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Abstract

Let $\alpha $ be a totally positive algebraic integer of degree $d$, i.e., all of its conjugates $\alpha _1= \alpha , \ldots ,\alpha _d$ are positive real numbers. We study the set ${\cal R}_2$ of the quantities $(\prod _{i=1}^d (1 + \alpha _i^2)^{1/2})^{1/d}$. We first show that $\sqrt 2$ is the smallest point of ${\cal R}_2$. Then, we prove that there exists a number $l$ such that ${\cal R}_2$ is dense in $(l, \infty )$. Finally, using the method of auxiliary functions, we find the six smallest points of ${\cal R}_2$ in $(\sqrt 2, l)$. The polynomials involved in the auxiliary function are found by a recursive algorithm.