Model theory of fields and heights la théorie des modéles des corps et des hauteurs
Başlık çevirisi mevcut değil.
- Tez No: 402926
- Danışmanlar: DR. AMADOR MARTIN-PIZARRO, DR. FRANK WAGNER
- Tez Türü: Doktora
- Konular: Matematik, Mathematics
- Anahtar Kelimeler: Belirtilmemiş.
- Yıl: 2015
- Dil: İngilizce
- Üniversite: Unıversıte Claude Bernard (lyon I)
- Enstitü: Yurtdışı Enstitü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: Belirtilmemiş.
Özet
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Özet (Çeviri)
In this thesis, we deal with the model theory of algebraically closed elds 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 eld with a distinguished multiplicative subgroup satisfying the Mann property. The model theory of this pair was rst studied in the papers of B. Zilber and L. van den Dries - A. G unayd n 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 nite rank in any eld of characteristic zero has the Mann property. In this chapter, we characterize the independence which enables us to characterize de nable and interpretable groups. In Chapter 3, we study algebraically closed elds expanded by two unary predicates denoting an algebraically closed sub eld and a multiplicative subgroup. This will be a proper extension of algebraically closed elds with a group satisfying the Mann property, and also pairs of algebraically closed elds. Then we characterize de nable and interpretable groups in the triple. Another goal of this thesis is to study the eld 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 coe cients 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.
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