On the Absolute Mahler Measure of Polynomials Having All Zeros in a Sector

Abstract

Let α be an algebraic integer of degree d, not 0 or a root of unity, all of whose conjugates α i are confined to a sector |arg z| ≤ θ. We compute the greatest lower bound c(θ) of the absolute Mahler measure (Π i=1 d max(1, |α i |)) 1/d of α, for θ belonging to nine subintervals of [0, 2π/3]. In particular, we show that c(π/2) = 1.12933793, from which it follows that any integer α ¬= 1 and α ¬= e ±iπ/3 all of whose conjugates have positive real part has absolute Mahler measure at least c(π/2). This value is achieved for α satisfying α + 1/α = β 0 2 , where β 0 = 1.3247... is the smallest Pisot number (the real root of β 0 3 = β 0 + 1)