Use this URL to cite or link to this record in EThOS: | https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.665151 |
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Title: | Local-global principles for linear spaces on hypersurfaces | ||||
Author: | Brandes, Julia |
ISNI:
0000 0004 5347 084X
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Awarding Body: | University of Bristol | ||||
Current Institution: | University of Bristol | ||||
Date of Award: | 2014 | ||||
Availability of Full Text: |
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Abstract: | |||||
In this thesis we study various aspects of the problem of finding rational linear
spaces on hypersurfaces. This problem can be approached by the Hardy-Littlewood
circle method, establishing a Local-Global Principle provided that
the hypersurface is 'sufficiently non-singular' and the number of variables is
large enough. However, the special structure of the linear spaces allows us to
obtain some improvement over previous approaches. A generalised version is
also addressed, which allows us to count linear spaces under somewhat more
flexible conditions.
We then investigate the local solubility. In particular, by adopting a new
approach to the analysis of the density of p-adic solutions arising in applications
of the circle method, we show that under modest conditions the existence of
non-trivial p-adic solutions suffices to establish positivity of the singular series.
This improves on earlier approaches due to Davenport, Schmidt and others,
which require the existence of non-singular p-adic solutions.
Finally, we exhibit the strength of our methods by deriving unconditional
results concerning the existence of linear spaces on systems of cubic and quintic
equations.
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Supervisor: | Not available | Sponsor: | Not available | ||
Qualification Name: | Thesis (Ph.D.) | Qualification Level: | Doctoral | ||
EThOS ID: | uk.bl.ethos.665151 | DOI: | Not available | ||
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