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

Abstract

Let α \alpha be an algebraic integer of degree d, not 0 or a root of unity, all of whose conjugates α i {\alpha _i} , are confined to a sector | arg ⁡ z | ≤ θ |\arg z| \leq \theta . We compute the greatest lower bound c ( θ ) c(\theta ) of the absolute Mahler measure ( ∏ i = 1 d max ( 1 , | α i | ) ) 1 / d (\prod \nolimits _{i = 1}^d {\max (1,|{\alpha _i}|){)^{1/d}}} of α \alpha , for θ \theta belonging to nine subintervals of [ 0 , 2 π / 3 ] [0,2\pi /3] . In particular, we show that c ( π / 2 ) = 1.12933793 c(\pi /2) = 1.12933793 , from which it follows that any integer α ≠ 1 \alpha \ne 1 and α ≠ e ± i π / 3 \alpha \ne {e^{ \pm i\pi /3}} all of whose conjugates have positive real part has absolute Mahler measure at least c ( π / 2 ) c(\pi /2) . This value is achieved for α \alpha satisfying α + 1 / α = β 0 2 \alpha + 1/\alpha = \beta _0^2 , where β 0 = 1.3247 … {\beta _0} = 1.3247 \ldots is the smallest Pisot number (the real root of β 0 3 = β 0 + 1 \beta _0^3 = {\beta _0} + 1 ).