Independence in exponential fields

Zilber constructed a class of exponential�fields CFSK,CCP whose models have exponential-algebraic properties similar to the classical complex field with exponentiation Cexp. In this thesis we study this class and the more general classes ECFSK, also defined by Zilber, and ECF, studied by Zilber and Kirby. We investigate stable-like behaviour modulo arithmetic in these classes by developing a unique independence relation for each class, and in ECF we use this relation to examine types. We provide an exposition of exponential fields that is more model theoretic and type-oriented than preceding work. We then investigate the types in ECF that are orthogonal to the kernel. New ideas presented include a characterisation of these types, and the definition of a grounding set; these results allow us to�find su�fficient conditions to prove that a type over a set uniquely extends to a type over the smallest strong ELA-sub�field containing that set. For each class we define a ternary relation on subsets, and prove that these relations are independence relations, with properties akin to non-forking independence in first order theories. Applying work of Kangas, Hyttinen and Kes�al�a, we prove that in ECFSK our independence notion is the unique independence relation for this class, and that our independence notion in ECFSK,CCP is exactly the canonical independence relation for this class derived from the pre-geometry. Assuming the conjecture known as CIT, we use our independence relation in ECF to prove that types orthogonal to the kernel are exactly the generically stable types.

Name dc:type.qualificationname

phd

Level dc:type.qualificationlevel

doctoral

Grantor dc:publisher.institution

University of East Anglia

Year dc:date.issued

2014

Author dc:creator

• Henderson, Robert S.

Language dc:language

en