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

Abstract

Let $\alpha$ be a nonzero algebraic integer of degree $d$ with conjugates ${\alpha }_1=\alpha ,\dots ,{\alpha }_d$. It is well known that $S_k\left(\alpha \right)={\sum }_{i=1}^d{\alpha }_i^k$. Here, we define $S_k^{\prime }\left(\alpha \right)={\sum }_{i=1}^d|{\alpha }_i{|}^k$ and $s_k^{\prime }\left(\alpha \right)=S_k^{\prime }\left(\alpha \right)∕d$. Then, we focus our attention on the case $k=2$ when $\alpha$ is an algebraic integer all of whose conjugates lie in a sector $|argz|\le 𝜃$, . We compute the greatest lower bound $c\left(𝜃\right)$ of ${s_2}^{\prime }\left(\alpha \right)$ for $𝜃$ belonging to eleven subintervals of $\left[0,\frac{\pi }{2}\right)$. 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.