On the Absolute Trace of Polynomials Having All Zeros in a Sector

Abstract

Let $\alpha$ be an algebraic integer all of whose conjugates lie in a sector $ | \operatorname{arg {\ z | \leq \theta$} with $ 0 \leq \theta <90^\circ$. Using the method of explicit auxiliary functions, we compute the greatest lower bound $v(\theta)$ of the absolute trace of $\alpha$, for $\theta$ belonging to seven subintervals of $[0,90^\circ)$. The polynomials involved in the auxiliary functions are found by Wu's algorithm.