The main goal of this thesis is to demonstrate the adaptation of the Hardy-Littlewood circle method to the function field setting. Particular emphasis is placed on counting points on rather general varieties defined over function fields. With suitable notions for integral points and heights in this setting, asymptotic formulae are obtained for the number of integral points on these varieties with bounded heights, subject to certain conditions on the varieties and function fields at hand. Under the same conditions, weak approximation is also established for those varieties that are smooth. This counting problem is then specialised to the case of a cubic hypersurface, in which similar results are obtained, with further refinements available in terms of the invariant of the cubic involved. The thesis also addresses the particular case of Waring's problem for cubes over the integers. The representability of positive integers as the sum of four cubes, two of which are small, is investigated. A lower bound is obtained for how small these two cubes call be without impeding the representation of almost all natural numbers this way. An asymptotic formula is finally established for the number of such representations for almost all positive integers.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:619268 |
| Date | January 2013 |
| Creators | Lee, Siu-lun Alan |
| Publisher | University of Bristol |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
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