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

Abstract:

Let $\alpha$ be an algebraic integer of degree $d$, not $0$ or a root of unity, all of whose conjugates $\alpha _i$ lie in a sector $\vert \arg z \vert \leq \theta$. In 1995, G. Rhin and C. Smyth computed the greatest lower bound $c(\theta )$ of the absolute Mahler measure ( $\prod _{i=1}^d \max (1, | \alpha _i |))^{1/d}$ of $\alpha$, for $\theta$ belonging to nine subintervals of $[0, 2 \pi /3]$. More recently, in 2004, G. Rhin and Q. Wu improved the result to thirteen subintervals of $[0, \pi ]$ and extended some existing subintervals. In this paper, for the first time we find a complete subinterval where $c(\theta )$ is known exactly, as well as a fourteenth subinterval. Moreover, we slightly extend further all the existing subintervals.