TY - THES

T1 - On Mahler Measures and Digraphs

AU - Coyston, Joshua

PY - 2022

Y1 - 2022

N2 - This thesis is based in the intersection of number theory and graph theory, introducing a link between the Mahler measure of a polynomial and specific types of graphs, which we refer to as digraphs. One of the main aims of this thesis is to find “small” Mahler measure values “from” digraphs. Our other aims are to further extend our knowledge of the digraphs presented, the theory linking Mahler measures and graphs more broadly and, finally, understanding when polynomials may share the same Mahler measure.As such, this thesis has two distinctive themes which are intertwined together. Inparts of the thesis, primarily Chapters 1, 3 and 6, our focus is more theoretical, and we aim to introduce and survey existing ideas, both from the topics of Mahler measures and digraph theory, as well as build upon these and present new concepts. Other parts, particularly Chapters 2 and 5, are focused on calculating Mahler measure values from a practical viewpoint, including introducing a new method for calculating the Mahler measures of two-variable polynomials, as well as detailing how exactly we performed our experiments.Meanwhile, Chapter 4 is a blend of these themes: whilst it focuses on developing our theoretical knowledge, it is done in a way to prepare us for our experiments. We round off the thesis in Chapter 7 by summarizing potential future work.In most circumstances, we present Mahler measure values to 8 decimal places, though this convention is broken on occasions where appropriate.

AB - This thesis is based in the intersection of number theory and graph theory, introducing a link between the Mahler measure of a polynomial and specific types of graphs, which we refer to as digraphs. One of the main aims of this thesis is to find “small” Mahler measure values “from” digraphs. Our other aims are to further extend our knowledge of the digraphs presented, the theory linking Mahler measures and graphs more broadly and, finally, understanding when polynomials may share the same Mahler measure.As such, this thesis has two distinctive themes which are intertwined together. Inparts of the thesis, primarily Chapters 1, 3 and 6, our focus is more theoretical, and we aim to introduce and survey existing ideas, both from the topics of Mahler measures and digraph theory, as well as build upon these and present new concepts. Other parts, particularly Chapters 2 and 5, are focused on calculating Mahler measure values from a practical viewpoint, including introducing a new method for calculating the Mahler measures of two-variable polynomials, as well as detailing how exactly we performed our experiments.Meanwhile, Chapter 4 is a blend of these themes: whilst it focuses on developing our theoretical knowledge, it is done in a way to prepare us for our experiments. We round off the thesis in Chapter 7 by summarizing potential future work.In most circumstances, we present Mahler measure values to 8 decimal places, though this convention is broken on occasions where appropriate.

KW - Mahler measure

KW - Mahler's measure

KW - Digraph

KW - Number Theory

KW - Graph Theory

KW - Royal Holloway

KW - Joshua Coyston

KW - Josh Coyston

KW - Coyston

KW - Kronecker-cyclotomic

KW - Cyclotomic graph

KW - Cyclotomic digraph

KW - Graph

KW - Equivalent digraphs

KW - Pendant paths

KW - Charged pendant paths

KW - Bridged digraphs

KW - Small Mahler Measures

KW - Small limit points

KW - Same Mahler Measure

KW - Equivalent Mahler Measures

M3 - Doctoral Thesis

ER -