The total distance for algebraic integers having all their conjugates in a sector

Abstract

Let α be a nonzero algebraic integer of degree d whose all conjugates α1=α,α2,…,αd lie in a sector |arg⁡z|≤θ, 0<θ<90°. In 2013, K. Stulov and R. Yang defined the total distance of α as td(α)=∑i=1d||αi|−1|. This paper is organized in two parts. First, we compute the greatest lower bound c(θ) of the absolute total distance of α, for α belonging to twenty one subintervals of (0,90). Among these subintervals, eighteen are complete. Moreover, there are two sequences of seven then nine both complete and consecutive subintervals. It is the first time that these phenomena appear: such a large number of subintervals, such a large number of complete subintervals and such a large number of complete and consecutive subintervals. These phenomena are keeping with a conjecture of G. Rhin and C. J. Smyth on the nature of the function c(θ). Secondly, using the previous results and assuming that the above conjecture is true, we give upper bounds and lower bounds for the total distance involving the Mahler measure. Mostly these bounds improve those of K. Stulov and R. Yang. All computations are done by using the method of explicit auxiliary functions. The polynomials involved in these functions are found by our recursive algorithm.