On the absolute length of polynomials having all zeros in a sector

Abstract

Let α be an algebraic integer whose all conjugates lie in a sector |argz|≤θ, 0≤θ<90°. Using the method of auxiliary functions, we first improve the known lower bounds of the absolute length of totally positive algebraic integers, i.e., when θ is equal to 0. Then, for 0<θ<90°, we compute the greatest lower bound c(θ) of the absolute length of α, for θ belonging to eight subintervals of [0,90°). Moreover, we have a complete subinterval, i.e., an interval on which the function c(θ) describing the minimum on the sector |arg(z)|≤θ is constant, with jump discontinuities at each end. Finally, we obtain an upper bound for the integer transfinite diameter of the interval [0,1] from the lower bound of the absolute length. The polynomials involved in the auxiliary functions are found by our recursive algorithm.