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### Browsing Department of Mathematical Sciences by browse.metadata.advisor "Boxall, Gareth John"

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- ItemAlgebraic points in tame expansions of fields(Stellenbosch : Stellenbosch University, 2021-12) Harrison-Migochi, Andrew; Boxall, Gareth John; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.
Show more ENGLISH ABSTRACT: We investigate the behaviour of algebraic points in several expansions of the real, complex and p-adic fields. We build off the work of Eleftheriou, Günaydin and Hieronymi in [17] and [18] to prove a Pila-Wilkie result for a p-adic subanalytic structure with a predicate for either a dense elementary substructure or a dense dcl-independent set. In the process we prove a structure theorem for p-minimal structures with a predicate for a dense independent set. We then prove quantifier reduction results for the complex field with a predicate for the singular moduli and the real field with an exponentially transcendental power function and a predicate for the algebraic numbers using a Schanuel property proved by Bays, Kirby and Wilkie [5]. Finally we adapt a theorem by Ax [2] about exponential fields, key to the proof of the Schanuel property for power functions, to power functions.Show more - ItemContributions to the theory of Beidleman near-vector spaces(Stellenbosch : Stellenbosch University, 2019-12) Djagba, Prudence; Howell, Karin-Therese; Boxall, Gareth John; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.
Show more ENGLISH SUMMARY: (Please refer to the abstract on the full text for symbols that did not translate well into this abstract). The study of nearfields was started in 1905 by L.E. Dickson. This thesis is a first step toward a detailed study of J.C. Beidleman near-vector spaces, as first introduced by Beidleman in 1966. Recalling well-known results, we conduct a detailed study of finite nearfields by showing how to construct a finite Dickson nearfield and presenting the center of a finite Dickson nearfield that arises from the Dickson pair (q, n). Furthermore, as main results of this thesis, we present the following. We characterise the finite dimensional Beidleman near-vector spaces. We develop an algorithm called EGE (Expanded Gaussian Elimination) which determines the smallest R-subgroup containing a given finite set of vectors v1, . . . , vk 2 Rm where R is a proper nearfield and k,m are positive integers, defined as gen(v1, . . . , vk). We also classify all the subspaces of Rm by designing an algorithm called the Adjustment of the EGE algorithm. We study the concept of seed number of an R-subgroup T (i.e., the minimal cardinality of all the possible finite sets of vectors that generate T) and R-dimension of gen(v1, . . . , vk) (i.e., the number of vectors obtained after the implementation of the EGE algorithm on the finite set of vectors v1, . . . , vk). We evaluate the seed number of Rm for some positive integer m satisfying m jRj +1. Furthermore from the EGE algorithm we also study, for a given pair (a, b) in R2, the generalized distributive set defined as D(a, b) = l 2 R : (a + b) l = a l + b l , where ” ” is the multiplication of the nearfield. We find that in contrast to the situation of D(R) = fl 2 R : (a+ b) l = a l+ b l for all a, b 2 Rg from the work of Zemmer in 1964, the generalized distributive set D(a, b) is not always a subnearfield of R where R is a finite Dickson nearfield arising from the Dickson pair (q, n). We find a sufficient condition on a and b such that D(a, b) is a subfield of the finite field of order qn and develop an algorithm that tests whether D(a, b) is a subfield of Fqn or not. We then investigate D(a, b) where a, b and a + b are all in distinct g qi1 q1 H (where g is a generator of F q n and H is the subgroup generated by gn) and we obtain a construction of a subfield of Fqn by making use of D(a, b).Show more - ItemThe definable (p,q)-theorem for dense pairs of certain geometric structures(Stellenbosch : Stellenbosch University, 2021-12) Rakotonarivo, Tsinjo Odilon; Boxall, Gareth John; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences. Division Mathematics.
Show more ENGLISH ABSTRACT: The definable (p, q)-conjecture is a model-theoretic version of a (p, q)- theorem in combinatorics, which was expressed in the form of a question by A. Chernikov and P. Simon in 2015. Researchers have proved that the property holds for certain classes of structures. Based on those existing results, the main objective of the present thesis is to show that the definable (p, q)-conjecture holds for a dense pair of geometric distal structures that satisfies the following condition: algebraic closure and definable closure are the same in sense of the original geometric structure. Independently, we also explore a different approach to prove that under some conditions, the definable (p, q)-conjecture holds in certain cases for dense pairs of real closed ordered fields.Show more - ItemGeometry 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
Show more 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.Show more - ItemImaginaries in dense pairs of real-closed fields(Stellenbosch : Stellenbosch University, 2017-03) Rakotonarivo, Tsinjo Odilon; Boxall, Gareth John; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences
Show more ENGLISH ABSTRACT : Imaginaries are definable equivalence classes, which play an important role in model theory. In this thesis, we are interested in imaginaries of dense pairs of real-closed fields. More precisely, we consider the following problem: is acleq equal to dcleq in dense pairs of real-closed fields? To answer this question, we first present some results about real-closed fields, which are basically completeness, quantifier elimination and elimination of imaginaries. Then, we concentrate on the completeness and near model-completeness for the theory of dense pairs of real-closed fields. And finally, we present the key point of the thesis. Namely, we demonstrate that acleq(∅) = dcleq(∅) but there exists A such that acleq(A) 6= dcleq(A)Show more - ItemNevanlinna 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
Show more 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.Show more - ItemOn the definable generalized Bohr compactification of SL(2,Qp)(Stellenbosch : Stellenbosch University, 2018-12) Pillay, Nathan Lingamurthi; Boxall, Gareth John; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences. Division Mathematics.
Show more ENGLISH ABSTRACT : This paper provides an overview of existing knowledge regarding the socalled definable generalized Bohr compactification of the group SL(2, Qp) of 2 × 2 matrices with determinant 1 and entries in Qp. The (open) question of whether this definable generalized Bohr compactification coincides with the Ellis group of the action of SL(2, Qp) on its type space is also studied in detail. This includes a discussion on the topologies associated with the space of complete types over Qp concentrating on SL(2, Qp), as well as an investigation of the possibility of first-countability of this type space.Show more - ItemUltraproducts and Los’s Theorem: A Category-Theoretic Analysis(Stellenbosch : Stellenbosch University, 2017-03) Chimes, Mark Jonathan; Boxall, Gareth John; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences
Show more ENGLISH ABSTRACT : Ultraproducts are an important construction in model theory, especially as applied to algebra. Given some family of structures of a certain type, an ultraproduct of this family is a single structure which, in some sense, captures the important aspects of the family, where “important” is defined relative to a set of sets called an ultrafilter, which encodes which subfamilies are considered “large”. This follows from Lo´s’s Theorem, namely, the Fundamental Theorem of Ultraproducts, which states that every first-order sentence is true of the ultraproduct if, and only if, there is some “large” subfamily of the family such that it is true of every structure in this subfamily. In this dissertation, ultraproducts are examined both from the standard model-theoretic, as well as from the category-theoretic view. Some potential problems with the categorytheoretic definition of ultraproducts are pointed out, and it is argued that these are not as great an issue as first perceived. A general version of Lo´s’s Theorem is shown to hold for category-theoretic ultraproducts in general. This makes use of the concept of injectivity of a (compact) tree, which is intended to generalize truth of first-order formulae (under given assignments of variables), and, in the category of relational structures, corresponds exactly to first-order formulae. This type of thinking leads to a means of characterizing fields in the category of rings, and a new proof that every ultraproduct of fields is a field, which takes place entirely in the category of rings (along with the inclusion of the category of fields). Finally, the family of all (category-theoretic) ultraproducts on a given family is shown to arise from the “codensity monad" of the functor which includes the category of finite families into the category of families. In this sense, it is shown that ultraproducts are a rather natural construction category-theoretically speaking.Show more