Masters Degrees (Mathematical Sciences)

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Browsing Masters Degrees (Mathematical Sciences) by browse.metadata.advisor "Breuer, Florian"

Now showing 1 - 6 of 6
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    Cyclotomic polynomials (in the parallel worlds of number theory)
    (Stellenbosch : Stellenbosch University, 2011-12) Bamunoba, Alex Samuel; Breuer, Florian; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.
    ENGLISH ABSTRACT: It is well known that the ring of integers Z and the ring of polynomials A = Fr[T] over a finite field Fr have many properties in common. It is due to these properties that almost all the famous (multiplicative) number theoretic results over Z have analogues over A. In this thesis, we are devoted to utilising this analogy together with the theory of Carlitz modules. We do this to survey and compare the analogues of cyclotomic polynomials, the size of their coefficients and cyclotomic extensions over the rational function field k = Fr(T).
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    Drinfeld modules and their application to factor polynomials
    (Stellenbosch : Stellenbosch University, 2012-12) Randrianarisoa, Tovohery Hajatiana; Breuer, Florian; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.
    ENGLISH ABSTRACT: Major works done in Function Field Arithmetic show a strong analogy between the ring of integers Z and the ring of polynomials over a nite eld Fq[T]. While an algorithm has been discovered to factor integers using elliptic curves, the discovery of Drinfeld modules, which are analogous to elliptic curves, made it possible to exhibit an algorithm for factorising polynomials in the ring Fq[T]. In this thesis, we introduce the notion of Drinfeld modules, then we demonstrate the analogy between Drinfeld modules and Elliptic curves. Finally, we present an algorithm for factoring polynomials over a nite eld using Drinfeld modules.
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    Elliptic curve cryptography
    (Stellenbosch : Stellenbosch University, 2016-12) Louw, Gerard Jacques; Breuer, Florian; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences
    ENGLISH ABSTRACT : In this thesis we present a selection of Diffie-Hellman cryptosystems, which were classically formulated using the multiplicative group of a finite field, but which may be generalised to use other group varieties such as elliptic curves. We also describe known attacks on special cases of such cryptosystems, which manifest as solutions to the discrete logarithm problem for group varieties, and the elliptic curve discrete logarithm problem in particular. We pursue a computational approach throughout, with a focus on the development of practical algorithms.
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    Geometric actions of the absolute Galois group
    (Stellenbosch : University of Stellenbosch, 2006-03) Joubert, Paul; Breuer, Florian; University of Stellenbosch. Faculty of Science. Dept. of Mathematical Sciences.
    This thesis gives an introduction to some of the ideas originating from A. Grothendieck's 1984 manuscript Esquisse d'un programme. Most of these ideas are related to a new geometric approach to studying the absolute Galois group over the rationals by considering its action on certain geometric objects such as dessins d'enfants (called stick figures in this thesis) and the fundamental groups of certain moduli spaces of curves. I start by defining stick figures and explaining the connection between these innocent combinatorial objects and the absolute Galois group. I then proceed to give some background on moduli spaces. This involves describing how Teichmuller spaces and mapping class groups can be used to address the problem of counting the possible complex structures on a compact surface. In the last chapter I show how this relates to the absolute Galois group by giving an explicit description of the action of the absolute Galois group on the fundamental group of a particularly simple moduli space. I end by showing how this description was used by Y. Ihara to prove that the absolute Galois group is contained in the Grothendieck-Teichmuller group.
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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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    Riemann hypothesis for the zeta function of a function field over a finite field
    (Stellenbosch : Stellenbosch University, 2013-12) Ranorovelonalohotsy, Marie Brilland Yann; Breuer, Florian; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.
    ENGLISH ABSTRACT: See the full text for the abstract