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Nevanlinna Theory and Rational Values of Meromorphic Functions
ENGLISH ABSTRACT: In this thesis, we are concerned with the problem of counting algebraic points of bounded height and degree on graphs of certain transcendental holomorphic and meromorphic functions. Adopting a Nevanlinna theoretic approach for the latter, we attain bounds of the form C(d)(log H...
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| Format: | Thesis |
| Language: | en_ZA |
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Stellenbosch : University of Stellenbosch
2020
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| Summary: | ENGLISH ABSTRACT: In this thesis, we are concerned with the problem of counting algebraic
points of bounded height and degree on graphs of certain transcendental
holomorphic and meromorphic functions. Adopting a Nevanlinna theoretic
approach for the latter, we attain bounds of the form C(d)(log H)b for the
number of algebraic points of height at most H and degree at most d on the
restrictions to compact subsets of domains of holomorphy of meromorphic
functions with certain growth/decay conditions. In the second half of the
thesis, we turn our attention to counting points on graphs of certain analytic
functions with growth behaviour stricter than finite order and positive
lower order. For these functions, we are able to relax the need to restrict
them to compact subsets of C, and indeed, to count points either on the
whole graph or nearly all of it. For these functions we also attain a bound
of the form C(d)(log H)h. We end this work with several pointers towards
possible extensions of our results. The results in this thesis can be seen as
extensions of the work of Boxall and Jones on algebraic values of certain
analytic functions. |
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