Given an o-minimal structure on the real field, we consider an elementary extension to a non-archimedean field R, and interpret the algebraically closed field K=R[sqrt(-1)] on this extension. We construct two pregeometries on K: one by considering images under C-definable holomorphic functions, and the other by considering images under proper restrictions of C-definable holomorphic functions together with algebraic functions (i.e. zeros of polynomials).We show that these two pregeometries are the same, generalising a result of A. Wilkie for complex holomorphic functions. We also do some work towards generalising another result of his on local definability of complex holomorphic functions to our non-archimedean setting.
| Date of Award | 1 Aug 2015 |
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| Original language | English |
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| Awarding Institution | - The University of Manchester
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| Supervisor | Alex Wilkie (Supervisor) & Marcus Tressl (Supervisor) |
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- model theory
- complex analysis
Model theory of holomorphic Functions in an o-minimal setting
Utreras Alarcon, J. A. (Author). 1 Aug 2015
Student thesis: Phd