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UBC Theses and Dissertations

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UBC Theses and Dissertations

Points of small height on affine varieties defined over function fields of finite transcendence degree Nguyen, Dac Nhan Tam

Abstract

The problem of this thesis concerns points of small height on affine varieties defined over arbitrary function fields, and is based on published work with Prof. Dragos Ghioca (see [GN20]). The main result is as follows: the points lying outside the largest subvariety defined over the constant field cannot have arbitrarily small height. Prior results of this type include [Ghi09], [Ghi14]. In particular, [Ghi14] answers this question for function fields of transcendence degree 1. It also captures the history of the subject and features an argument that was initially used by the author of this thesis to extend [Ghi14] to varieties defined over function fields of arbitrary (finite) transcendence degree. The content of this thesis and the associated published paper not only extends [Ghi14] to arbitrary transcendence degree, but also provides a sharp lower bound for points which are not contained in the largest subvariety defined over the constant field. The argument here works directly with the defining polynomials of the variety (compare with [Ghi14]), and the lower bound only depends on their degrees.

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