A generalization of the Schur-Siegel-Smyth trace problem

Let α be a totally positive algebraic integer, and define its absolute trace to be Tr(α)deg(α), the trace of α divided by the degree of α. Elementary considerations show that the absolute trace is always at least one, while it is plausible that for any ϵ>0, the absolute trace is at least 2−ϵ with only finitely many exceptions. This is known as the Schur-Siegel-Smyth trace problem. Our aim in this paper is to show that the Schur-Siegel-Smyth trace problem can be considered as a special case of a more general problem.