A Generalization of the Schur-Siegel-Smyth Trace Problem

Abstract

Let $\alpha$ be a totally positive algebraic integer, and define its absolute trace to be $\frac{Tr(\alpha )}{\text{deg}(\alpha )}$, the trace of $\alpha$ divided by the degree of $\alpha$. Elementary considerations show that the absolute trace is always at least one, while it is plausible that for any $\epsilon >0$, the absolute trace is at least $2-\epsilon$ 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.

There is an error in the proof of Theorem 1.4, and Theorem 1.4 is not correct