A Generalization of the Schur-Siegel-Smyth Trace Problem

TL;DR

This paper generalizes the Schur-Siegel-Smyth trace problem, which concerns the lower bounds of the average trace of totally positive algebraic integers, by framing it within a broader mathematical context.

Contribution

It demonstrates that the classical trace problem is a special case of a more general problem, expanding the scope of the original question.

Findings

The classical trace problem is a specific instance of a broader mathematical framework.

Elementary bounds show the absolute trace is always at least one.

The paper suggests a new perspective for approaching the 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.