Mahler’s Measure and Möbius Transformations

Abstract

This thesis is about establishing a lower bound on the Mahler measure of a non-cyclotomic polynomial. There is a famous, decade-old conjecture about this, due to Lehmer. In the 1990s, a result of Shou-Wou Zhang in Arakelov geometry proved an equivariant analogue of this, namely, a bound for the sum of the (log) Mahler measures of f(X) and f(X+1). Later, Zagier gave an “elementary” proof (using the theory of harmonic functions) and Dresden extended the method to triples f(X), f(1/(1-X)) and f(1-1/X). After some expositional chapters, the thesis (in Chapter 6) generalizes this result to orbits of polynomials under all finite groups of rational Möbius transformations. A classification result (Lemma 6.2.4) is proven for such finite groups into three categories, classifies the first type (which contains the case of Lehmer's original problem, which is still open), and proves the required lower bound in the second case (analogue of results of Zhang, Zagier and Dresden) and in the third case (entirely a new method). Also, the lower bounds are expressed through solutions of high-degree algebraic equations that turn out to be solvable in radicals.