TY - JOUR

T1 - A generalization of the Schur-Siegel-Smyth trace problem

AU - Pratt, Kyle

AU - Shakan, George

AU - Zaharescu, Alexandru

N1 - Funding Information:
The first author is supported by the National Science Foundation Graduate Research Fellowship Program under Grant Number DGE-1144245 . The authors used SAGE [14] version 6.7 for computations. We thank Michael Zieve and the anonymous referee for helpful comments.
Publisher Copyright:
© 2015.

PY - 2016/4/1

Y1 - 2016/4/1

N2 - 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.

AB - 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.

KW - Minimal polynomial

KW - Schur-Siegel-Smyth trace problem

KW - Totally positive algebraic number

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U2 - 10.1016/j.jmaa.2015.12.003

DO - 10.1016/j.jmaa.2015.12.003

M3 - Article

AN - SCOPUS:84953228553

VL - 436

SP - 489

EP - 500

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 1

ER -