Browsing by Author "Chalebgwa, Taboka Prince"

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    Geometry of Complex Polynomials: On Sendov's Conjecture
    (Stellenbosch : Stellenbosch University, 2016-12) Chalebgwa, Taboka Prince; Boxall, Gareth John; Breuer, Florian; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences
    ENGLISH ABSTRACT : Sendov’s conjecture states that if all the zeroes of a complex polynomial P(z) of degree at least two lie in the unit disk, then within a unit distance of each zero lies a critical point of P(z). In a paper that appeared in 2014, Dégot proved that, for each α ε (0, 1), there is an integer N such that for any polynomial P(z) with degree greater than N, P(a) = 0 and all zeroes inside the unit disk, the disk │z- α│ ≤ 1 contains a critical point of P(z). Basing on this result, we derive an explicit formula N(a) for each α ε (0, 1) and, furthermore, obtain a uniform bound N for all a ε [α,β] where 0 < α < β < 1. This addresses the questions posed in Dégot’s paper.
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    Nevanlinna Theory and Rational Values of Meromorphic Functions
    (Stellenbosch : University of Stellenbosch, 2019-04) Chalebgwa, Taboka Prince; Boxall, Gareth John; University of Stellenbosch. Faculty of Science. Dept. of Mathematical Sciences
    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.