A N O N - A R C H I M E D E A N D E F I N A B L E C H OW T H E O R E M

oswal, abhishek (2022) A N O N - A R C H I M E D E A N D E F I N A B L E C H OW T H E O R E M. Doctoral thesis, UIN SAIZU Purwokerto.

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Abstract

O-minimality has had some striking applications to number theory. The utility of o-minimal structures originates from the remarkably tame topological properties satisfied by sets definable in such structures. Despite the rigidity that it imposes, the theory is sufficiently flexible to allow for a range of analytic constructions. An illustration of this ‘tame’ property is the following surprising generalization of Chow’s theorem proved by Peterzil and Starchenko - A closed analytic subset of a complex algebraic variety that is also definable in an o-minimal structure, is in fact algebraic. While the o-minimal machinery aims to capture the archimedean order topology of the real line, it is natural to wonder if such a machinery can be set up over non-archimedean fields. In this thesis, we explore a non�archimedean analogue of an o-minimal structure and prove a version of the definable Chow theorem in this context.

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