A variant S2 measure for algebraic integers all of whose conjugates lie in a sector

Abstract

Let α be a nonzero algebraic integer of degree d with conjugates α1=α,…,αd. It is well known that Sk(α)= ∑i=1dαik. Here, we define Sk′(α)= ∑i=1d|αi|k and sk′(α)=Sk′(α)∕d. Then, we focus our attention on the case k=2 when α is an algebraic integer all of whose conjugates lie in a sector |argz|≤𝜃, 0<𝜃<π2. We compute the greatest lower bound c(𝜃) of s2′(α) for 𝜃 belonging to eleven subintervals of [0,π2). Moreover, among these subintervals, there are twice two consecutive and complete subintervals. We use the method of explicit auxiliary functions for which the involved polynomials are found by our recursive algorithm.