An analog to the Schur-Siegel-Smyth trace problem

Abstract:

If $\alpha$ denotes a totally positive algebraic integer of degree $d$, i.e., its conjugates $\alpha _1=\alpha , \ldots , \alpha _d$ are positive real numbers, we define ${\mathrm {S}}_k(\alpha )= \sum _{i=1}^{d} \alpha _i^k$. ${\mathrm {S}}_1(\alpha )$ is the usual trace of $\alpha$ and has been studied by many authors throughout the years. In this paper, we focus our attention on the values of ${\mathrm {S}}_2(\alpha )$, and our purpose is to establish for ${\mathrm {S}}_2$ the same kinds of results as for the trace.