On the absolute Mahler measure of polynomials having all zeros in a sector. II

Abstract:

Let $\alpha$ be an algebraic integer of degree $d$, not $0$ or a root of unity, all of whose conjugates $\alpha _{i}$ are confined to a sector $\vert \operatorname {arg} z \vert \le \theta$. In the paper On the absolute Mahler measure of polynomials having all zeros in a sector, G. Rhin and C. Smyth compute the greatest lower bound $c(\theta )$ of the absolute Mahler measure ($\prod _{i=1}^{d} \max (1, \vert \alpha _{i} \vert ))^{1/d}$ of $\alpha$, for $\theta$ belonging to nine subintervals of $[0, 2\pi /3]$. In this paper, we improve the result to thirteen subintervals of $[0,\pi ]$ and extend some existing subintervals.