THE ABSOLUTE LENGTH OF ALGEBRAIC INTEGERS WITH POSITIVE REAL PARTS

Abstract

Let $\alpha$ be a nonzero algebraic integer of degree $d$, all of whose conjugates $\alpha_i$ are confined to a sector $\left|\arg(\alpha_i)\right|\leq\theta$ with $0\lt \theta\lt \pi /2$. Let $P=X^d+b_1X^{d-1}+\cdots +b_d$ be the minimal polynomial of $\alpha$. We give in this paper the greatest lower bounds $\rho_{\mathcal {L}}(\theta)$ of the absolute length $\mathcal{L}(P) = (1+\sum_{i=1}^d |b_i|)^{1/d}$ of all but finitely many such $\alpha$, for ten different values of $\theta$.