 par Haydar G¨oral devant le Jury compos´ede: Mme Fran¸coise DELON CNRS Institut de Math´ematiques de Jussieu, Paris Rapporteuse M. Ayhan GUNAYDIN¨ Universit´e des Beaux-Arts Mimar-Sinan, Istanbul Examinateur M. Piotr KOWALSKI Universit´edeWroclaw, Pologne Rapporteur M. Amador MARTIN-PIZARRO CNRS, Institut Camille Jordan, Villeurbanne Directeur de th`ese M. Frank WAGNER Universit´e Claude Bernard Lyon 1 Directeur de th`ese Model Theory of Fields and Heights La Th´eorie des Mod`eles des Corps et des Hauteurs 3 Juillet 2015 Haydar G¨oral Th`ese de doctorat Model Theory of Fields and Heights Abstract : In this thesis, we deal with the model theory of algebraically closed fields expanded by predicates to denote either elements of small height or multiplicative subgroups satisfying a diophantine condition. The questions we consider belong to the area of stability theory. First, we investigate an algebraically closed field with a distinguished multiplicative subgroup satisfying the Mann property. We characterize the independence which enables us to characterize definable and interpretable groups in the pair. Then we study the model theory of algebraically closed fields expanded by a proper algebraically closed subfield and a multiplicative subgroup. We characterize definable and interpretable groups in this triple. We also consider the set of algebraic numbers with elements of small height and we show that this theory is not simple and has the independence property. We then relate the simplicity of a certain pair with Lehmer’s conjecture. Finally, we apply nonstandard analysis to prove the existence of certain height bounds on the complexity of the coefficients of some polynomials. This allows us to characterize the ideal membership of a given polynomial. Moreover, we obtain a bound for the logarithmic height function, which enables us to test the primality of an ideal. Keywords : Model Theory ; Stability ; Field Theory ; Height Functions. La Th´eorie des Mod`eles des Corps et des Hauteurs R´esum´e:Dans cette th`ese, nous traitons la th´eorie des mod`eles de corps alg´ebriquement clos ´etendu par pr´edicats pour d´esigner soit des ´el´ements de hauteur born´ee, soit des sous-groupes multiplicatifs satisfaisant une condition diophantienne. Les questions que nous consid´erons appartiennent au domaine de la th´eorie de la stabilit´e. Tout d’abord, nous examinons un corps alg´ebriquement clos avec un sous-groupe multiplicatif distingu´e qui satisfait la propri´et´e Mann. Ainsi, nous caract´erisons l’ind´ependance qui nous permet de caract´eriser les groupes d´efinissables et les groupes interpr´etables dans la paire. Ensuite, nous ´etudions les corps alg´ebriquement clos ´etendu par un sous-corps propre alg´ebriquement clos et un sous- groupe multiplicatif. Nous caract´erisons les groupes d´efinissables et interpr´etables dans cette triple. Nous consid´erons aussi l’ensemble des nombres alg´ebriques avec des ´el´ements de petite hauteur et nous montrons que cette th´eorie n’est pas simple et a la propri´et´e d’ind´ependance. Puis, nous nous rapportons `ala simplicit´e d’une certaine paire avec la conjecture de Lehmer. Enfin, nous appliquons l’analyse nonstandard pour prouver l’existence de certaines bornes de hauteur de la complexit´e des coefficients de certains polynˆomes. Cela nous permet de caract´eriser l’appartenance id´eale d’un polynˆome donn´e. De plus, nous obtenons une borne pour la fonction de la hauteur logarithmique, ce qui nous permet de tester la primalit´e d’un id´eal. Mots cl´es : Th´eorie des Mod`eles ; Stabilit´e; Th´eorie de Corps ; Fonctions Hauteur. Image en couverture : Ecriture `a la main d´ecrivant ma th`ese. Cr´edit image : Olu¸sArı¨oz. Abstract In this thesis, we deal with the model theory of algebraically closed fields expanded by predicates to denote either elements of small height or multiplicative subgroups satisfying a diophantine condition. The questions we consider belong to the area of model theory and stability theory. In Chapter 2, we investigate an algebraically closed field with a distinguished multiplicative subgroup satisfying the Mann property. The model theory of this pair was first studied in the papers of B. Zilber and L. van den Dries - A. G¨unaydın respectively. In 1965, H. Mann showed that the set of complex roots of unity has the Mann Property. Later, it was proved that any multiplicative group of finite rank in any field of characteristic zero has the Mann property. In this chapter, we characterize the independence which enables us to characterize definable and interpretable groups. In Chapter 3, we study algebraically closed fields expanded by two unary predicates denoting an algebraically closed subfield and a multiplicative subgroup. This will be a proper extension of algebraically closed fields with a group satisfying the Mann property, and also pairs of algebraically closed fields. Then we characterize definable and interpretable groups in the triple. Another goal of this thesis is to study the field of algebraic numbers with elements of small height. In Chapter 4, we show that this theory is not simple and has the independence property. We also relate the simplicity of a certain pair with Lehmer’s conjecture. In Chapter 5, we apply nonstandard analysis to prove the existence of certain height bounds on the complexity of the coefficients of some polynomials. This allows us to characterize the ideal membership of a given polynomial. Moreover, we obtain a bound for the logarithmic height function, which enables us to test the primality of an ideal. iii iv R´esum´e Dans cette th`ese, nous traitons la th´eorie des mod`eles des corps alg´ebriquement clos ´etendu par pr´edicats pour d´esigner soit des ´el´ements de hauteur born´ee, soit des sous- groupes multiplicatifs satisfaisant une condition diophantienne. Les questions que nous consid´erons appartiennent au domaine de la th´eorie des mod`eles et la th´eorie de la stabilit´e. Dans le Chapitre 2, nous examinons un corps alg´ebriquement clos avec un sous-groupe multiplicatif distingu´e qui satisfait la propri´et´e Mann. La th´eorie des mod`eles de cette paire ´etait d’abord ´etudi´ee dans les articles de B. Zilber et L. van den Dries - A. G¨unaydın respectivement. En 1965, H. Mann a montr´e que l’ensemble des racines de l’unit´e a la propri´et´e Mann. Plus tard, il ´etait prouv´e que tout groupe multiplicatif de rang fini dans tout corps de caract´eristique z´eroalapropri´et´e Mann. Dans ce chapitre, nous caract´erisons l’ind´ependance qui nous permet de caract´eriser les groupes d´efinissables et les groupes interpr´etables. Dans le Chapitre 3, nous ´etudions les corps alg´ebriquement clos ´etendu par deux pr´edicats unaires qui d´enotent un sous- corps alg´ebriquement clos et un sous-groupe multiplicatif. Ce sera une extension propre du corps alg´ebriquement clos avec un groupe satisfaisant la propri´et´e Mann, et aussi les paires des corps alg´ebriquement clos. Ensuite, nous caract´erisons les groupes d´efinissables et interpr´etables dans le triple. Un autre but de cette th`ese est d’´etudier la th´eoriedecorpsdesnombresalg´ebriques avec des ´el´ements de petite hauteur. Dans le Chapitre 4, nous montrons que cette th´eorien’estpassimpleetalapropri´et´e d’ind´ependance. Nous nous rapportons aussi `a la simplicit´e de la certaine paire avec la conjecture de Lehmer. Dans le Chapitre 5, nous appliquons l’analyse nonstandard pour prouver l’existence de certaines bornes de hauteur de la complexit´e des coefficients de certains polynˆomes. Cela nous permet de caract´eriser l’appartenance id´eale d’un polynˆome donn´e. De plus, nous obtenons une borne pour la fonction de la hauteur logarithmique, ce qui nous permet de tester la primalit´ed’unid´eal. v vi ”Our greatest glory is not in never falling, but in rising every time we fall.” Confucius Dedication To Ali and Z¨uleyha G¨oral. vii viii Acknowledgements First and foremost I offer my sincerest gratitude to my supervisors Amador Martin-Pizarro and Frank Wagner. They taught me the topic from zero. I am indebted to them for their support and infinite patience. Without them, this thesis would not exist. I am grateful to Fran¸coise Delon, Piotr Kowalski and Ayhan G¨unaydın for accept- ing to be a part of my jury and for their invaluable corrections. I would like to thank also Ali Nesin and Oleg Belegradek, for their support since I started to mathematics.
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    More Model Theory Notes Miscellaneous information, loosely organized. 1. Kinds of Models A countable homogeneous model M is one such that, for any partial elementary map f : A ! M with A ⊆ M finite, and any a 2 M, f extends to a partial elementary map A [ fag ! M. As a consequence, any partial elementary map to M is extendible to an automorphism of M. Atomic models (see below) are homogeneous. A prime model of T is one that elementarily embeds into every other model of T of the same cardinality. Any theory with fewer than continuum-many types has a prime model, and if a theory has a prime model, it is unique up to isomorphism. Prime models are homogeneous. On the other end, a model is universal if every other model of its size elementarily embeds into it. Recall a type is a set of formulas with the same tuple of free variables; generally to be called a type we require consistency. The type of an element or tuple from a model is all the formulas it satisfies. One might think of the type of an element as a sort of identity card for automorphisms: automorphisms of a model preserve types. A complete type is the analogue of a complete theory, one where every formula of the appropriate free variables or its negation appears. Types of elements and tuples are always complete. A type is principal if there is one formula in the type that implies all the rest; principal complete types are often called isolated. A trivial example of an isolated type is that generated by the formula x = c where c is any constant in the language, or x = t(¯c) where t is any composition of appropriate-arity functions andc ¯ is a tuple of constants.