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

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 $|\arg z| \leq \theta$. We compute the greatest lower bound $c(\theta )$ of the absolute Mahler measure $(\prod \nolimits _{i = 1}^d {\max (1,|{\alpha _i}|){)^{1/d}}}$ of $\alpha$, for $\theta$ belonging to nine subintervals of $[0,2\pi /3]$. In particular, we show that $c(\pi /2) = 1.12933793$, from which it follows that any integer $\alpha \ne 1$ and $\alpha \ne {e^{ \pm i\pi /3}}$ all of whose conjugates have positive real part has absolute Mahler measure at least $c(\pi /2)$. This value is achieved for $\alpha$ satisfying $\alpha + 1/\alpha = \beta _0^2$, where ${\beta _0} = 1.3247 \ldots$ is the smallest Pisot number (the real root of $\beta _0^3 = {\beta _0} + 1$).